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|
||
<span class="md-ellipsis">
|
||
7.2 Binary tree traversal
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.3 Array Representation of tree
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.4 Binary Search tree
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
||
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.5 AVL tree *
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/summary/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
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7.6 Summary
|
||
</span>
|
||
|
||
|
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</a>
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</nav>
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
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<span class="md-ellipsis">
|
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Chapter 8. Heap
|
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Chapter 8. Heap
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<span class="md-ellipsis">
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8.1 Heap
|
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</span>
|
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|
||
|
||
</a>
|
||
</li>
|
||
|
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<span class="md-ellipsis">
|
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8.2 Building a heap
|
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</span>
|
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|
||
|
||
</a>
|
||
</li>
|
||
|
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<span class="md-ellipsis">
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8.3 Top-k problem
|
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</span>
|
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|
||
|
||
</a>
|
||
</li>
|
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|
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8.4 Summary
|
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|
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|
||
|
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<span class="md-ellipsis">
|
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Chapter 9. Graph
|
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|
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|
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</a>
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Chapter 9. Graph
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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9.1 Graph
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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9.2 Basic graph operations
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
9.3 Graph traversal
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/summary/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
||
9.4 Summary
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
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</nav>
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|
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|
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|
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|
||
|
||
|
||
|
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|
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|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
||
<span class="md-ellipsis">
|
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Chapter 10. Searching
|
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|
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|
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</a>
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Chapter 10. Searching
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|
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10.1 Binary search
|
||
</span>
|
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|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
||
10.2 Binary search insertion
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
||
10.3 Binary search boundaries
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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|
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|
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<span class="md-ellipsis">
|
||
10.4 Hashing optimization strategies
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
|
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10.5 Search algorithms revisited
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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<span class="md-ellipsis">
|
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10.6 Summary
|
||
</span>
|
||
|
||
|
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</a>
|
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</li>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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Chapter 11. Sorting
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<span class="md-nav__icon md-icon"></span>
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Chapter 11. Sorting
|
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|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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11.1 Sorting algorithms
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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11.2 Selection sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
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||
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||
|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
||
11.3 Bubble sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
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||
|
||
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|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
11.4 Insertion sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
11.5 Quick sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
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|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.6 Merge sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
11.7 Heap sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
11.8 Bucket sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.9 Counting sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.10 Radix sort
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
||
|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
||
11.11 Summary
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__container">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
Chapter 12. Divide and conquer
|
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</span>
|
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|
||
|
||
</a>
|
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|
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|
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<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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Chapter 12. Divide and conquer
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
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|
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<span class="md-ellipsis">
|
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12.1 Divide and conquer algorithms
|
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</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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12.2 Divide and conquer search strategy
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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12.3 Building binary tree problem
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
|
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12.4 Tower of Hanoi Problem
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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12.5 Summary
|
||
</span>
|
||
|
||
|
||
</a>
|
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</li>
|
||
|
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|
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|
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|
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</ul>
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</nav>
|
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|
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|
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|
||
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||
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||
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||
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||
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||
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||
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||
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||
|
||
|
||
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||
|
||
|
||
|
