From 441de1ebf5a18fee26cadc159d88fd6fecdf4059 Mon Sep 17 00:00:00 2001 From: youngyangyang04 <826123027@qq.com> Date: Wed, 9 Dec 2020 17:32:26 +0800 Subject: [PATCH] Update --- README.md | 5 + pics/62.不同路径.png | Bin 0 -> 83043 bytes pics/62.不同路径1.png | Bin 0 -> 39910 bytes pics/62.不同路径2.png | Bin 0 -> 18229 bytes problems/0062.不同路径.md | 176 ++++++++++++++++++++++++++++++++++ problems/导读.md | 9 +- 6 files changed, 189 insertions(+), 1 deletion(-) create mode 100644 pics/62.不同路径.png create mode 100644 pics/62.不同路径1.png create mode 100644 pics/62.不同路径2.png create mode 100644 problems/0062.不同路径.md diff --git a/README.md b/README.md index 34624ae1..524973ee 100644 --- a/README.md +++ b/README.md @@ -51,6 +51,10 @@ * [上海有这些互联网公司,你都知道么?](https://mp.weixin.qq.com/s/msqbX6eR2-JBQOYFfec4sg) * [成都有这些互联网公司,你都知道么?](https://mp.weixin.qq.com/s/Y9Qg22WEsBngs8B-K8acqQ) +* 算法性能分析 + * [关于时间复杂度,你不知道的都在这里!](https://mp.weixin.qq.com/s/LWBfehW1gMuEnXtQjJo-sw) + * [O(n)的算法居然超时了,此时的n究竟是多大?](https://mp.weixin.qq.com/s/73ryNsuPFvBQkt6BbhNzLA) + * 数组 * [必须掌握的数组理论知识](https://mp.weixin.qq.com/s/X7R55wSENyY62le0Fiawsg) * [数组:每次遇到二分法,都是一看就会,一写就废](https://mp.weixin.qq.com/s/fCf5QbPDtE6SSlZ1yh_q8Q) @@ -243,6 +247,7 @@ |[0056.合并区间](https://github.com/youngyangyang04/leetcode/blob/master/problems/0056.合并区间.md) |数组 |中等| **贪心** 以为是模拟题,其实是贪心| |[0057.插入区间](https://github.com/youngyangyang04/leetcode/blob/master/problems/0057.插入区间.md) |数组 |困难| **模拟** 是一道数组难题| |[0059.螺旋矩阵II](https://github.com/youngyangyang04/leetcode/blob/master/problems/0059.螺旋矩阵II.md) |数组 |中等|**模拟**| +|[0062.不同路径](https://github.com/youngyangyang04/leetcode/blob/master/problems/0062.不同路径.md) |数组、动态规划 |中等|**深搜** **动态规划** **数论**| |[0070.爬楼梯](https://github.com/youngyangyang04/leetcode/blob/master/problems/0070.爬楼梯.md) |动态规划|简单|**动态规划** dp里求排列| |[0077.组合](https://github.com/youngyangyang04/leetcode/blob/master/problems/0077.组合.md) |回溯 |中等|**回溯**| |[0078.子集](https://github.com/youngyangyang04/leetcode/blob/master/problems/0078.子集.md) |回溯/数组 |中等|**回溯**| diff --git a/pics/62.不同路径.png b/pics/62.不同路径.png new file mode 100644 index 0000000000000000000000000000000000000000..d82c6ed03af356554843e6266936f03ed36adb77 GIT binary patch literal 83043 zcmZ@=2RxPi`$no`#W5memX(<@B4k!pWbc*DLH1VmmI@J(kgc+hb$fKh!1pi=%peO4Z0JY8FG}zW1LNE~OpJ5jcTDK- z-eLXquh?Vn&i(a0mJ#%W>R%^^Ffhb09!TC(am8GEdl5<6J9tbx-Q^KCCd6`|%tLl* zpwQUo{f~x*cnbZ^4y#*l8Z=@lDy837J4;DMJR1;8zYoWrm#_FWHr=uFJK4#3uxhEw z#l@_3b=PxtWo7Am67M)Kuge_2BJ0j_>G(=fmD+o0f*g$^21K`QrORr`RUzjxXY;Qw+SPK5dYeEDf$5sw*#{QL|H z_Wj>q>At@HrKDm#Op;>Q|ND#XHTVe)hCOyH2D<TW22?