From 9b6cac0a2fa3e33d463f5d0ac3eb902818faf10c Mon Sep 17 00:00:00 2001 From: ShusenTang Date: Tue, 7 Apr 2020 12:17:17 +0800 Subject: [PATCH] add 85. Maximal Rectangle :beer: --- README.md | 1 + solutions/85. Maximal Rectangle.md | 122 +++++++++++++++++++++++++++++ solutions/img/85.png | Bin 0 -> 205083 bytes 3 files changed, 123 insertions(+) create mode 100644 solutions/85. Maximal Rectangle.md create mode 100644 solutions/img/85.png diff --git a/README.md b/README.md index 55e832a..893d196 100644 --- a/README.md +++ b/README.md @@ -85,6 +85,7 @@ My LeetCode solutions with Chinese explanation. 我的LeetCode中文题解。 | 82 |[Remove Duplicates from Sorted List II](https://leetcode.com/problems/remove-duplicates-from-sorted-list-ii/)|[C++](https://github.com/ShusenTang/LeetCode/blob/master/solutions/82.%20Remove%20Duplicates%20from%20Sorted%20List%20II.md)|Medium| | | 83 |[Remove Duplicates from Sorted List](https://leetcode.com/problems/remove-duplicates-from-sorted-list)|[C++](https://github.com/ShusenTang/LeetCode/blob/master/solutions/83.%20Remove%20Duplicates%20from%20Sorted%20List.md)|Easy| | | 84 |[Largest Rectangle in Histogram](https://leetcode.com/problems/largest-rectangle-in-histogram/)|[C++](solutions/84.%20Largest%20Rectangle%20in%20Histogram.md)|Hard| | +| 85 |[Maximal Rectangle](https://leetcode.com/problems/maximal-rectangle/)|[C++](solutions/85.%20Maximal%20Rectangle.md)|Hard| | | 86 |[Partition List](https://leetcode.com/problems/partition-list/)|[C++](https://github.com/ShusenTang/LeetCode/blob/master/solutions/86.%20Partition%20List.md)|Medium| | | 88 |[Merge Sorted Array](https://leetcode.com/problems/merge-sorted-array)|[C++](https://github.com/ShusenTang/LeetCode/blob/master/solutions/88.%20Merge%20Sorted%20Array.md)|Easy| | | 89 |[Gray Code](https://leetcode.com/problems/gray-code/)|[C++](https://github.com/ShusenTang/LeetCode/blob/master/solutions/89.%20Gray%20Code.md)|Medium| | diff --git a/solutions/85. Maximal Rectangle.md b/solutions/85. Maximal Rectangle.md new file mode 100644 index 0000000..53e1510 --- /dev/null +++ b/solutions/85. Maximal Rectangle.md @@ -0,0 +1,122 @@ +# [85. Maximal Rectangle](https://leetcode.com/problems/maximal-rectangle/) + +# 思路 + +## 思路一 + +对于二维矩阵的第 i 行,这一行及上面的部分可以看成是一个直方图,而在直方图里求最大矩形即[84. Largest Rectangle in Histogram](https://leetcode.com/problems/largest-rectangle-in-histogram/),我们可以用单调栈在线性复杂度内求解84题,详见[84题解](84.%20Largest%20Rectangle%20in%20Histogram.md)。 + +所以对于此题,我们只需要开辟一个表示直方图的一维数组`heights`,然后对于每一行,建立直方图,然后再用84题思路(代码中采用的是[84题解思路二](84.%20Largest%20Rectangle%20in%20Histogram.md))进行求解。我们可以根据上一行的直方图`heights`在线性时间复杂度内建立当前行的直方图`heights`: +* 如果`matrix[i][j] == '0'`,则`heights[j] = 0`; +* 否则,`heights[j] += 1`; + +时间复杂度O(MN),空间复杂度O(N);其中M和N分别为高和宽。 + +## 思路二 + +对于每个点,我们会通过以下步骤计算出一个矩形: + +* 不断向上方遍历,直到遇到“0”,以此找到矩形的高度。 +* 向左右两边扩展,直到无法容纳这个高度。 + +例如,找到黄色点对应的矩形([图片来源](https://leetcode-cn.com/problems/maximal-rectangle/solution/zui-da-ju-xing-by-leetcode/)): + +
+85img +