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<div class="md-nav__link md-nav__container">
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<a href="../" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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|
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<span class="md-ellipsis">
|
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Chapter 13. Backtracking
|
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</span>
|
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|
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</a>
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<label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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Chapter 13. Backtracking
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../backtracking_algorithm/" class="md-nav__link">
|
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|
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<span class="md-ellipsis">
|
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13.1 Backtracking algorithms
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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<span class="md-ellipsis">
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13.2 Permutation problem
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-ellipsis">
|
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13.2 Permutation problem
|
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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<span class="md-nav__icon md-icon"></span>
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Table of contents
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<a href="#1321-cases-without-duplicate-elements" class="md-nav__link">
|
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<span class="md-ellipsis">
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13.2.1 Cases without duplicate elements
|
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</span>
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</a>
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|
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<nav class="md-nav" aria-label="13.2.1 Cases without duplicate elements">
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<a href="#1-repeated-choice-pruning" class="md-nav__link">
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<span class="md-ellipsis">
|
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1. Repeated-choice pruning
|
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</a>
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|
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|
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<li class="md-nav__item">
|
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<a href="#2-code-implementation" class="md-nav__link">
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2. Code implementation
|
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|
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13.2.2 Considering duplicate elements
|
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|
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<nav class="md-nav" aria-label="13.2.2 Considering duplicate elements">
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|
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1. Equal-element pruning
|
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|
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|
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2. Code implementation
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|
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|
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3. Comparing the two pruning methods
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13.3 Subset sum problem
|
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</span>
|
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|
||
|
||
</a>
|
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</li>
|
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|
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13.4 n queens problem
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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<span class="md-ellipsis">
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13.5 Summary
|
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</span>
|
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|
||
|
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</a>
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<h1 id="132-permutation-problem">13.2 Permutation problem<a class="headerlink" href="#132-permutation-problem" title="Permanent link">¶</a></h1>
|
||
<p>The permutation problem is a typical application of the backtracking algorithm. It involves finding all possible arrangements (permutations) of elements from a given set, such as an array or a string.</p>
|
||
<p>Table 13-2 shows several examples, including input arrays and their corresponding permutations.</p>
|
||
<p align="center"> Table 13-2 Permutation examples </p>
|
||
|
||
<div class="center-table">
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th style="text-align: left;">Input array</th>
|
||
<th style="text-align: left;">Permutations</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1]\)</span></td>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1]\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1, 2]\)</span></td>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1, 2], [2, 1]\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1, 2, 3]\)</span></td>
|
||
<td style="text-align: left;"><span class="arithmatex">\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\)</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
<h2 id="1321-cases-without-duplicate-elements">13.2.1 Cases without duplicate elements<a class="headerlink" href="#1321-cases-without-duplicate-elements" title="Permanent link">¶</a></h2>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">Question</p>
|
||
<p>Given an integer array with no duplicate elements, return all possible permutations.</p>
|
||
</div>
|
||
<p>From a backtracking perspective, <strong>we can view the process of generating permutations as a series of choices.</strong> Suppose the input array is <span class="arithmatex">\([1, 2, 3]\)</span>. If we choose <span class="arithmatex">\(1\)</span> first, then <span class="arithmatex">\(3\)</span>, and finally <span class="arithmatex">\(2\)</span>, we get the permutation <span class="arithmatex">\([1, 3, 2]\)</span>. "Backtracking" means undoing a previous choice and exploring alternative options.</p>
|
||
<p>From a coding perspective, the candidate set <code>choices</code> consists of all elements in the input array, while <code>state</code> holds the elements selected so far. Since each element can only be chosen once, <strong>all elements in <code>state</code> must be unique</strong>.</p>
|
||
<p>As illustrated in Figure 13-5, we can expand the search process into a recursive tree, where each node represents the current <code>state</code>. Starting from the root node, after three rounds of selections, we reach the leaf nodes—each corresponding to a permutation.</p>
|
||
<p><a class="glightbox" href="../permutations_problem.assets/permutations_i.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Permutation recursive tree" class="animation-figure" src="../permutations_problem.assets/permutations_i.png" /></a></p>
|
||
<p align="center"> Figure 13-5 Permutation recursive tree </p>
|
||
|
||
<h3 id="1-repeated-choice-pruning">1. Repeated-choice pruning<a class="headerlink" href="#1-repeated-choice-pruning" title="Permanent link">¶</a></h3>
|
||
<p>To ensure each element is selected only once, we introduce a boolean array <code>selected</code>, where <code>selected[i]</code> indicates whether <code>choices[i]</code> has been chosen. We then base our pruning steps on this array:</p>
|
||
<ul>
|
||
<li>After choosing <code>choice[i]</code>, set <code>selected[i]</code> to <span class="arithmatex">\(\text{True}\)</span> to mark it as chosen.</li>
|
||
<li>While iterating through <code>choices</code>, skip all elements marked as chosen (i.e., prune those branches).</li>
|
||
</ul>
|
||
<p>As shown in Figure 13-6, suppose we choose 1 in the first round, then 3 in the second round, and finally 2 in the third round. We need to prune the branch for element 1 in the second round and the branches for elements 1 and 3 in the third round.</p>
|
||
<p><a class="glightbox" href="../permutations_problem.assets/permutations_i_pruning.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Permutation pruning example" class="animation-figure" src="../permutations_problem.assets/permutations_i_pruning.png" /></a></p>
|
||
<p align="center"> Figure 13-6 Permutation pruning example </p>
|
||
|
||
<p>From the figure, we can see that this pruning process reduces the search space from <span class="arithmatex">\(O(n^n)\)</span> to <span class="arithmatex">\(O(n!)\)</span>.</p>
|
||
<h3 id="2-code-implementation">2. Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">¶</a></h3>
|
||