}-{~e_Xncu4$>O*N#Vb*PLZkHI=2WD9u?rkZh?9$|De$G(M zQESUqFR*o5=&>i+eD=7?eJfWg>~i0`d(U%+^qr(n=0qoibKkYK-Cn8SKQnwNhOutl z6n=%zc}atc-!`c&MRdHxW?1bRhelVc`bXu)p<*kua~JRnMZESj_P3YcYN)3*eJMN{ zqu4#7=ruK>`rNmq|6YB+1BODYe$K?Z9wqYe+;vO}N#>6;6^e_k`b|xkOKnFNhTgZy z(0cW(JzE_7oVm3!I_&=Ir(vFUX-OKxsO1wA?w(-i<4puH5BXa*FqL=yGvsRKLXRC= 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+这道题目,刚一看最直观的想法就是用图论里的深搜,来枚举出来有多少种路径。 + +注意题目中说机器人每次只能向下或者向右移动一步,那么其实**机器人走过的路径可以抽象为一颗二叉树,而叶子节点就是终点!** + +如图举例: + +![62.不同路径](https://img-blog.csdnimg.cn/20201209113602700.png) + +此时问题就可以转化为求二叉树叶子节点的个数,代码如下: + +```C++ +class Solution { +private: + int dfs(int i, int j, int m, int n) { + if (i > m || j > n) return 0; // 越界了 + if (i == m && j == n) return 1; // 找到一种方法,相当于找到了叶子节点 + return dfs(i + 1, j, m, n) + dfs(i, j + 1, m, n); + } +public: + int uniquePaths(int m, int n) { + return dfs(1, 1, m, n); + } +}; +``` + +大家如果提交了代码就会发现超时了! + +来分析一下时间复杂度,这个深搜的算法,其实就是要遍历整个二叉树。 + +这颗树的深度其实就是m+n-1(深度按从1开始计算)。 + +那二叉树的节点个数就是 2^(m + n - 1) - 1。可以理解深搜的算法就是遍历了整个满二叉树(其实没有把搜索节点都遍历到,只是近似而已) + +所以上面深搜代码的时间复杂度为O(2^(m + n - 1) - 1),可以看出,这是指数级别的时间复杂度,是非常大的。 + +## 动态规划 + +机器人从(0 , 0) 位置触发,到(m - 1, n - 1)终点。 + +按照动规三部曲来分析: + +* dp数组表述啥 + +这里设计一个dp二维数组,dp[i][j] 表示从(0 ,0)出发,到(i, j) 有几条不同的路径。 + +* dp数组的初始化 + +如何初始化呢,首先dp[i][0]一定都是1,因为从(0, 0)的位置到(i, 0)的路径只有一条,那么dp[0][j]也同理。 + +所以初始化代码为: + +``` +for (int i = 0; i < m; i++) dp[i][0] = 1; +for (int j = 0; j < n; j++) dp[0][j] = 1; +``` + +* 递推公式 + +想要求dp[i][j],只能有两个方向来推导出来,即dp[i - 1][j] 和 dp[i][j - 1]。 + +此时在回顾一下 dp[i-1][j] 表示啥,是从(0, 0)的位置到(i-1, j)有几条路径,dp[i][j - 1]同理。 + +那么很自然,dp[i][j] = dp[i-1][j] + dp[i][j - 1],因为dp[i][j]只有这两个方向过来。 + +如图所示: + +![62.不同路径1](https://img-blog.csdnimg.cn/20201209113631392.png) + +C++代码如下: + +```C++ +class Solution { +public: + int uniquePaths(int m, int n) { + vector> dp(m, vector(n, 0)); + for (int i = 0; i < m; i++) dp[i][0] = 1; + for (int j = 0; j < n; j++) dp[0][j] = 1; + for (int i = 1; i < m; i++) { + for (int j = 1; j < n; j++) { + dp[i][j] = dp[i - 1][j] + dp[i][j - 1]; + } + } + return dp[m - 1][n - 1]; + } +}; +``` +* 时间复杂度:O(m * n) +* 空间复杂度:O(m * n) + +其实用一个一维数组(也可以理解是滚动数组)就可以了,但是不利于理解,可以优化点空间,建议先理解了二维,在理解一维,C++代码如下: + +```C++ +class Solution { +public: + int uniquePaths(int m, int n) { + vector dp(n); + for (int i = 0; i < n; i++) dp[i] = 1; + for (int j = 1; j < m; j++) { + for (int i = 1; i < n; i++) { + dp[i] += dp[i - 1]; + } + } + return dp[n - 1]; + } +}; +``` +* 时间复杂度:O(m * n) +* 空间复杂度:O(n) + +# 数论方法 + +在这个图中,可以看出一共 m,n的话,无论怎么走,走到终点都需要 m + n - 2 步。 + +![62.不同路径](https://img-blog.csdnimg.cn/20201209113602700.png) + +在这m + n - 2 步中,一定有 m - 1 步是要向下走的,不用管什么时候向下走。 + +那么有几种走法呢? 可以转化为,给你m + n - 2个不同的数,随便取m - 1个数,有几种取法。 + +那么这就是一个组合问题了。 + +那么答案,如图所示: + +![62.不同路径2](https://img-blog.csdnimg.cn/20201209113725324.png) + +**求组合的时候,要防止两个int相乘溢出!** 所以不能把算式的分子都算出来,分母都算出来再做除法。 + +``` +class Solution { +public: + int uniquePaths(int m, int n) { + int numerator = 1, denominator = 1; + int count = m - 1; + int t = m + n - 2; + while (count--) numerator *= (t--); // 计算分子,此时分子就会溢出 + for (int i = 1; i <= m - 1; i++) denominator *= i; // 计算分母 + return numerator / denominator; + } +}; + +``` + +需要在计算分子的时候,不算除以分母,代码如下: + +``` +class Solution { +public: + int uniquePaths(int m, int n) { + long long numerator = 1; // 分子 + int denominator = m - 1; // 分母 + int count = m - 1; + int t = m + n - 2; + while (count--) { + numerator *= (t--); + while (denominator != 0 && numerator % denominator == 0) { + numerator /= denominator; + denominator--; + } + } + return numerator; + } +}; +``` + +计算组合问题的代码还是有难度的,特别是处理溢出的情况! + +最后这个代码还有点复杂了,还是可以优化,我就不继续优化了,有空在整理一下,哈哈,就酱! + + + + diff --git a/problems/导读.md b/problems/导读.md index 500dd2a8..4b842b66 100644 --- a/problems/导读.md +++ b/problems/导读.md @@ -12,7 +12,10 @@ # 文章篇 -* 求职 +* 代码风格 + * [看了这么多代码,谈一谈代码风格!](https://mp.weixin.qq.com/s/UR9ztxz3AyL3qdHn_zMbqw) + +* 求职 * [程序员的简历应该这么写!!(附简历模板)](https://mp.weixin.qq.com/s/nCTUzuRTBo1_R_xagVszsA) * [BAT级别技术面试流程和注意事项都在这里了](https://mp.weixin.qq.com/s/815qCyFGVIxwut9I_7PNFw) * [深圳原来有这么多互联网公司,你都知道么?](https://mp.weixin.qq.com/s/Yzrkim-5bY0Df66Ao-hoqA) @@ -20,6 +23,10 @@ * [上海有这些互联网公司,你都知道么?](https://mp.weixin.qq.com/s/msqbX6eR2-JBQOYFfec4sg) * [成都有这些互联网公司,你都知道么?](https://mp.weixin.qq.com/s/Y9Qg22WEsBngs8B-K8acqQ) +* 算法性能分析 + * [关于时间复杂度,你不知道的都在这里!](https://mp.weixin.qq.com/s/LWBfehW1gMuEnXtQjJo-sw) + * [O(n)的算法居然超时了,此时的n究竟是多大?](https://mp.weixin.qq.com/s/73ryNsuPFvBQkt6BbhNzLA) + * 数组 * [必须掌握的数组理论知识](https://mp.weixin.qq.com/s/X7R55wSENyY62le0Fiawsg) * [数组:每次遇到二分法,都是一看就会,一写就废](https://mp.weixin.qq.com/s/fCf5QbPDtE6SSlZ1yh_q8Q)