+ +如果知道了高height和左右边界left、right,这样就可以计算出面积等于`height * (right - left + 1)`。 + +求height的方法和思路一是一样的,下面讨论求left和right。 + +设第`i-1`行的left和right已经求出来了分别为`old_left`和`old_right`,当前点为`matrix[i][j]`为`"1"`, +1. 求left +设`left_zero`为`matrix[i][j]`向左看第一个0的列坐标,那么易知`new_left[j] = max(old_left[j], left_zero + 1)`; + +2. 求right +类似的,设`right_zero`为`matrix[i][j]`向右看第一个0的列坐标,那么易知`new_right[j] = min(old_right[j], right_zero - 1)`。 + +所以我们可以开辟一个一维数组`left`和`right`,然后对于每一行:从左往后更新`left`,从右往左更新`right`(就是动态规划的滚动数组思路)。 + +时间复杂度O(MN),空间复杂度O(N);其中M和N分别为高和宽。 + + +# C++ +## 思路一 +``` C++ +class Solution { +public: + int maximalRectangle(vector>& matrix) { + int m = matrix.size(); + if(!m) return 0; + int n = matrix[0].size(); + + int max_res = 0, res = 0; + vectorheights(n + 1, 0); // 注意大小是n+1, 见84题解 + for(int i = 0; i < m; i++){ + for(int j = 0; j < n; j++){ + if(matrix[i][j] == '0') heights[j] = 0; + else heights[j] += 1; + } + + // 下面几行基本就是84题解 + stackascend_stk; res = 0; + for(int k = 0; k <= n; k++){ + while(!ascend_stk.empty() && heights[ascend_stk.top()] >= heights[k]){ + int height = heights[ascend_stk.top()]; ascend_stk.pop(); + int width = ascend_stk.empty() ? k : k - 1 - ascend_stk.top(); + res = max(res, height * width); + } + ascend_stk.push(k); + } + + max_res = max(res, max_res); + } + return max_res; + } +}; +``` + +## 思路二 +``` C++ +class Solution { +public: + int maximalRectangle(vector>& matrix) { + int m = matrix.size(); + if(!m) return 0; + int n = matrix[0].size(); + + int res = 0, left_zero, right_zero; + vectorheights(n, 0), left(n, -1), right(n, n); + for(int i = 0; i < m; i++){ + // 更新高度 + for(int j = 0; j < n; j++) + if(matrix[i][j] == '0') heights[j] = 0; + else heights[j] += 1; + + // 更新左边界(可以和上面更新高度合在一个循环内) + left_zero = -1; + for(int j = 0; j < n; j++){ + if(matrix[i][j] == '0') { + left_zero = j; + left[j] = -1; + } + else left[j] = max(left_zero + 1, left[j]); + } + + // 更新右边界 + right_zero = n; + for(int j = n-1; j >= 0; j--){ // 逆向循环!!! + if(matrix[i][j] == '0') { + right_zero = j; + right[j] = n; + } + else right[j] = min(right_zero - 1, right[j]); + res = max(res, heights[j] * (right[j] - left[j] + 1)); + } + } + + return res; + } +}; +``` \ No newline at end of file diff --git a/solutions/img/85.png b/solutions/img/85.png new file mode 100644 index 0000000000000000000000000000000000000000..4d9a5d2cda9144b68118ec45e7f87990618c2de5 GIT binary patch literal 205083 zcmeFZRa9Kj)-H$zNN{%#5ZntdfaH!z!TDZHrQ@C}`?Ry`3 zjQ;z*$2f1>mh83W{HAfK30jR(Bf5`) zA-E)x3s{gcbY$7hLO@s#cw)PPocIbJY?LMCI_U5!d zmhJ7Zm&=0~;Ds*BIwr#hH;A(FoAeuF@4>e5Xt!^k1Y^h z0+8k$_EU5xt7r!`RLO@G&50R~`5UI}PxydL5q1k8KjNb*OYjNx)_ds$FwK+GJ95gM z@dVl3ogInq4PkcKx7oQd{?{3Rlcb&@VheNj5328Rzw|9;k-S+7zBcTbjfCxS2FEfd zg!BzNr_Qsn@av6z8?=`jl{22Aiah%2V^dA=>m))uuK*}Q5Vudy35=`#U7|B@JQe}y zJ!q-^iyA0T!ssPvm`?mo8{Wl{A3$;rm`F!KL*3;7ilpt=7&3@hAJO zlXP+K@{C*x=6SFLC~0p1BlZ*;9IceWf!XWHbveB7<5Y31k=Pu)ku3s_6&or2GqNt} z;Jb{^3v<&Gi2h985^yMTpOH5ZpTwaP5g(=FQGwf^Y3|8zspP^3-Z6VUP*WXspxdUX 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