<p>With this understanding, we can "fill in the blanks" of our framework code. To keep the overall code concise, we won’t implement each part of the framework separately but instead expand everything in the <code>backtrack()</code> function:</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="1:14"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">permutations_i.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">):</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="sd">"""Backtracking algorithm: Permutation I"""</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="c1"># When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">return</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="c1"># Traverse all choices</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="c1"># Pruning: do not allow repeated selection of elements</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="c1"># Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="c1"># Proceed to the next round of selection</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="c1"># Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="k">def</span> <span class="nf">permutations_i</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]:</span>
|
||
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="sd">"""Permutation I"""</span>
|
||
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="p">[],</span> <span class="n">choices</span><span class="o">=</span><span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">),</span> <span class="n">res</span><span class="o">=</span><span class="n">res</span><span class="p">)</span>
|
||
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="k">return</span> <span class="n">res</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking algorithm: Permutation I */</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="o">&</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="c1">// Traverse all choices</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="c1">// Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// Proceed to the next round of selection</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="c1">// Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="p">}</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="cm">/* Permutation I */</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsI</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">;</span>
|
||
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">selected</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
|
||
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking algorithm: Permutation I */</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="c1">// Traverse all choices</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w"> </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="c1">// Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w"> </span><span class="c1">// Proceed to the next round of selection</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w"> </span><span class="c1">// Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="p">}</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="cm">/* Permutation I */</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="nf">permutationsI</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="p">();</span>
|
||
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">permutations_i</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">permutations_i</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">PermutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">backtrackI</span><span class="p">}</span>
|
||
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
|
||
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
|
||
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
|
||
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
|
||
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
|
||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
|
||
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutations_i</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a>
|
||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
|
||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a>
|
||
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">permutations_i</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_i.zig</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>
|
||
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsI</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<h2 id="1322-considering-duplicate-elements">13.2.2 Considering duplicate elements<a class="headerlink" href="#1322-considering-duplicate-elements" title="Permanent link">¶</a></h2>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">Question</p>
|
||
<p>Given an integer array**that may contain duplicate elements**, return all unique permutations.</p>
|
||
</div>
|
||
<p>Suppose the input array is <span class="arithmatex">\([1, 1, 2]\)</span>. To distinguish between the two identical elements <span class="arithmatex">\(1\)</span>, we label the second one as <span class="arithmatex">\(\hat{1}\)</span>.</p>
|
||
<p>As shown in Figure 13-7, half of the permutations produced by this method are duplicates:</p>
|
||
<p><a class="glightbox" href="../permutations_problem.assets/permutations_ii.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Duplicate permutations" class="animation-figure" src="../permutations_problem.assets/permutations_ii.png" /></a></p>
|
||
<p align="center"> Figure 13-7 Duplicate permutations </p>
|
||
|
||
<p>So how can we eliminate these duplicate permutations? One direct approach is to use a hash set to remove duplicates after generating all permutations. However, this is less elegant <strong>because branches that produce duplicates are inherently unnecessary and should be pruned in advance,</strong> thus improving the algorithm’s efficiency.</p>
|
||
<h3 id="1-equal-element-pruning">1. Equal-element pruning<a class="headerlink" href="#1-equal-element-pruning" title="Permanent link">¶</a></h3>
|
||
<p>Looking at Figure 13-8, in the first round, choosing <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(\hat{1}\)</span> leads to the same permutations, so we prune <span class="arithmatex">\(\hat{1}\)</span>.</p>
|
||
<p>Similarly, after choosing <span class="arithmatex">\(2\)</span> in the first round, choosing <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(\hat{1}\)</span> in the second round also leads to duplicate branches, so we prune <span class="arithmatex">\(\hat{1}\)</span> then as well.</p>
|
||
<p>Essentially, <strong>our goal is to ensure that multiple identical elements are only selected once per round of choices.</strong></p>
|
||
<p><a class="glightbox" href="../permutations_problem.assets/permutations_ii_pruning.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Duplicate permutations pruning" class="animation-figure" src="../permutations_problem.assets/permutations_ii_pruning.png" /></a></p>
|
||
<p align="center"> Figure 13-8 Duplicate permutations pruning </p>
|
||
|
||
<h3 id="2-code-implementation_1">2. Code implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">¶</a></h3>
|
||
<p>Based on the code from the previous problem, we introduce a hash set <code>duplicated</code> in each round. This set keeps track of elements we have already attempted, so we can prune duplicates:</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="2:14"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label><label for="__tabbed_2_14">Zig</label></div>
|
||
<div class="tabbed-content">
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
|
||
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
|
||
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="p">):</span>
|
||
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w"> </span><span class="sd">"""Backtracking algorithm: Permutation II"""</span>
|
||
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a> <span class="c1"># When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
|
||
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a> <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
|
||
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="k">return</span>
|
||
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a> <span class="c1"># Traverse all choices</span>
|
||
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a> <span class="n">duplicated</span> <span class="o">=</span> <span class="nb">set</span><span class="p">[</span><span class="nb">int</span><span class="p">]()</span>
|
||
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
|
||
<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a> <span class="c1"># Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
|
||
<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">and</span> <span class="n">choice</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">duplicated</span><span class="p">:</span>
|
||
<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a> <span class="c1"># Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a> <span class="n">duplicated</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span> <span class="c1"># Record selected element values</span>
