From c449518a5ec8af21ff98dc394277de41efa37111 Mon Sep 17 00:00:00 2001 From: youngyangyang04 <826123027@qq.com> Date: Sat, 14 Nov 2020 13:05:55 +0800 Subject: [PATCH] Update --- README.md | 1 + pics/90.子集II1.png | Bin 0 -> 120943 bytes pics/90.子集II2.png | Bin 0 -> 122873 bytes problems/0090.子集II.md | 33 +++ ...溯算法去重问题的另一种写法.md | 240 ++++++++++++++++++ problems/贪心算法理论基础.md | 13 +- 6 files changed, 280 insertions(+), 7 deletions(-) create mode 100644 pics/90.子集II1.png create mode 100644 pics/90.子集II2.png create mode 100644 problems/回溯算法去重问题的另一种写法.md diff --git a/README.md b/README.md index 55af7c35..cc15e2a3 100644 --- a/README.md +++ b/README.md @@ -147,6 +147,7 @@ * [回溯算法:递增子序列](https://mp.weixin.qq.com/s/ePxOtX1ATRYJb2Jq7urzHQ) * [回溯算法:排列问题!](https://mp.weixin.qq.com/s/SCOjeMX1t41wcvJq49GhMw) * [回溯算法:排列问题(二)](https://mp.weixin.qq.com/s/9L8h3WqRP_h8LLWNT34YlA) + * [本周小结!(回溯算法系列三)](https://mp.weixin.qq.com/s/tLkt9PSo42X60w8i94ViiA) (持续更新中....) diff --git a/pics/90.子集II1.png 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zaCCMjS25%7wa9(57*4AiCae{FjNqzQ1?Ho?!=C2HVho+Q;ibnh3y#43~@-WBs+- z^3M+yTCikIpM+j5&o%h}GxIf?sm%X#=03{zy3eZg^C8iHv|ST^ kq+?SynCkiegTd$IP50$voW@a8IM|n(inel@qGjlR04EkD(EtDd literal 0 HcmV?d00001 diff --git a/problems/0090.子集II.md b/problems/0090.子集II.md index 7a57eaba..7fe4bfc1 100644 --- a/problems/0090.子集II.md +++ b/problems/0090.子集II.md @@ -77,6 +77,39 @@ public: ``` +使用set去重的版本。 +``` +class Solution { +private: + vector> result; + vector path; + void backtracking(vector& nums, int startIndex, vector& used) { + result.push_back(path); + unordered_set uset; + for (int i = startIndex; i < nums.size(); i++) { + if (uset.find(nums[i]) != uset.end()) { + continue; + } + uset.insert(nums[i]); + path.push_back(nums[i]); + backtracking(nums, i + 1, used); + path.pop_back(); + } + } + +public: + vector> subsetsWithDup(vector& nums) { + result.clear(); + path.clear(); + vector used(nums.size(), false); + sort(nums.begin(), nums.end()); // 去重需要排序 + backtracking(nums, 0, used); + return result; + } +}; + +``` + # 总结 其实这道题目的知识点,我们之前都讲过了,如果之前讲过的子集问题和去重问题都掌握的好,这道题目应该分分钟AC。 diff --git a/problems/回溯算法去重问题的另一种写法.md b/problems/回溯算法去重问题的另一种写法.md new file mode 100644 index 00000000..a002704f --- /dev/null +++ b/problems/回溯算法去重问题的另一种写法.md @@ -0,0 +1,240 @@ + + +> 在 [本周小结!(回溯算法系列三)](https://mp.weixin.qq.com/s/tLkt9PSo42X60w8i94ViiA) 中一位录友对 整颗树的本层和同一节点的本层有疑问,也让我重新思考了一下,发现这里确实有问题,所以专门写一篇来纠正,感谢录友们的积极交流哈! + +接下来我再把这块再讲一下。 + +在[回溯算法:求子集问题(二)](https://mp.weixin.qq.com/s/WJ4JNDRJgsW3eUN72Hh3uQ)中的去重和 [回溯算法:递增子序列](https://mp.weixin.qq.com/s/ePxOtX1ATRYJb2Jq7urzHQ)中的去重 都是 同一父节点下本层的去重。 + +[回溯算法:求子集问题(二)](https://mp.weixin.qq.com/s/WJ4JNDRJgsW3eUN72Hh3uQ)也可以使用set针对同一父节点本层去重,但子集问题一定要排序,为什么呢? + +我用没有排序的集合{2,1,2,2}来举例子画一个图,如图: + +![90.子集II2](https://img-blog.csdnimg.cn/2020111316440479.png) + +图中,大家就很明显的看到,子集重复了。 + +那么下面我针对[回溯算法:求子集问题(二)](https://mp.weixin.qq.com/s/WJ4JNDRJgsW3eUN72Hh3uQ) 给出使用set来对本层去重的代码实现。 + +# 90.