From 7343c01b560e25ede6fd98ea555d9aa51e314dfc Mon Sep 17 00:00:00 2001 From: youngyangyang04 <826123027@qq.com> Date: Sun, 25 Oct 2020 21:16:59 +0800 Subject: [PATCH] Update --- README.md | 3 +- pics/77.组合1.png | Bin 0 -> 125426 bytes pics/77.组合2.png | Bin 0 -> 112594 bytes pics/77.组合3.png | Bin 0 -> 111810 bytes problems/0053.最大子序和.md | 3 - problems/0077.组合.md | 162 +++++++++++++++------- problems/0104.二叉树的最大深度.md | 5 +- problems/0559.N叉树的最大深度.md | 6 +- problems/二叉树总结.md | 147 ++++++++++++++++++++ problems/回溯算法理论基础.md | 118 ++++++++++++++++ problems/链表总结篇.md | 3 + 11 files changed, 386 insertions(+), 61 deletions(-) create mode 100644 pics/77.组合1.png create mode 100644 pics/77.组合2.png create mode 100644 pics/77.组合3.png create mode 100644 problems/二叉树总结.md create mode 100644 problems/回溯算法理论基础.md diff --git a/README.md b/README.md index 6d92a5d5..4a7b43b8 100644 --- a/README.md +++ b/README.md @@ -123,7 +123,8 @@ * [二叉树:搜索树中的删除操作](https://mp.weixin.qq.com/s/-p-Txvch1FFk3ygKLjPAKw) * [二叉树:修剪一棵搜索树](https://mp.weixin.qq.com/s/QzmGfYUMUWGkbRj7-ozHoQ) * [二叉树:构造一棵搜索树](https://mp.weixin.qq.com/s/sy3ygnouaZVJs8lhFgl9mw) - + * [二叉树:搜索树转成累加树](https://mp.weixin.qq.com/s/hZtJh4T5lIGBarY-lZJf6Q) + * [二叉树:总结篇!(需要掌握的二叉树技能都在这里了)](https://mp.weixin.qq.com/s/-ZJn3jJVdF683ap90yIj4Q) (持续更新中....) diff --git a/pics/77.组合1.png b/pics/77.组合1.png new file mode 100644 index 0000000000000000000000000000000000000000..a6a4a2721755c422a5ba5c8958574edaae2c773f GIT binary patch literal 125426 zcmeFZWmuGJ*9I)1f(Q&PEufTiOG}6d2#9opl*E8^4k(R)sFXA)pma$yG)f9c$2fF% z3`~7DdvAUB`|NlB`o3S^@%-RmhPmdRd)BqCbDe9ga}lNWNQL+s-L*@XE)lCfP@jNJW;YSP_@<2xWo-SCb)zX;cyB6VhQj= z2mAnU&BnQe2mHpl_**vauU9W?W#j#Nj0d_{NECQg^U@{xOR7rup7`KwWe_%b>YyVg zHUpdU-*J8d3Eh85Nj`jwh4>AKmK&oS{l<3?