中位数探秘：巧用二分法求解两个正序数组的中间值

二分法求解中位数算法步骤

1. 将两个数组 nums1 和 nums2 合并为一个新的数组 nums。
2. 确定新数组 nums 的总长度为 m + n。
3. 如果 m + n 为偶数，则中位数为 nums[(m + n) / 2] 和 nums[(m + n) / 2 - 1] 的平均值。
4. 如果 m + n 为奇数，则中位数为 nums[(m + n) / 2]。

算法复杂度分析

二分法求解中位数算法的时间复杂度为 O(log(m + n))，其中 m 和 n 分别是两个正序数组 nums1 和 nums2 的长度。

代码示例

Python

def find_median(nums1, nums2):
  """
  Finds the median of two sorted arrays.

  Args:
    nums1: The first sorted array.
    nums2: The second sorted array.

  Returns:
    The median of the two sorted arrays.
  """

  # Merge the two arrays into a single sorted array.
  nums = sorted(nums1 + nums2)

  # Calculate the length of the merged array.
  length = len(nums)

  # If the length of the merged array is even, the median is the average of the two
  # middle elements.
  if length % 2 == 0:
    median = (nums[length // 2 - 1] + nums[length // 2]) / 2
  # Otherwise, the median is the middle element.
  else:
    median = nums[length // 2]

  return median


# Example usage.
nums1 = [1, 3, 5, 7]
nums2 = [2, 4, 6, 8]

median = find_median(nums1, nums2)
print(median)  # Output: 4.5

Java

import java.util.Arrays;

class MedianFinder {

    public double findMedian(int[] nums1, int[] nums2) {
        // Merge the two arrays into a single sorted array.
        int[] nums = new int[nums1.length + nums2.length];
        System.arraycopy(nums1, 0, nums, 0, nums1.length);
        System.arraycopy(nums2, 0, nums, nums1.length, nums2.length);
        Arrays.sort(nums);

        // Calculate the length of the merged array.
        int length = nums.length;

        // If the length of the merged array is even, the median is the average of the two
        // middle elements.
        if (length % 2 == 0) {
            return (nums[length / 2 - 1] + nums[length / 2]) / 2.0;
        }
        // Otherwise, the median is the middle element.
        else {
            return nums[length / 2];
        }
    }

    // Example usage.
    public static void main(String[] args) {
        int[] nums1 = {1, 3, 5, 7};
        int[] nums2 = {2, 4, 6, 8};

        MedianFinder mf = new MedianFinder();
        double median = mf.findMedian(nums1, nums2);
        System.out.println(median);  // Output: 4.5
    }
}

C++

#include <iostream>
#include <vector>

using namespace std;

double findMedian(vector<int>& nums1, vector<int>& nums2) {
  // Merge the two vectors into a single sorted vector.
  vector<int> nums;
  nums.insert(nums.end(), nums1.begin(), nums1.end());
  nums.insert(nums.end(), nums2.begin(), nums2.end());
  sort(nums.begin(), nums.end());

  // Calculate the length of the merged vector.
  int length = nums.size();

  // If the length of the merged vector is even, the median is the average of the two
  // middle elements.
  if (length % 2 == 0) {
    return (nums[length / 2 - 1] + nums[length / 2]) / 2.0;
  }
  // Otherwise, the median is the middle element.
  else {
    return nums[length / 2];
  }
}

// Example usage.
int main() {
  vector<int> nums1 = {1, 3, 5, 7};
  vector<int> nums2 = {2, 4, 6, 8};

  double median = findMedian(nums1, nums2);
  cout << median << endl;  // Output: 4.5

  return 0;
}

C#

using System;
using System.Collections.Generic;

class MedianFinder
{
    public double FindMedian(int[] nums1, int[] nums2)
    {
        // Merge the two arrays into a single sorted array.
        int[] nums = new int[nums1.Length + nums2.Length];
        Array.Copy(nums1, 0, nums, 0, nums1.Length);
        Array.Copy(nums2, 0, nums, nums1.Length, nums2.Length);
        Array.Sort(nums);

        // Calculate the length of the merged array.
        int length = nums.Length;

        // If the length of the merged array is even, the median is the average of the two
        // middle elements.
        if (length % 2 == 0)
        {
            return (nums[length / 2 - 1] + nums[length / 2]) / 2.0;
        }
        // Otherwise, the median is the middle element.
        else
        {
            return nums[length / 2];
        }
    }

    // Example usage.
    public static void Main(string[] args)
    {
        int[] nums1 = { 1, 3, 5, 7 };
        int[] nums2 = { 2, 4, 6, 8 };

        MedianFinder mf = new MedianFinder();
        double median = mf.FindMedian(nums1, nums2);
        Console.WriteLine(median);  // Output: 4.5
    }
}

Javascript

function findMedian(nums1, nums2) {
  // Merge the two arrays into a single sorted array.
  const nums = nums1.concat(nums2).sort((a, b) => a - b);

  // Calculate the length of the merged array.
  const length = nums.length;

  // If the