From dd5fb4bae5c3e327689f8051641758ddd7f06be5 Mon Sep 17 00:00:00 2001 From: xiaowei wan Date: Tue, 6 Aug 2019 16:54:21 -0700 Subject: [PATCH] LC 4 median of two sorted array --- README.en.md | 2 + README.md | 1 + .../4.median-of-two-sorted-array-1.jpg | Bin 0 -> 74040 bytes .../4.median-of-two-sorted-array-2.jpg | Bin 0 -> 148112 bytes .../4.median-of-two-sorted-array-3.png | Bin 0 -> 817987 bytes .../4.median-of-two-sorted-array-4.png | Bin 0 -> 178267 bytes .../4.median-of-two-sorted-array-5.png | Bin 0 -> 938089 bytes problems/4.median-of-two-sorted-array.md | 173 ++++++++++++++++++ 8 files changed, 176 insertions(+) create mode 100644 assets/problems/4.median-of-two-sorted-array-1.jpg create mode 100644 assets/problems/4.median-of-two-sorted-array-2.jpg create mode 100644 assets/problems/4.median-of-two-sorted-array-3.png create mode 100644 assets/problems/4.median-of-two-sorted-array-4.png create mode 100644 assets/problems/4.median-of-two-sorted-array-5.png create mode 100644 problems/4.median-of-two-sorted-array.md diff --git a/README.en.md b/README.en.md index ac3e651cd..e537b095e 100644 --- a/README.en.md +++ b/README.en.md @@ -195,6 +195,8 @@ The data structures mainly includes: - [1031.maximum-sum-of-two-non-overlapping-subarrays](./problems/1031.maximum-sum-of-two-non-overlapping-subarrays.md) 🆕 #### Hard + +- [0004.median-of-two-sorted-array](./problems/4.median-of-two-sorted-array.md) 🆕 - [0023.merge-k-sorted-lists](./problems/23.merge-k-sorted-lists.md) - [0032.longest-valid-parentheses](./problems/32.longest-valid-parentheses.md) 🆕 - [0042.trapping-rain-water](./problems/42.trapping-rain-water.md) diff --git a/README.md b/README.md index a332ec672..d93a5c465 100644 --- a/README.md +++ b/README.md @@ -215,6 +215,7 @@ leetcode 题解,记录自己的 leetcode 解题之路。 #### 困难难度 +- [0004.median-of-two-sorted-array](./problems/4.median-of-two-sorted-array.md) 🆕 - [0023.merge-k-sorted-lists](./problems/23.merge-k-sorted-lists.md) - [0032.longest-valid-parentheses](./problems/32.longest-valid-parentheses.md) 🆕 - [0042.trapping-rain-water](./problems/42.trapping-rain-water.md) diff --git a/assets/problems/4.median-of-two-sorted-array-1.jpg b/assets/problems/4.median-of-two-sorted-array-1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..22a90d9b8309ec48cae9a71a1b803ebf5b942937 GIT binary patch literal 74040 zcmeFZ2|Sej-Y|Y?h>!`%ZYo41gcLGKkt9?oYbARKkt{P3Nnt{WVhGt{vXe4)LfQ9y z?EBcpFx&r9_kEvppVN8HIsf;4KJR%y?{o1nuDmn*w^v9a-CtHr-1Ylt2AM?2@9%f*+nGjN1WxJMIzMoBvvt0IQ0}-4 zaO#|v4*kyR;PvCZ=#RJiIt4@eP(U92tlys33&e8>S_Qz)v}+%O41|sy*u_oZ>r=-45 zOV7y6%P%M_DlRGgR8?J5TUX!kwXwaUv#YzOx37N;H$E{rH9a#sx3apnzCqX|Zf)*B3Sm58Eko^r@AQI?y(bGfd8F%2K+x2hPE{QxV3 z4rC?>I{*hL{OBc%r_zc(7>*Hd)jQCo4@4mZ-PHSnI;)^>A_waiU zHfYVSp;xk0Q?L2Yn76yfU#ZsjpW+W14pYiK=F|Ecy zyh*svGx~rlcI#uo|5-%-+#N~l`Hw}X4Q>sYJUg54VH&1(9hKZ@n6Gj&eLI30#iE;G zaxPhPLJzLQYG0<7zYy03W-UYk-z?{R3 zdZfds`vLWVfM>S2ZtHe`J{DbS3x2Fa~)F^<#D>2?OQa! 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z=S5rpoznRKbR7DhnpX&*=!m?1Z(<}?^|9_1ilRuA^EV&q7O~{^$M*H#C$h*D zbO`?BTHfaG|8;d5?f1dB_ssGC8`%H B[j]`, 则把`B[j]` 放入新的数组中,j往后移一位,即 `j+1`. +3. 重复步骤#1 和 #2,直到`i`移到`A`最后,或者`j`移到`B`最后。 +4. 如果`j`移动到`B`数组最后,那么直接把剩下的所有`A`依次放入新的数组中. +5. 如果`i`移动到`A`数组最后,那么直接把剩下的所有`B`依次放入新的数组中. + +Merge的过程如下图。 +![merge two sorted array](../assets/problems/4.median-of-two-sorted-array-2.jpg) + + +*时间复杂度: `O(m+n) - m is length of A, n is length of B`* + +*空间复杂度: `O(m+n)`* + +#### 解法二 - 二分查找 (Binary Search) +由于题中给出的数组都是排好序的,在排好序的数组中查找很容易想到可以用二分查找(Binary Search), 这里对数组长度小的做二分, +保证数组A 和 数组B 做partition 之后 + +`len(Aleft)+len(Bleft)=(m+n+1)/2 - m是数组A的长度, n是数组B的长度` + +对数组A的做partition的位置是区间`[0,m]` + +如图: +![partition A,B](../assets/problems/4.median-of-two-sorted-array-3.png) + +下图给出几种不同情况的例子(注意但左边或者右边没有元素的时候,左边用`INF_MIN`,右边用`INF_MAX`表示左右的元素: +![median examples](../assets/problems/4.median-of-two-sorted-array-5.png) + +下图给出具体做的partition 解题的例子步骤, +![median partition example](../assets/problems/4.median-of-two-sorted-array-4.png) + +*时间复杂度: `O(log(min(m, n)) - m is length of A, n is length of B`* + +*空间复杂度: `O(1)` - 这里没有用额外的空间* + +## 关键点分析 +1. 暴力求解,在线性时间内merge两个排好序的数组成一个数组。 +2. 二分查找,关键点在于 + - 要partition两个排好序的数组成左右两等份,partition需要满足`len(Aleft)+len(Bleft)=(m+n+1)/2 - m是数组A的长度, n是数组B的长度` + + - 并且partition后 A左边最大(`maxLeftA`), A右边最小(`minRightA`), B左边最大(`maxLeftB`), B右边最小(`minRightB`) 满足 +`(maxLeftA <= minRightB && maxLeftB <= minRightA)` + +有了这两个条件,那么median就在这四个数中,根据奇数或者是偶数, +``` +奇数: +median = max(maxLeftA, maxLeftB) +偶数: +median = (max(maxLeftA, maxLeftB) + min(minRightA, minRightB)) / 2 +``` + +## 代码(Java code) +*解法一 - 暴力解法(Brute force)* +```java +class MedianTwoSortedArrayBruteForce { + public double findMedianSortedArrays(int[] nums1, int[] nums2) { + int[] newArr = mergeTwoSortedArray(nums1, nums2); + int n = newArr.length; + if (n % 2 == 0) { + // even + return (double) (newArr[n / 2] + newArr[n / 2 - 1]) / 2; + } else { + // odd + return (double) newArr[n / 2]; + } + } + private int[] mergeTwoSortedArray(int[] nums1, int[] nums2) { + int m = nums1.length; + int n = nums2.length; + int[] res = new int[m + n]; + int i = 0; + int j = 0; + int idx = 0; + while (i < m && j < n) { + if (nums1[i] <= nums2[j]) { + res[idx++] = nums1[i++]; + } else { + res[idx++] = nums2[j++]; + } + } + while (i < m) { + res[idx++] = nums1[i++]; + } + while (j < n) { + res[idx++] = nums2[j++]; + } + return res; + } +} +``` +*解法二 - 二分查找(Binary Search* +```java +class MedianSortedTwoArrayBinarySearch { + public static double findMedianSortedArraysBinarySearch(int[] nums1, int[] nums2) { + // do binary search for shorter length array, make sure time complexity log(min(m,n)). + if (nums1.length > nums2.length) { + return findMedianSortedArraysBinarySearch(nums2, nums1); + } + int m = nums1.length; + int n = nums2.length; + int lo = 0; + int hi = m; + while (lo <= hi) { + // partition A position i + int i = lo + (hi - lo) / 2; + // partition B position j + int j = (m + n + 1) / 2 - i; + + int maxLeftA = i == 0 ? Integer.MIN_VALUE : nums1[i - 1]; + int minRightA = i == m ? Integer.MAX_VALUE : nums1[i]; + + int maxLeftB = j == 0 ? Integer.MIN_VALUE : nums2[j - 1]; + int minRightB = j == n ? Integer.MAX_VALUE : nums2[j]; + + if (maxLeftA <= minRightB && maxLeftB <= minRightA) { + // total length is even + if ((m + n) % 2 == 0) { + return (double) (Math.max(maxLeftA, maxLeftB) + Math.min(minRightA, minRightB)) / 2; + } else { + // total length is odd + return (double) Math.max(maxLeftA, maxLeftB); + } + } else if (maxLeftA > minRightB) { + // binary search left half + hi = i - 1; + } else { + // binary search right half + lo = i + 1; + } + } + return 0.0; + } +} +```

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