分治算法

一，分治算法的思想

将一个规模为n的问题分解为k个规模较小的子问题，这些子问题相互独立且与原问题相同。递归求解这些子问题，然后将各子问题的解合并得到原问题的解。

二，常见的使用分治算法求解的问题

1，二分搜索技术

int BinarySearch(vector<int> &nums, int x){
    int low = 0, mid, high = nums.size() - 1;
    while(low <= high){
        mid = (low + high) / 2;
        if(nums[mid] == x){
            return mid;
        } else if(nums[mid] > x){
            high = mid - 1;
        } else {
            low = mid + 1;
        }
    }
    return -1;
}

2，快速排序

void quickSort(vector<int> &nums, int s, int t){
    if(s < t){
        int temp = nums[s];
        int i = s, j = t;
        while(i < j){
            while(i < j && nums[j] >= temp){
                j --;
            }
            nums[i] = nums[j];
            while(i < j && nums[i] <= temp){
                i ++;
            }
            nums[j] = nums[i];
        }
        nums[i] = temp;
        quickSort1(nums, s, i - 1);
        quickSort1(nums, i + 1, t);
    }
}

3，归并排序

void Merge(int nums[], int low, int mid, int high){
    int i = low, j = mid + 1, k = 0;
    int *temp;
    temp = new int[high - low + 1];
    while(i <= mid && j <= high){
        if(nums[i] < nums[j]){
            temp[k ++] = nums[i];
            i ++;
        } else {
            temp[k ++] = nums[j];
            j ++;
        }
    }
    while(i <= mid){
        temp[k ++] = nums[i];
        i ++;
    }
    while(j <= high){
        temp[k ++] = nums[j];
        j ++;
    }
    for(i = low, k = 0; i <= high; k ++, i ++){
        nums[i] = temp[k];
    }
    delete temp;
}

void MergePass(int nums[], int length, int n){
    int i;
    for(i = 0; i + 2 * length - 1 < n; i = i + 2 * length){
        Merge(nums, i, i + length - 1, i + 2 *length - 1);
    }
    if(i + length - 1 < n - 1){
        Merge(nums, i, i + length - 1, n - 1);
    }
}

void MergeSort(int nums[], int numsSize){
    for(int length = 1; length < numsSize; length = 2 * length){
        MergePass(nums, length, numsSize);
    }
}