Merge Sort

- 图片来源与翁恺2025程算授课课件

Core idea

1. Divide
   Split the array into two halves.

2. Conquer
   Recursively sort the left half and the right half.

3. Combine
   Merge the two sorted halves into one sorted array.

Implementation

• recursion (Top-Down)

#difine MAX_N 10000
int tmp[MAX_N];

void Merge(int arr[],int tmp[], int low,int m,int high){
    int i=low, j=m+1;
    for(int k = low; k <= high; k++){
        if(i <= m && (j > high || arr[i] <= arr[j])){
            tmp[k] = arr[i++];
        }else{
            tmp[k] = arr[j++];
        }
    }
    for(int k = low; k <= high; k++){
        arr[k] = tmp[k];
    }
}

void merge_sort(int arr[], int low, int high) {
    if(low < high){
        int mid = (low + high) / 2;
        merge_sort(arr, low, mid);
        merge_sort(arr, mid+1, high);
        Merge(arr, tmp, low, mid, high);
    }
}

Complexity Analysis

• the length of the array each time we process:

1 → 2 → 4 → 8 → … → n

• iteration (Bottom-Up)

void merge_sort(int arr[], int n) {
    int sz;
    for (sz = 1; sz < n; sz *= 2) {       // 每次子区间大小翻倍
        int i;
        for (i = 0; i + sz < n; i += 2*sz) {  // 每次合并两个大小为 sz 的区间
            int mid = i + sz;
            int right = (i + 2*sz < n) ? i + 2*sz : n;
            merge(arr, i, mid, right);   // 使用你之前的 merge 函数
        }
    }
}

Time complexity - the outer loop executes log(2)n times - the inner loop iterate through the entire array(n) each time - the final time complexity is nlogn

Space complexity - recursion depth O(log n) - temp -> O(n) - final space complexity: O(n) + O(logn) = O(n)