子集II + +used数组去重版本: [回溯算法:求子集问题(二)](https://mp.weixin.qq.com/s/WJ4JNDRJgsW3eUN72Hh3uQ) + +使用set去重的版本如下: + +```C++ +class Solution { +private: + vector> result; + vector path; + void backtracking(vector& nums, int startIndex, vector& used) { + result.push_back(path); + unordered_set uset; // 定义set对同一节点下的本层去重 + for (int i = startIndex; i < nums.size(); i++) { + if (uset.find(nums[i]) != uset.end()) { // 如果发现出现过就pass + continue; + } + uset.insert(nums[i]); // set跟新元素 + path.push_back(nums[i]); + backtracking(nums, i + 1, used); + path.pop_back(); + } + } + +public: + vector> subsetsWithDup(vector& nums) { + result.clear(); + path.clear(); + vector used(nums.size(), false); + sort(nums.begin(), nums.end()); // 去重需要排序 + backtracking(nums, 0, used); + return result; + } +}; + +``` + +针对留言区录友们的疑问,我再补充一些常见的错误写法, + + +## 错误写法一 + +把uset定义放到类成员位置,然后模拟回溯的样子 insert一次,erase一次。 + +例如: + +```C++ +class Solution { +private: + vector> result; + vector path; + unordered_set uset; // 把uset定义放到类成员位置 + void backtracking(vector& nums, int startIndex, vector& used) { + result.push_back(path); + + for (int i = startIndex; i < nums.size(); i++) { + if (uset.find(nums[i]) != uset.end()) { + continue; + } + uset.insert(nums[i]); // 递归之前insert + path.push_back(nums[i]); + backtracking(nums, i + 1, used); + path.pop_back(); + uset.erase(nums[i]); // 回溯再erase + } + } + +``` + +在树形结构中,**如果把unordered_set uset放在类成员的位置(相当于全局变量),就把树枝的情况都记录了,不是单纯的控制某一节点下的同一层了**。 + +如图: + +![90.子集II1](https://img-blog.csdnimg.cn/202011131625054.png) + +可以看出一旦把unordered_set uset放在类成员位置,它控制的就是整棵树,包括树枝。 + +**所以这么写不行!** + +## 错误写法二 + +有同学把 unordered_set uset; 放到类成员位置,然后每次进入单层的时候用uset.clear()。 + +代码如下: + +```C++ +class Solution { +private: + vector> result; + vector path; + unordered_set uset; // 把uset定义放到类成员位置 + void backtracking(vector& nums, int startIndex, vector& used) { + result.push_back(path); + uset.clear(); // 到每一层的时候,清空uset + for (int i = startIndex; i < nums.size(); i++) { + if (uset.find(nums[i]) != uset.end()) { + continue; + } + uset.insert(nums[i]); // set记录元素 + path.push_back(nums[i]); + backtracking(nums, i + 1, used); + path.pop_back(); + } + } +``` +uset已经是全局变量,本层的uset记录了一个元素,然后进入下一层之后这个uset(和上一层是同一个uset)就被清空了,也就是说,层与层之间的uset是同一个,那么就会相互影响。 + +**所以这么写依然不行!** + +**组合问题和排列问题,其实也可以使用set来对同一节点下本层去重,下面我都分别给出实现代码**。 + +# 40. 组合总和 II + +使用used数组去重版本:[回溯算法:求组合总和(三)](https://mp.weixin.qq.com/s/_1zPYk70NvHsdY8UWVGXmQ) + +使用set去重的版本如下: + +```C++ +class Solution { +private: + vector> result; + vector path; + void backtracking(vector& candidates, int target, int sum, int startIndex) { + if (sum == target) { + result.push_back(path); + return; + } + unordered_set uset; // 控制某一节点下的同一层元素不能重复 + for (int i = startIndex; i < candidates.size() && sum + candidates[i] <= target; i++) { + if (uset.find(candidates[i]) != uset.end()) { + continue; + } + uset.insert(candidates[i]); // 记录元素 + sum += candidates[i]; + path.push_back(candidates[i]); + backtracking(candidates, target, sum, i + 1); + sum -= candidates[i]; + path.pop_back(); + } + } + +public: + vector> combinationSum2(vector& candidates, int target) { + path.clear(); + result.clear(); + sort(candidates.begin(), candidates.end()); + backtracking(candidates, target, 0, 0); + return result; + } +}; +``` + +# 47. 