|
||
<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
|
||
<a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
|
||
<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a> <span class="c1"># Proceed to the next round of selection</span>
|
||
<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
||
<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a> <span class="c1"># Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
|
||
<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a>
|
||
<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="k">def</span> <span class="nf">permutations_ii</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]:</span>
|
||
<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w"> </span><span class="sd">"""Permutation II"""</span>
|
||
<a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="p">[],</span> <span class="n">choices</span><span class="o">=</span><span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">),</span> <span class="n">res</span><span class="o">=</span><span class="n">res</span><span class="p">)</span>
|
||
<a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a> <span class="k">return</span> <span class="n">res</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.cpp</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Backtracking algorithm: Permutation II */</span>
|
||
<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="o">&</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w"> </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
|
||
<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||
<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w"> </span><span class="c1">// Traverse all choices</span>
|
||
<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w"> </span><span class="n">unordered_set</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">duplicated</span><span class="p">;</span>
|
||
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
|
||
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w"> </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
|
||
<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">find</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">end</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w"> </span><span class="c1">// Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record selected element values</span>
|
||
<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
|
||
<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
|
||
<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w"> </span><span class="c1">// Proceed to the next round of selection</span>
|
||
<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w"> </span><span class="c1">// Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
|
||
<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
|
||
<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="p">}</span>
|
||
<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a>
|
||
<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="cm">/* Permutation II */</span>
|
||
<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsII</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">;</span>
|
||
<a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">selected</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
|
||
<a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Backtracking algorithm: Permutation II */</span>
|
||
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w"> </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
|
||
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
|
||
<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w"> </span><span class="c1">// Traverse all choices</span>
|
||
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w"> </span><span class="n">Set</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">();</span>
|
||
<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
|
||
<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w"> </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
|
||
<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w"> </span><span class="c1">// Attempt: make a choice, update the state</span>
|
||
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record selected element values</span>
|
||
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
|
||
<a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
|
||
<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w"> </span><span class="c1">// Proceed to the next round of selection</span>
|
||
<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w"> </span><span class="c1">// Retract: undo the choice, restore to the previous state</span>
|
||
<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
|
||
<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
|
||
<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="p">}</span>
|
||
<a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a>
|
||
<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="cm">/* Permutation II */</span>
|
||
<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="nf">permutationsII</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="p">();</span>
|
||
<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
||
<a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">permutations_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a>
|
||
<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">permutations_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">PermutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">backtrackII</span><span class="p">}</span>
|
||
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a>
|
||
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.swift</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a>
|
||
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a>
|
||
<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a>
|
||
<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a>
|
||
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a>
|
||
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutations_ii</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a>
|
||
<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.kt</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a>
|
||
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.rb</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a>
|
||
<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">permutations_ii</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">permutations_ii.zig</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a>
|
||
<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">permutationsII</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Assuming all elements are distinct, there are <span class="arithmatex">\(n!\)</span> (factorial) permutations of <span class="arithmatex">\(n\)</span> elements. Recording each result requires copying a list of length <span class="arithmatex">\(n\)</span>, which takes <span class="arithmatex">\(O(n)\)</span> time. <strong>Hence, the total time complexity is <span class="arithmatex">\(O(n!n)\)</span>.</strong></p>
|
||
<p>The maximum recursion depth is <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> stack space. The <code>selected</code> array also requires <span class="arithmatex">\(O(n)\)</span> space. Because there can be up to <span class="arithmatex">\(n\)</span> separate <code>duplicated</code> sets at any one time, they collectively occupy <span class="arithmatex">\(O(n^2)\)</span> space. <strong>Therefore, the space complexity is <span class="arithmatex">\(O(n^2)\)</span>.</strong></p>
|
||
<h3 id="3-comparing-the-two-pruning-methods">3. Comparing the two pruning methods<a class="headerlink" href="#3-comparing-the-two-pruning-methods" title="Permanent link">¶</a></h3>
|
||
<p>Although both <code>selected</code> and <code>duplicated</code> serve as pruning mechanisms, they target different issues:</p>
|
||
<ul>
|
||
<li><strong>Repeated-choice pruning</strong>(via <code>selected</code>): There is a single <code>selected</code> array for the entire search, indicating which elements are already in the current state. This prevents the same element from appearing more than once in <code>state</code>.</li>
|
||
<li><strong>Equal-element pruning</strong>(via <code>duplicated</code>): Each call to the <code>backtrack</code> function uses its own <code>duplicated</code> set, recording which elements have already been chosen in that specific iteration (<code>for</code> loop). This ensures that equal elements are selected only once per round of choices.</li>
|
||
</ul>
|
||
<p>Figure 13-9 shows the scope of these two pruning strategies. Each node in the tree represents a choice; the path from the root to any leaf corresponds to one complete permutation.</p>
|
||
<p><a class="glightbox" href="../permutations_problem.assets/permutations_ii_pruning_summary.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Scope of the two pruning conditions" class="animation-figure" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></a></p>
|
||
<p align="center"> Figure 13-9 Scope of the two pruning conditions </p>
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