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@ -25,7 +25,7 @@
<title>13.2.   DP 问题特性（New） - Hello 算法</title>
<title>14.2.   DP 问题特性（New） - Hello 算法</title>
@ -79,7 +79,7 @@
<div data-md-component="skip">
<a href="#132" class="md-skip">
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跳转至
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@ -113,7 +113,7 @@
<div class="md-header__topic" data-md-component="header-topic">
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13.2. &nbsp; DP 问题特性（New）
14.2. &nbsp; DP 问题特性（New）
</span>
</div>
@ -1771,6 +1771,87 @@
<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_divide_and_conquer/">12. &nbsp; &nbsp; 分治</a>
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12. &nbsp; &nbsp; 分治
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<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
12.1. &nbsp; 分治算法（New）
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12.2. &nbsp; 构建树问题（New）
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@ -1782,18 +1863,18 @@
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12. &nbsp; &nbsp; 回溯
13. &nbsp; &nbsp; 回溯
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@ -1806,7 +1887,7 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法
13.1. &nbsp; 回溯算法
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@ -1820,7 +1901,7 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
12.2. &nbsp; 全排列问题
13.2. &nbsp; 全排列问题
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@ -1834,7 +1915,7 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题
13.3. &nbsp; 子集和问题
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@ -1848,7 +1929,7 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.4. &nbsp; N 皇后问题
13.4. &nbsp; N 皇后问题
</a>
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@ -1862,7 +1943,7 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/summary/" class="md-nav__link">
12.5. &nbsp; 小结
13.5. &nbsp; 小结
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@ -1890,7 +1971,7 @@
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@ -1917,18 +1998,18 @@
<div class="md-nav__link md-nav__link--index ">
<a href="../">13. &nbsp; &nbsp; 动态规划</a>
<a href="../">14. &nbsp; &nbsp; 动态规划</a>
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13. &nbsp; &nbsp; 动态规划
14. &nbsp; &nbsp; 动态规划
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@ -1941,7 +2022,7 @@
<li class="md-nav__item">
<a href="../intro_to_dynamic_programming/" class="md-nav__link">
13.1. &nbsp; 初探动态规划（New）
14.1. &nbsp; 初探动态规划（New）
</a>
</li>
@ -1964,12 +2045,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
13.2. &nbsp; DP 问题特性（New）
14.2. &nbsp; DP 问题特性（New）
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
13.2. &nbsp; DP 问题特性（New）
14.2. &nbsp; DP 问题特性（New）
</a>
@ -1988,15 +2069,15 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<a href="#1321" class="md-nav__link">
13.2.1. &nbsp; 最优子结构
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14.2.1. &nbsp; 最优子结构
</a>
</li>
<li class="md-nav__item">
<a href="#1322" class="md-nav__link">
13.2.2. &nbsp; 无后效性
<a href="#1422" class="md-nav__link">
14.2.2. &nbsp; 无后效性
</a>
</li>
@ -2017,7 +2098,7 @@
<li class="md-nav__item">
<a href="../dp_solution_pipeline/" class="md-nav__link">
13.3. &nbsp; DP 解题思路（New）
14.3. &nbsp; DP 解题思路（New）
</a>
</li>
@ -2031,7 +2112,7 @@
<li class="md-nav__item">
<a href="../knapsack_problem/" class="md-nav__link">
13.4. &nbsp; 0-1 背包问题（New）
14.4. &nbsp; 0-1 背包问题（New）
</a>
</li>
@ -2045,7 +2126,7 @@
<li class="md-nav__item">
<a href="../unbounded_knapsack_problem/" class="md-nav__link">
13.5. &nbsp; 完全背包问题（New）
14.5. &nbsp; 完全背包问题（New）
</a>
</li>
@ -2059,7 +2140,7 @@
<li class="md-nav__item">
<a href="../edit_distance_problem/" class="md-nav__link">
13.6. &nbsp; 编辑距离问题（New）
14.6. &nbsp; 编辑距离问题（New）
</a>
</li>
@ -2073,76 +2154,7 @@
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13.7. &nbsp; 小结（New）
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14.1. &nbsp; 编程环境安装
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14.2. &nbsp; 一起参与创作
14.7. &nbsp; 小结（New）
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@ -2173,6 +2185,75 @@
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15. &nbsp; &nbsp; 附录
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15.1. &nbsp; 编程环境安装
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15.2. &nbsp; 一起参与创作
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@ -2185,8 +2266,8 @@
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参考文献
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@ -2227,15 +2308,15 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1321" class="md-nav__link">
13.2.1. &nbsp; 最优子结构
<a href="#1421" class="md-nav__link">
14.2.1. &nbsp; 最优子结构
</a>
</li>
<li class="md-nav__item">
<a href="#1322" class="md-nav__link">
13.2.2. &nbsp; 无后效性
<a href="#1422" class="md-nav__link">
14.2.2. &nbsp; 无后效性
</a>
</li>
@ -2263,10 +2344,10 @@
<h1 id="132">13.2. &nbsp; 动态规划问题特性<a class="headerlink" href="#132" title="Permanent link">&para;</a></h1>
<h1 id="142">14.2. &nbsp; 动态规划问题特性<a class="headerlink" href="#142" title="Permanent link">&para;</a></h1>
<p>在上节中，我们学习了动态规划问题的暴力解法，从递归树中观察到海量的重叠子问题，以及了解到动态规划是如何通过记录解来优化时间复杂度的。</p>
<p>实际上，动态规划最常用来求解最优方案问题，例如寻找最短路径、最大利润、最少时间等。<strong>这类问题不仅包含重叠子问题，往往还具有另外两大特性：最优子结构、无后效性</strong>。</p>
<h2 id="1321">13.2.1. &nbsp; 最优子结构<a class="headerlink" href="#1321" title="Permanent link">&para;</a></h2>
<h2 id="1421">14.2.1. &nbsp; 最优子结构<a class="headerlink" href="#1421" title="Permanent link">&para;</a></h2>
<p>我们对爬楼梯问题稍作改动，使之更加适合展示最优子结构概念。</p>
<div class="admonition question">
<p class="admonition-title">爬楼梯最小代价</p>
@ -2484,7 +2565,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
</div>
</div>
</div>
<h2 id="1322">13.2.2. &nbsp; 无后效性<a class="headerlink" href="#1322" title="Permanent link">&para;</a></h2>
<h2 id="1422">14.2.2. &nbsp; 无后效性<a class="headerlink" href="#1422" title="Permanent link">&para;</a></h2>
<p>「无后效性」是动态规划能够有效解决问题的重要特性之一，定义为：<strong>给定一个确定的状态，它的未来发展只与当前状态有关，而与当前状态过去所经历过的所有状态无关</strong>。</p>
<p>以爬楼梯问题为例，给定状态 <span class="arithmatex">\(i\)</span> ，它会发展出状态 <span class="arithmatex">\(i+1\)</span> 和状态 <span class="arithmatex">\(i+2\)</span> ，分别对应跳 <span class="arithmatex">\(1\)</span> 步和跳 <span class="arithmatex">\(2\)</span> 步。在做出这两种选择时，我们无需考虑状态 <span class="arithmatex">\(i\)</span> 之前的状态，即它们对状态 <span class="arithmatex">\(i\)</span> 的未来没有影响。</p>
<p>然而，如果我们向爬楼梯问题添加一个约束，情况就不一样了。</p>
@ -2713,7 +2794,7 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
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@ -2722,20 +2803,20 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
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