全排列 II + +使用used数组去重版本:[回溯算法:排列问题(二)](https://mp.weixin.qq.com/s/9L8h3WqRP_h8LLWNT34YlA) + +使用set去重的版本如下: + +```C++ +class Solution { +private: + vector> result; + vector path; + void backtracking (vector& nums, vector& used) { + if (path.size() == nums.size()) { + result.push_back(path); + return; + } + unordered_set uset; // 控制某一节点下的同一层元素不能重复 + for (int i = 0; i < nums.size(); i++) { + if (uset.find(nums[i]) != uset.end()) { + continue; + } + if (used[i] == false) { + uset.insert(nums[i]); // 记录元素 + used[i] = true; + path.push_back(nums[i]); + backtracking(nums, used); + path.pop_back(); + used[i] = false; + } + } + } +public: + vector> permuteUnique(vector& nums) { + result.clear(); + path.clear(); + sort(nums.begin(), nums.end()); // 排序 + vector used(nums.size(), false); + backtracking(nums, used); + return result; + } +}; +``` + +# 两种写法的性能分析 + +需要注意的是:**使用set去重的版本相对于used数组的版本效率都要低很多**,大家在leetcode上提交,能明显发现。 + +原因在[回溯算法:递增子序列](https://mp.weixin.qq.com/s/ePxOtX1ATRYJb2Jq7urzHQ)中也分析过,主要是因为程序运行的时候对unordered_set 频繁的insert,unordered_set需要做哈希映射(也就是把key通过hash function映射为唯一的哈希值)相对费时间,而且insert的时候其底层的符号表也要做相应的扩充,也是费时的。 + +**而使用used数组在时间复杂度上几乎没有额外负担!** + +**使用set去重,不仅时间复杂度高了,空间复杂度也高了**,在[本周小结!(回溯算法系列三)](https://mp.weixin.qq.com/s/tLkt9PSo42X60w8i94ViiA)中分析过,组合,子集,排列问题的空间复杂度都是O(n),但如果使用set去重,空间复杂度就变成了O(n^2),因为每一层递归都有一个set集合,系统栈空间是n,每一个空间都有set集合。 + +那有同学可能疑惑 用used数组也是占用O(n)的空间啊? + +used数组可是全局变量,每层与每层之间公用一个used数组,所以空间复杂度是O(n + n),最终空间复杂度还是O(n)。 + +# 总结 + +本篇本打算是对[本周小结!(回溯算法系列三)](https://mp.weixin.qq.com/s/tLkt9PSo42X60w8i94ViiA)的一个点做一下纠正,没想到又写出来这么多! + +**这个点都源于一位录友的疑问,然后我思考总结了一下,就写着这一篇,所以还是得多交流啊!** + +如果大家对「代码随想录」文章有什么疑问,尽管打卡留言的时候提出来哈,或者在交流群里提问。 + +**其实这就是相互学习的过程,交流一波之后都对题目理解的更深刻了,我如果发现文中有问题,都会在评论区或者下一篇文章中即时修正,保证不会给大家带跑偏!** + +就酱,「代码随想录」一直都是干货满满,公众号里的一抹清流,值得推荐给身边的每一位同学朋友! + diff --git a/problems/贪心算法理论基础.md b/problems/贪心算法理论基础.md index 22d7abb9..98253614 100644 --- a/problems/贪心算法理论基础.md +++ b/problems/贪心算法理论基础.md @@ -12,20 +12,19 @@ 在举一个例子如果是 有一堆盒子,你有一个背包体积为n,如何把背包尽可能装满,如果还每次选最大的盒子,一定不行。这时候就需要动态规划。动态规划的问题在下一个系列会详细讲解。 - 很多同学做贪心的题目的时候,想不出来是贪心,想知道有没有什么套路可以一看看出来是贪心,说实话贪心算法并没有固定的套路。 所以唯一的难点就是如何通过局部最优,推出整体最优。 -那么如何能看出局部最优是否能退出整体最优呢?有没有什么固定策略呢? +那么如何能看出局部最优是否能退出整体最优呢?有没有什么固定策略或者套路呢? -不好意思,也没有,靠自己手动模拟,如果模拟可行,就可以试一试贪心策略,如果不可行,可能需要动态规划。 +**不好意思,没有!** 靠自己手动模拟,如果模拟可行,就可以试一试贪心策略,如果不可行,可能需要动态规划。 验证可不可以用贪心最好用的策略就是举反例,如果想不到反例,那么就试一试贪心吧。 那又有同学认为手动模拟,举例子得出的结论不靠谱,想要严格的数学证明。 -数学证明一般可以是 +数学证明一般可以是: * 数学归纳法 * 反证法 @@ -34,13 +33,13 @@ 所以做了贪心题目的时候大家就会发现,如果啥都要数学证明一下,就是把简单问题搞复杂了。 -面试中基本不会让面试者现场证明贪心的合理性,代码写出来跑过测试用例即可,或者自己能自圆其说就行。 +**面试中基本不会让面试者现场证明贪心的合理性,代码写出来跑过测试用例即可,或者自己能自圆其说就行**。 举一个不太恰当的例子:我要用一下1+1 = 2,但我要先证明1+1 为什么等于2。严谨是严谨了,但没必要。 -虽然这个例子有点极端,但可以表达这么个意思,就是手动模拟一下感觉可以局部最优推出整体最优,那么就试一试贪心。 +虽然这个例子有点极端,但可以表达这么个意思:就是手动模拟一下感觉可以局部最优推出整体最优,而且想不到反例,那么就试一试贪心。 -例如刚刚举的拿钞票的例子,就是模拟一下每次那做大的,最后就能拿到最多的钱,这还要数学证明的话,是不是感觉有点怪怪的。 +**例如刚刚举的拿钞票的例子,就是模拟一下每次那做大的,最后就能拿到最多的钱,这还要数学证明的话,其实就不在算法面试的范围内了,可以看看专业的数学书籍!** 所以这也是为什么有的通过AC(accept)了一些贪心的题目,但都不知道自己用了贪心算法,因为贪心有时候就是常识性的推导,所以会认为本就应该这么做!