^pK3)<|`!?Yt>Vp0iiGuw~)=s@-j3t zaQ=8c@N{l<)_rGgb<2MzaD3H$cgl{`(P!KUS&W{89vi3tf$GCSBOSEfA0;r?;phUG8gQ)=A_wx|5J1+9o%*};;sn`4hv z|Lwa{fwbgHF6C&0wT4hjbEFaTb<;)FJW&fWZpuBU$z(d~d`?EOIRzp-GV-*5tRsm{%DGlD9n 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zE$smEN_n_0+5dpe0sg0Dz+6XynemG2mPq)ITLK+`;d@v3RPi5M{5|*#AW;q}E8m)^ z{QKehyT1fhVDbFUh8AF)Tt_3>)7GccOxp5@8^$pQ`r%7R^xz0X{4dUk3b2cbdDG`PS_bK z4Ba<~9dY;#aD={7@f)p=cx_Mr0DL#$!~<1MfX#{WhyNs_edxS5<}ld?psTKKf4=x( zZM4SD#XoI*KYyScsb?{A{N-QZjIVap0)g6pC!rJ1l>xtGzw><5UpD=}6=dJ;WWfxM zd(q^f>H>F-FwJTK7&YFyfNOWjT$pd)#Qu<>m<;TxfL~XQ}!r-DrEwoCbh(u79=w?lN2WD&F%6;qm83(TjHh6%f;L`KPVFkCNqU zzlYQT*pQ3FTTqD2EYFJfi$MCXKpVZ*(ilS3A=|PG2A}Umf6uRaZqC#5y-0fUJ@b$I ze>-3^pK@!0>G+RV*xxDhGz^&XQKot&|MUEN3|wW!iuL~@GXK+b)*-van1<*j-r_F* zc~is((ud@u{|schJyurfmc0tnSNcm+{>RRNIf08zO>yQQ8~p7n<*l6-8=Ugp|I{sn zYzi 笔者在BAT从事技术研发多年,利用工作之余重刷leetcode,更多原创技术文章欢迎关注「代码随想录」,校招社招求职内推欢迎通过公众号「代码随想录」联系我,度厂很缺人! - -> 笔者在BAT从事技术研发多年,利用工作之余重刷leetcode,希望结合自己多年的实践经验,把算法讲的更清楚,更多原创文章欢迎关注公众号「代码随想录」。 ## 题目地址 https://leetcode-cn.com/problems/maximum-subarray/ diff --git a/problems/0077.组合.md b/problems/0077.组合.md index 6400a75c..7d154367 100644 --- a/problems/0077.组合.md +++ b/problems/0077.组合.md @@ -1,4 +1,8 @@ + # 第77题. 组合 + +题目链接:https://leetcode-cn.com/problems/combinations/ + 给定两个整数 n 和 k,返回 1 ... n 中所有可能的 k 个数的组合。 示例: @@ -13,6 +17,7 @@ [1,4], ] + # 思路 这是回溯法的经典题目。 @@ -33,65 +38,122 @@ 那么就三层for循环,代码如下: ``` - for (int i = 1; i <= n; i++) { - for (int j = i + 1; j <= n; j++) { - for (int u = j + 1; u <=n; n++) { +for (int i = 1; i <= n; i++) { + for (int j = i + 1; j <= n; j++) { + for (int u = j + 1; u <=n; n++) { - } } } -``` -**如果n为 100,k为50呢,那就50层for循环,是不是开始窒息。** - -那么回溯法就能解决这个问题了。 - -回溯是用来做选择的,递归用来节点层叠嵌套,**每一次的递归是层叠嵌套的关系,可以用于解决多层嵌套循环的问题。** - -其实子集和组合问题都可以抽象为一个树形结构,如下: - - - - -可以看一下这个棵树,一开始集合是 1,2,3,4, 从左向右去数,取过的数,不在重复取。 - -第一取1,集合变为2,3,4 ,因为k为2,我们只需要去一个数就可以了,分别取,2,3,4, 得到集合[1,2] [1,3] [1,4],以此类推。 - -**其实这就转化成从集合中选取子集的问题,可选择的范围随着选择的进行而限缩,于是做剪枝,调整可选择的范围** - -如何在这个树上遍历,然后收集到我们要的结果集呢,用的就是回溯搜索法,**可以发现,每次搜索到了叶子节点,我们就找到了一个结果。