题目描述：Median of Two Sorted Arrays
There are two sorted arrays nums1 and nums2 of size m and n respectively.
Find the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)).
题目是《算法导论》上的一道习题，不过已多次出现在面试题当中。注意，此题中两个数组的长度是相等的。当然，长度不等的话也可以做，只是要多些判断条件。参考leetcode题目 Median of Two Sorted Arrays
方法1 直接遍历
直接的解法是遍历两个数组并计数，类似归并排序里面的有序数组的合并，复杂度为O（n）。代码如下：
1 def getMedian(nums1, nums2, n): 2 ans1, ans2 = -1, -1 3 i = j = 0 4 for count in range(n+1): 5 if i < n and (nums1[i] < nums2[j] or j >= n): 6 ans1 = ans2 7 ans2 = nums1[i] 8 i += 1 9 else : 10 ans1 = ans2 11 ans2 = nums2[j] 12 j += 1 13 return (ans1 + ans2)/2.0 14 15 if __name__ == '__main__': 16 nums1 = [1, 12, 15, 26, 38] 17 nums2 = [2, 13, 17, 30, 45] 18 n1 = len(nums1) 19 n2 = len(nums2) 20 if n1 == n2: 21 print("Median is %s", getMedian(nums1, nums2, n1)) 22 else: 23 print("Can't work for nums of unequal size")
方法2 分治法
需要用分治法求解，时间复杂度为O(log (m+n))。
1.假设数组A的中位数为m1，数组B为m2，例如：nums1 = {1, 12, 15, 26, 38}， nums2 = {2, 13, 17, 30, 45}，m1 = 15 ，m2 = 17 。
2.由于m1<m2，则可以确定中位数即为下面两个子数组的中位数 ：[15, 26, 38] 和 [2, 13, 17]
3.重复步骤1、2，可以得到 m1 = 26， m2 = 13. 得到两个子数组：[15, 26] 和[13, 17]
4.这时，由于n=2，无法在继续分下去了。可以直接计算得：
median = (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1]))/2
= (max(15, 13) + min(26, 17))/2
= (15 + 17)/2
= 16
代码如下：
1 def median(nums, n): 2 3 if n%2 == 0: 4 return (nums[n//2] + nums[n//2-1])/2.0 5 else: 6 return nums[n//2] 7 8 def getMedian(nums1, nums2, n): 9 m1, m2 = -1, -1 10 if n <= 0: 11 return -1 12 elif n == 1: 13 return (nums1[0] + nums2[0]) / 2 14 elif n == 2: 15 return (max(nums1[0], nums2[0]) + min(nums1[1], nums2[1])) / 2 16 m1 = median(nums1, n) 17 m2 = median(nums2, n) 18 # /* 相等可直接返回 */ 19 if m1 == m2: 20 return m1 21 if m1 < m2: 22 if n%2 == 0: 23 return getMedian(nums1[n//2-1:], nums2[:n//2+1], n//2 + 1) 24 else: 25 return getMedian(nums1[n//2:], nums2[:n//2+1], n//2 + 1) 26 else: 27 if n%2 == 0: 28 return getMedian(nums2[n//2-1:], nums1[:n//2+1], n//2 + 1) 29 else: 30 return getMedian(nums2[n//2:], nums1[:n//2+1], n//2 + 1) 31 32 if __name__ == "__main__": 33 nums1 = [1, 12, 15, 26, 38] 34 nums2 = [2, 13, 17, 30, 45] 35 n1 = len(nums1) 36 n2 = len(nums2) 37 if n1 == n2: 38 print("Median is %s", getMedian(nums1, nums2, n1)) 39 else: 40 print("Doesn't work for arrays of unequal size")