关于求中位数Median的相关LeetCode题目

关于我的 Leetcode 题目解答，代码前往 Github：https://github.com/chenxiangcyr/leetcode-answers

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如何理解中位数 Median:

Dividing a set into two equal length subsets, that one subset is always greater than the other.
中位数 Median 可以将一个集合分为长度相等的两个子集合，其中一个子集合的元素都大于另一个子集合。

LeetCode题目：4. 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)).
假设集合A的长度为m，集合B的长度为n。
将集合A从位置 i 处分开，得到左右两个部分：

left_A = A[0]....A[i - 1] 长度为 i
right_A = A[i]....A[m - 1] 长度为 m - i

将集合B从位置 j 处分开，得到左右两个部分：

left_B = B[0]....B[j - 1] 长度为 j
right_B = B[j]....B[n - 1] 长度为 n - j

令 left_part = left_A + left_B, right_part = right_A + right_B。
如果我们可以确保如下两个条件成立：

• 条件1：len(left_part)=len(right_part)
• 条件2：max(left_part) ≤ min(right_part)

则可以得到中位数Median = (max(left_part) + min(right_part)) / 2

要确保如上两个条件成立，则需要保证：

• 条件1：i + j == m + n - i - j
• 条件2：A[i - i] <= B[j] && B[j - 1] <= A[i]

具体代码如下：

class Solution {
    public double findMedianSortedArrays(int[] A, int[] B) {
        int m = A.length;
        int n = B.length;
        
        // 确保数组A长度小于数组B长度，因此确保 j 不会为负数
        if (m > n) {
            int[] temp = A;
            A = B;
            B = temp;
            
            int tmp = m;
            m = n;
            n = tmp;
        }
        
        int iMin = 0;
        int iMax = m;
        int halfLen = (m + n + 1) / 2;
        
        while (iMin <= iMax) {
            // 初始从数组A的中间分隔
            int i = (iMin + iMax) / 2;
            
            // 确保条件1：`i + j == m + n - i - j`
            int j = halfLen - i;
            
            // i is too small
            if (i < iMax && B[j-1] > A[i]){
                iMin = iMin + 1;
            }
            // i is too big
            else if (i > iMin && A[i-1] > B[j]) {
                iMax = iMax - 1;
            }
            // 确保条件2：`A[i - i] <= B[j] && B[j - 1] <= A[i]`
            else {
                int maxLeft = 0;
                if (i == 0) {
                    maxLeft = B[j-1];
                }
                else if (j == 0) {
                    maxLeft = A[i-1];
                }
                else {
                    maxLeft = Math.max(A[i-1], B[j-1]);
                }
                
                if ( (m + n) % 2 == 1 ) {
                    return maxLeft;
                }

                int minRight = 0;
                if (i == m) {
                    minRight = B[j];
                }
                else if (j == n) {
                    minRight = A[i];
                }
                else {
                    minRight = Math.min(B[j], A[i]);
                }

                return (maxLeft + minRight) / 2.0;
            }
        }
        
        return 0.0;
    }
}

LeetCode题目：295. Find Median from Data Stream
Design a data structure that supports the following two operations:
void addNum(int num) - Add a integer number from the data stream to the data structure.
double findMedian() - Return the median of all elements so far.

基本思想：

• 使用一个最大堆 PriorityQueue 存储左半部分
• 使用一个最小堆 PriorityQueue 存储右半部分
• 保证 maxHeap.size() - minHeap.size() <= 1

具体代码如下：

class MedianFinder {
    // store the smaller half of the input numbers
    private PriorityQueue maxHeap;
    
    // store the larger half of the input numbers
    private PriorityQueue minHeap;
    
    
    /** initialize your data structure here. */
    public MedianFinder() {
        // Should provide comparator to support max heap
        maxHeap = new PriorityQueue(Collections.reverseOrder());
        
        
        // by default, PriorityQueue is a min heap
        minHeap = new PriorityQueue();
    }
    
    public void addNum(int num) {
        
        maxHeap.add(num);
        
        // balancing
        minHeap.add(maxHeap.peek());
        maxHeap.poll();
        
        // maintain size property
        if(maxHeap.size() < minHeap.size()) {
            maxHeap.add(minHeap.peek());
            minHeap.poll();
        }
    }
    
    public double findMedian() {
        if((maxHeap.size() + minHeap.size()) % 2 == 0) {
            return (maxHeap.peek() + minHeap.peek()) / 2.0;
        }
        else {
            return maxHeap.peek();
        }
    }
}