** - -分析完过程,我们来看一下 回溯算法的模板框架如下: -``` -backtracking() { - if (终止条件) { - 存放结果; - } - - for (选择:选择列表(可以想成树中节点孩子的数量)) { - 递归,处理节点; - backtracking(); - 回溯,撤销处理结果 - } } ``` -分析模板: +**如果n为100,k为50呢,那就50层for循环,是不是开始窒息**。 -什么是达到了终止条件,树中就可以看出,搜到了叶子节点了,就找到了一个符合题目要求的答案,就把这个答案存放起来。 +那么回溯法就能解决这个问题了。 -看一下这个for循环,这个for循环是做什么的,for 就是处理树中节点各个孩子的情况, 一个节点有多少个孩子,这个for循环就执行多少次。 +回溯是用来做选择,递归用来做节点层叠嵌套(可以理解是随便开K的for循环),**每一次的递归相当于嵌套一个for循环,可以用于解决多层嵌套循环的问题了**。 -最后就要看这个递归的过程了,注意这个backtracking就是自己调用自己,实现递归。 +**回溯问题都可以抽象为一棵树形结构!用树形结构来理解回溯就容易多了**。 -一些同学对递归操作本来就不熟练,递归上面又加上一个for循环,可能就更迷糊了, 我来给大家捋顺一下。 +那么我们把组合问题抽象为如下树形结构: -这个backtracking 其实就是向树的叶子节点方向遍历, for循环可以理解是横向遍历,backtracking 就是纵向遍历,这样就把这棵树全遍历完了。 + -那么backtracking就是一直往深处遍历,总会遇到叶子节点,遇到了叶子节点,就要返回,那么backtracking的下面部分就是回溯的操作了,撤销本次处理的结果。 +可以看出这个棵树,一开始集合是 1,2,3,4, 从左向右取数,取过的数,不在重复取。 -分析完模板,本题代码如下: +第一次取1,集合变为2,3,4 ,因为k为2,我们只需要再取一个数就可以了,分别取,2,3,4, 得到集合[1,2] [1,3] [1,4],以此类推。 + +**回溯的问题都可以抽象为一个树形结构,在求解组合问题的过程中,n相当于树的宽度,k相当于树的深度**。 + +**每次从集合中选组元素,可选择的范围随着选择的进行而限缩,调整可选择的范围** + +如何在这个树上遍历,然后收集到我们要的结果集呢? + +用的就是回溯搜索法,**可以发现,每次搜索到了叶子节点,我们就找到了一个结果**。 + + +**这份模板,大家可以要记住了,后面做回溯搜索的题目,都离不开这个模板**。 + +## 求组合 + +掌握了模板之后,我们再来看一下这道求组合的题目。 + +* 回溯函数返回值以及参数 + +在这里要定义两个全局变量,一个用来存放符合条件单一结果,一个用来存放符合条件结果的集合。 + +代码如下: + +``` +vector> result; // 存放符合条件结果的集合 +vector path; // 用来存放符合条件结果 +``` + +其实不定义这两个全局遍历也是可以的,把这两个变量放进回溯函数的参数里,但为了函数里参数太多影响可读性,所以我定义全局变量。 + +首先两个参数,集合n里面取k的数,是两个int型的变量。 + +然后还需要一个参数,也为int型变量startIndex,这个参数用来记录本层递归的中,集合从哪里开始遍历(集合就是[1,...,n] )。 + +为什么要有这个startIndex呢? + +从下图中红线部分可以看出,在集合[1,2,3,4]取1之后,下一层递归,就要在[2,3,4]中取数了,那么下一层递归如何知道从[2,3,4]中取数呢,靠的就是startIndex。 + + + +所以需要startIndex来记录下一层递归,搜索的起始位置。 + +那么整体代码如下: + +``` +vector> result; // 存放符合条件结果的集合 +vector path; // 用来存放符合条件单一结果 +void backtracking(int n, int k, int startIndex) +``` + +* 回溯函数终止条件 + +什么时候到达所谓的叶子节点了呢? + +就是path这个数组的大小如果达到k,说明我们找到了一个集合大小为k的组合了,在图中path存的就是根节点到叶子节点的路径。 + +如图红色部分: + + + +此时用result二维数组,把path保存起来,并终止本层递归。 + +所以终止条件代码如下: + +``` +if (path.size() == k) { + result.push_back(path); + return; +} +``` + + +* 回溯搜索的遍历过程 + +在如下如中,我们知道for循环用来横向遍历,递归的过程是纵向遍历。 + + +如此我们才遍历完图中的这棵树。 + +那么for循环每次就是从startIndex开始遍历,然后用path保存每次遍历到的节点。 + +代码如下: + +``` +for (int i = startIndex; i <= n; i++) { + path.push_back(i); // 处理节点 + backtracking(n, k, i + 1); // 注意下一层搜索要从i+1开始 + path.pop_back(); // 回溯,撤销处理的节点 +} +``` + +关键地方都讲完了,组合问题C++完整代码如下: -# C++ 代码 ``` class Solution { @@ -103,7 +165,6 @@ private: result.push_back(path); return; } - // 这个for循环有讲究,组合的时候 要用startIndex,排列的时候就要从0开始 for (int i = startIndex; i <= n; i++) { path.push_back(i); // 处理节点 backtracking(n, k, i + 1); @@ -111,8 +172,9 @@ private: } } public: - vector> combine(int n, int k) { + result.clear(); // 可以不写 + path.clear(); // 可以不写 backtracking(n, k, 1); return result; } @@ -145,7 +207,7 @@ for (int i = startIndex; i <= n; i++) for (int i = startIndex; i <= n - (k - path.size()) + 1; i++) ``` -整体代码如下: +优化后整体代码如下: ``` class Solution { @@ -157,7 +219,6 @@ private: result.push_back(path); return; } - // 这个for循环有讲究,组合的时候 要用startIndex,排列的时候就要从0开始 for (int i = startIndex; i <= n - (k - path.size()) + 1; i++) { path.push_back(i); // 处理节点 backtracking(n, k, i + 1); @@ -174,6 +235,3 @@ public: ``` - -# 观后感 -我来写一下观后感: 很厉害,转化成从集合中选取子集的问题,可选择的范围随着选择的进行而限缩,于是做剪枝,调整可选择的范围。 每一次的递归是层叠嵌套的关系,可以用于解决多层嵌套循环的问题。 每一层递归中,尽量节省循环次数,这样在后续的递归调用中,节省下来的循环会被以至少指数等级放大。 diff --git a/problems/0104.二叉树的最大深度.md b/problems/0104.二叉树的最大深度.md index aca59dde..91772b79 100644 --- a/problems/0104.二叉树的最大深度.md +++ b/problems/0104.二叉树的最大深度.md @@ -174,12 +174,13 @@ public: int depth = 0; while (!que.empty()) { int size = que.size(); + vector vec; depth++; // 记录深度 for (int i = 0; i < size; i++) { Node* node = que.front(); que.pop(); - for (int i = 0; i < node->children.size(); i++) { - if (node->children[i]) que.push(node->children[i]); + for (int j = 0; j < node->children.size(); j++) { + if (node->children[j]) que.push(node->children[j]); } } } diff --git a/problems/0559.N叉树的最大深度.md b/problems/0559.N叉树的最大深度.md index 