[二分查找] 一：子区间界限应当如何确定

普通的二分查找

先引入一个最简单的二分查找吧

int binarySearch(int* nums, int numsSize, int target){
	int left = 0, right = numsSize-1;
	while(left <= right){
		int mid = left + (right - left)/2;
		if(nums[mid] == target){
			return mid;
		}
		else if(nums[mid] > target){
			right = mid - 1;
		}
		else left = mid + 1;
	}
	return -1;
}

很明显，这么写法将每次执行的区间划分为
[left .....mid -1] (mid) [mid+1....right]

那换个写法呢（结果都是正确的）？

int binarySearch_2(int* nums, int numsSize, int target){
    int left = 0, right = numsSize;
    while(left < right){
        int mid = left + (right - left)/2;
        if(nums[mid] == target){
            return mid;
        }
        else if(nums[mid] > target){
            right = mid;
        }
        else left = mid + 1;
    }
    return -1;
}

这个写法将区间分为了左闭右开型，不同于上面的左右两边都是闭区间
[left....mid) (mid) [mid+1....right)

以上两个写法的while条件，一个是left <= right，一个则是left < right

说明第一种while循环的结束条件为left > right，第二种结束条件为left == right

其原因就是因为左闭右闭的区间[i, i]是包含元素nums[i]的，而左闭右开的区间[i, i) 相当于空集。

搞清楚while循环结束时的具体情况，对我们分析边界值非常重要。

当数组中存在多个目标值时，这种方法返回的是位于中间位置那个目标值的下标，但是如果我们想获得目标值第一次出现时的下标应该怎么办呢?

获取左侧边界的二分查找

int binarySearch_3(int *nums, int numsSize, int target){
	int left = 0, right = numsSize - 1;
	while(left <= right){
        int mid = left + (right - left)/2;
		if(nums[mid] == target){
			right = mid - 1;
		}
		else if(nums[mid] > target){
			right = mid - 1;
		}
		else left = mid + 1;
	}
	return nums[left] == target? left : -1;
}

此算法while循环的结束条件为left > right，或者说是left = right + 1，如果nums[left] == target则说明我们最终找到了目标值，而且此时的left是该目标值第一次出现的位置。这种也可以换成子区间左闭右开的写法

int binarySearch_4(int *nums, int numsSize, int target){
    int left = 0, right = numsSize;
    while(left < right){
        int mid = left + (right - left)/2;
        if(nums[mid] == target){
            right = mid;
        }
        else if(nums[mid] > target){
            right = mid;
        }
        else left = mid + 1;
    }
    return nums[left] == target? left : -1;
}

获取右侧边界的二分查找

原理同上，两种写法

int binarySearch_5(int *nums, int numsSize, int target){
    int left = 0, right = numsSize-1;
    while(left <= right){
        int mid = left + (right - left)/2;
        if(nums[mid] == target){
            left = mid + 1;
        }
        else if(nums[mid] > target){
            right = mid - 1;
        }
        else left = mid + 1;
    }
    return nums[right] == target? right : -1;
}

int binarySearch_6(int *nums, int numsSize, int target){
    int left = 0, right = numsSize;
    while(left < right){
        int mid = left + (right - left)/2;
        if(nums[mid] == target){
            left = mid + 1;
        }
        else if(nums[mid] > target){
            right = mid;
        }
        else left = mid + 1;
    }
    return nums[right-1] == target? right-1 : -1;
}

int main(){
    int nums[6] = {1,2,2,2,3,5};
    printf("binarySearch_1: %d\n", binarySearch_1(nums, 6, 2));
    printf("binarySearch_2: %d\n", binarySearch_2(nums, 6, 2));
    printf("binarySearch_3: %d\n", binarySearch_3(nums, 6, 2));
    printf("binarySearch_4: %d\n", binarySearch_4(nums, 6, 2));
    printf("binarySearch_5: %d\n", binarySearch_5(nums, 6, 2));
    printf("binarySearch_6: %d\n", binarySearch_6(nums, 6, 2));
    return 0;
}

“茴香豆”的“茴”字有四种写法，可这二分法有六种写法，孔乙己，得亏你生的早啊。

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