8ead9f27..ad424230 100644 --- a/problems/0559.N叉树的最大深度.md +++ b/problems/0559.N叉树的最大深度.md @@ -31,12 +31,12 @@ public: while (!que.empty()) { int size = que.size(); vector vec; - depth++; + depth++; // 记录深度 for (int i = 0; i < size; i++) { Node* node = que.front(); que.pop(); - for (int i = 0; i < node->children.size(); i++) { - if (node->children[i]) que.push(node->children[i]); + for (int j = 0; j < node->children.size(); j++) { + if (node->children[j]) que.push(node->children[j]); } } } diff --git a/problems/二叉树总结.md b/problems/二叉树总结.md new file mode 100644 index 00000000..c4eb0578 --- /dev/null +++ b/problems/二叉树总结.md @@ -0,0 +1,147 @@ +> 二叉树大总结! + +不知不觉二叉树已经和我们度过了**三十三天**,代码随想录里已经发了**三十三篇二叉树的文章**,详细讲解了**30+二叉树经典题目**,一直坚持下来的录友们一定会二叉树有深刻理解了。 + +在每一道二叉树的题目中,我都使用了**递归三部曲**来分析题目,相信大家以后看到二叉树,看到递归,都会想:返回值、参数是什么?终止条件是什么?单层逻辑是什么? + +而且**几乎每一道题目我都给出对应的迭代法**,可以用来进一步提高自己。 + +下面Carl把分析过的题目分门别类,可以帮助新录友循序渐进学习二叉树,也方便老录友面试前快速复习,看到一个标题,就回想一下对应的解题思路,这样很快就可以系统性的复习一遍二叉树了。 + +公众号的发文顺序,就是循序渐进的,所以如下分类基本就是按照文章发文顺序来的,我再做一个系统性的分类。 + +# 二叉树的理论基础 + +* [关于二叉树,你该了解这些!](https://mp.weixin.qq.com/s/_ymfWYvTNd2GvWvC5HOE4A):二叉树的种类、存储方式、遍历方式、定义方式 + +# 二叉树的遍历方式 + +* 深度优先遍历 + * [二叉树:前中后序递归法](https://mp.weixin.qq.com/s/PwVIfxDlT3kRgMASWAMGhA):递归三部曲初次亮相 + * [二叉树:前中后序迭代法(一)](https://mp.weixin.qq.com/s/c_zCrGHIVlBjUH_hJtghCg):通过栈模拟递归 + * [二叉树:前中后序迭代法(二)统一风格](https://mp.weixin.qq.com/s/WKg0Ty1_3SZkztpHubZPRg) +* 广度优先遍历 + * [二叉树的层序遍历](https://mp.weixin.qq.com/s/Gb3BjakIKGNpup2jYtTzog):通过队列模拟 + + +# 求二叉树的属性 + +* [二叉树:是否对称](https://mp.weixin.qq.com/s/Kgf0gjvlDlNDfKIH2b1Oxg) + * 递归:后序,比较的是根节点的左子树与右子树是不是相互翻转 + * 迭代:使用队列/栈将两个节点顺序放入容器中进行比较 +* [二叉树:求最大深度](https://mp.weixin.qq.com/s/guKwV-gSNbA1CcbvkMtHBg) + * 递归:后序,求根节点最大高度就是最大深度,通过递归函数的返回值做计算树的高度 + * 迭代:层序遍历 +* [二叉树:求最小深度](https://mp.weixin.qq.com/s/BH8-gPC3_QlqICDg7rGSGA) + * 递归:后序,求根节点最小高度就是最小深度,注意最小深度的定义 + * 迭代:层序遍历 +* [二叉树:求有多少个节点](https://mp.weixin.qq.com/s/2_eAjzw-D0va9y4RJgSmXw) + * 递归:后序,通过递归函数的返回值计算节点数量 + * 迭代:层序遍历 +* [二叉树:是否平衡](https://mp.weixin.qq.com/s/isUS-0HDYknmC0Rr4R8mww) + * 递归:后序,注意后序求高度和前序求深度,递归过程判断高度差 + * 迭代:效率很低,不推荐 +* [二叉树:找所有路径](https://mp.weixin.qq.com/s/Osw4LQD2xVUnCJ-9jrYxJA) + * 递归:前序,方便让父节点指向子节点,涉及回溯处理根节点到叶子的所有路径 + * 迭代:一个栈模拟递归,一个栈来存放对应的遍历路径 +* [二叉树:递归中如何隐藏着回溯](https://mp.weixin.qq.com/s/ivLkHzWdhjQQD1rQWe6zWA) + * 详解[二叉树:找所有路径](https://mp.weixin.qq.com/s/Osw4LQD2xVUnCJ-9jrYxJA)中递归如何隐藏着回溯 +* [二叉树:求左叶子之和](https://mp.weixin.qq.com/s/gBAgmmFielojU5Wx3wqFTA) + * 递归:后序,必须三层约束条件,才能判断是否是左叶子。 + * 迭代:直接模拟后序遍历 +* [二叉树:求左下角的值](https://mp.weixin.qq.com/s/MH2gbLvzQ91jHPKqiub0Nw) + * 递归:顺序无所谓,优先左孩子搜索,同时找深度最大的叶子节点。 + * 迭代:层序遍历找最后一行最左边 +* [二叉树:求路径总和](https://mp.weixin.qq.com/s/6TWAVjxQ34kVqROWgcRFOg) + * 递归:顺序无所谓,递归函数返回值为bool类型是为了搜索一条边,没有返回值是搜索整棵树。 + * 迭代:栈里元素不仅要记录节点指针,还要记录从头结点到该节点的路径数值总和 + + + +# 二叉树的修改与构造 + +* [翻转二叉树](https://mp.weixin.qq.com/s/6gY1MiXrnm-khAAJiIb5Bg) + * 递归:前序,交换左右孩子 + * 迭代:直接模拟前序遍历 +* [构造二叉树](https://mp.weixin.qq.com/s/7r66ap2s-shvVvlZxo59xg) + * 递归:前序,重点在于找分割点,分左右区间构造 + * 迭代:比较复杂,意义不大 +* [构造最大的二叉树](https://mp.weixin.qq.com/s/1iWJV6Aov23A7xCF4nV88w) + * 递归:前序,分割点为数组最大值,分左右区间构造 + * 迭代:比较复杂,意义不大 +* [合并两个二叉树](https://mp.weixin.qq.com/s/3f5fbjOFaOX_4MXzZ97LsQ) + * 递归:前序,同时操作两个树的节点,注意合并的规则 + * 迭代:使用队列,类似层序遍历 + +# 求二叉搜索树的属性 + +* [二叉搜索树中的搜索](https://mp.weixin.qq.com/s/vsKrWRlETxCVsiRr8v_hHg) + * 递归:二叉搜索树的递归是有方向的 + * 迭代:因为有方向,所以迭代法很简单 +* [是不是二叉搜索树](https://mp.weixin.qq.com/s/8odY9iUX5eSi0eRFSXFD4Q) + * 递归:中序,相当于变成了判断一个序列是不是递增的 + * 迭代:模拟中序,逻辑相同 +* [求二叉搜索树的最小绝对差](https://mp.weixin.qq.com/s/Hwzml6698uP3qQCC1ctUQQ) + * 递归:中序,双指针操作 + * 迭代:模拟中序,逻辑相同 +* [求二叉搜索树的众数](https://mp.weixin.qq.com/s/KSAr6OVQIMC-uZ8MEAnGHg) + * 递归:中序,清空结果集的技巧,遍历一遍便可求众数集合 + * 迭代:模拟中序,逻辑相同 +* [二叉搜索树转成累加树](https://mp.weixin.qq.com/s/hZtJh4T5lIGBarY-lZJf6Q) + * 递归:中序,双指针操作累加 + * 迭代:模拟中序,逻辑相同 + +# 二叉树公共祖先问题 + +* [二叉树的公共祖先问题](https://mp.weixin.qq.com/s/n6Rk3nc_X3TSkhXHrVmBTQ) + * 递归:后序,回溯,找到左子树出现目标值,右子树节点目标值的节点。 + * 迭代:不适合模拟回溯 +* [二叉搜索树的公共祖先问题](https://mp.weixin.qq.com/s/Ja9dVw2QhBcg_vV-1fkiCg) + * 递归:顺序无所谓,如果节点的数值在目标区间就是最近公共祖先 + * 迭代:按序遍历 + +# 二叉搜索树的修改与构造 + +* [二叉搜索树中的插入操作](https://mp.weixin.qq.com/s/lwKkLQcfbCNX2W-5SOeZEA) + * 递归:顺序无所谓,通过递归函数返回值添加节点 + * 迭代:按序遍历,需要记录插入父节点,这样才能做插入操作 +* [二叉搜索树中的删除操作](https://mp.weixin.qq.com/s/-p-Txvch1FFk3ygKLjPAKw) + * 递归:前序,想清楚删除非叶子节点的情况 + * 迭代:有序遍历,较复杂 +* [修剪二叉搜索树](https://mp.weixin.qq.com/s/QzmGfYUMUWGkbRj7-ozHoQ) + * 递归:前序,通过递归函数返回值删除节点 + * 迭代:有序遍历,较复杂 +* [构造二叉搜索树](https://mp.weixin.qq.com/s/sy3ygnouaZVJs8lhFgl9mw) + * 递归:前序,数组中间节点分割 + * 迭代:较复杂,通过三个队列来模拟 + +# 阶段总结 + +大家以上题目都做过了,也一定要看如下阶段小结。 + +**每周小结都会对大家的疑问做统一解答,并且对每周的内容进行拓展和补充,所以一定要看,将细碎知识点一网打尽!** + +* [本周小结!(二叉树系列一)](https://mp.weixin.qq.com/s/JWmTeC7aKbBfGx4TY6uwuQ) +* [本周小结!(二叉树系列二)](https://mp.weixin.qq.com/s/QMBUTYnoaNfsVHlUADEzKg) +* [本周小结!(二叉树系列三)](https://mp.weixin.qq.com/s/JLLpx3a_8jurXcz6ovgxtg) +* [本周小结!(二叉树系列四)](https://mp.weixin.qq.com/s/CbdtOTP0N-HIP7DR203tSg) + +# 最后总结 + +**在二叉树题目选择什么遍历顺序是不少同学头疼的事情,我们做了这么多二叉树的题目了,Carl给大家大体分分类**。 + +* 涉及到二叉树的构造,无论普通二叉树还是二叉搜索树一定前序,都是先构造中节点。 + +* 求普通二叉树的属性,一般是后序,一般要通过递归函数的返回值做计算。 + +* 求二叉搜索树的属性,一定是中序了,要不白瞎了有序性了。 + +注意在普通二叉树的属性中,我用的是一般为后序,例如单纯求深度就用前序, [二叉树:找所有路径](https://mp.weixin.qq.com/s/Osw4LQD2xVUnCJ-9jrYxJA)也用了前序,这是为了方便让父节点指向子节点。 + +所以求普通二叉树的属性还是要具体问题具体分析。 + +**最后,二叉树系列就这么完美结束了,估计这应该是最长的系列了,感谢大家33天的坚持与陪伴,接下来我们又要开始新的系列了「回溯算法」!** + + +**录友们打卡的时候也说一说自己的感想吧!哈哈** + diff --git a/problems/回溯算法理论基础.md b/problems/回溯算法理论基础.md new file mode 100644 index 00000000..0f6fe6d3 --- /dev/null +++ b/problems/回溯算法理论基础.md @@ -0,0 +1,118 @@ + +> + +# 什么是回溯法 +回溯法也可以叫做回溯搜索法,它是一种搜索的方式。 + +在二叉树系列中,我们已经不止一次,提到了回溯,例如[二叉树:以为使用了递归,其实还隐藏着回溯](https://mp.weixin.qq.com/s/ivLkHzWdhjQQD1rQWe6zWA)。 + +回溯是递归的副产品,只要有递归就会有回溯。 + +回溯法的性能如何呢,这里要很大家说清楚了,**虽然回溯法很难,很不好理解,但是回溯法并不是什么高效的算法**。 + +# 回溯法的效率 + +**因为回溯的本质是穷举,穷举所有可能,然后选出我们想要的答案**,如果想让回溯法高效一些,可以加一些剪枝的操作,但也改不了回溯法就是穷举的本质。 + +那么既然回溯法并不高效为什么还要用它呢? + +因为没得选,一些问题只能暴力搜,没有更优的算法了。 + +# 回溯法解决的问题 + + + +回溯算法,相信 + +**回溯的问题都可以抽象为一个树形结构,在求解组合问题的过程中,n相当于树的宽度,k相当于树的深度**。 + +# 题外话 + +一些同学可能分不清什么是组合,什么是排列? + +组合是不强调元素顺序的,排列是强调元素顺序的。 + +例如:{1, 2} 和 {2, 1} 在组合上,就是一个集合,因为不强调顺序,而要是排列的话,{1, 2} 和 {2, 1} 就是两个集合了。 + +## 回溯算法模板 + +这里将给出Carl总结的回溯算法模板,那么先来讲一讲回溯算法的模板伪代码。 + +既然回溯是递归的副产品,那么必然要有递归。 + +**所以以下讲解中,回溯函数也就是递归函数,指的都是一个函数**。 + +在讲[二叉树的递归](https://mp.weixin.qq.com/s/PwVIfxDlT3kRgMASWAMGhA)中我们说了递归三部曲,这里我再给大家列出回溯三部曲。 + +* 回溯函数模板返回值以及参数 + +回溯函数伪代码如下: + +``` +backtracking(参数) +``` + +在回溯算法中,我的习惯函数起名字为backtracking,回溯中递归返回值一般为void。 + +在来看一下参数,因为回溯算法需要的参数,可不像二叉树递归的时候那么容易一次性确定下来,所以一般是先写逻辑,需要什么参数,就填什么参数。 + +但后面的回溯题目的讲解中,为了方便理解,我在一开始就帮大家把参数确定下来。 + +* 回溯函数终止条件 + +回溯函数终止条件伪代码如下: +``` +if (终止条件) { + 存放结果; +} +``` + +既然是树形结构,那么我们在讲解[二叉树的递归](https://mp.weixin.qq.com/s/PwVIfxDlT3kRgMASWAMGhA)的时候,就知道遍历树形结构一定要有终止条件。 + +所以回溯也有要终止条件。 + +什么时候达到了终止条件,树中就可以看出,搜到了叶子节点了,就找到了一个符合题目要求的答案,就把这个答案存放起来。 + + +* 回溯搜索的遍历过程 + +回溯函数遍历过程伪代码如下: +``` +for (选择:选择列表(可以想成树中每个节点孩子的数量)) { + 递归,处理节点; + backtracking(路径,选择列表); + 回溯,撤销处理结果 +} +``` + +看一下这个for循环,这个for循环是做什么呢? + +for 就是处理树中每一个节点的各个孩子的情况, 一个节点有多少个孩子,这个for循环就执行多少次。 + +注意这个backtracking就是自己调用自己,实现递归。 + +**一些同学对递归操作本来就不熟练,递归上面又加上一个for循环,可能就更迷糊了**, 我再给大家捋顺一下。 + +这个backtracking(递归函数)是从根节点向树的叶子节点方向遍历,**for循环可以理解是横向遍历,backtracking(递归)就是纵向遍历**,这样就把这棵树全遍历完了,如图所示: + + + +backtracking一直往深处遍历,总会遇到叶子节点,遇到了叶子节点就要返回,那么backtracking的下面部分就是回溯的操作了,撤销本次处理的结果。 + +分析完过程,回溯算法模板框架如下: + +``` +backtracking(参数) { + if (终止条件) { + 存放结果; + } + + for (选择:选择列表(可以想成树中每个节点孩子的数量)) { + 递归,处理节点; + backtracking(路径,选择列表); + 回溯,撤销处理结果 + } +} + +``` + diff --git a/problems/链表总结篇.md b/problems/链表总结篇.md index d1339934..8632463e 100644 --- a/problems/链表总结篇.md +++ b/problems/链表总结篇.md @@ -1,4 +1,7 @@ +> 链表理论基础 + + * [关于链表,你该了解这些!](https://mp.weixin.qq.com/s/ntlZbEdKgnFQKZkSUAOSpQ) * [链表:听说用虚拟头节点会方便很多?](https://mp.weixin.qq.com/s/slM1CH5Ew9XzK93YOQYSjA) * [链表:一道题目考察了常见的五个操作!](https://mp.weixin.qq.com/s/Cf95Lc6brKL4g2j8YyF3Mg)