11.container-with-most-water.md

题目地址

https://leetcode.com/problems/container-with-most-water/description/

题目描述

Given n non-negative integers a1, a2, ..., an , where each represents a point at coordinate (i, ai). n vertical lines are drawn such that the two endpoints of line i is at (i, ai) and (i, 0). Find two lines, which together with x-axis forms a container, such that the container contains the most water.

Note: You may not slant the container and n is at least 2.

The above vertical lines are represented by array [1,8,6,2,5,4,8,3,7]. In this case, the max area of water (blue section) the container can contain is 49.

 

Example:

Input: [1,8,6,2,5,4,8,3,7]
Output: 49

思路

符合直觉的解法是，我们可以对两两进行求解，计算可以承载的水量。 然后不断更新最大值，最后返回最大值即可。 这种解法，需要两层循环，时间复杂度是O(n^2)

eg:

   // 这个解法比较暴力，效率比较低
    // 时间复杂度是O(n^2)
    let max = 0;
    for(let i = 0; i < height.length; i++) {
        for(let j = i + 1; j < height.length; j++) {
            const currentArea = Math.abs(i - j) * Math.min(height[i], height[j]);
            if (currentArea > max) {
                max = currentArea;
            }
        }
    }
    return max;

这种符合直觉的解法有点像冒泡排序， 大家可以稍微类比一下

那么有没有更加优的解法呢？我们来换个角度来思考这个问题，上述的解法是通过两两组合，这无疑是完备的， 那我门是否可以先计算长度为n的面积，然后计算长度为n-1的面积，... 计算长度为1的面积。 这样去不断更新最大值呢？ 很显然这种解法也是完备的，但是似乎时间复杂度还是O(n ^ 2), 不要着急。

考虑一下，如果我们计算n-1长度的面积的时候,是直接直接排除一半的结果的。

如图：

比如我们计算n面积的时候，假如左侧的线段高度比右侧的高度低，那么我们通过左移右指针来将长度缩短为n-1的做法是没有意义的， 因为新的形成的面积变成了(n-1) * heightOfLeft 这个面积一定比刚才的长度为n的面积nn * heightOfLeft 小

也就是说最大面积一定是当前的面积或者通过移动短的线段得到。

关键点解析

• 双指针优化时间复杂度

代码

• 语言支持：JS，C++

JavaScript Code:

/*
 * @lc app=leetcode id=11 lang=javascript
 *
 * [11] Container With Most Water
 *
 * https://leetcode.com/problems/container-with-most-water/description/
 *
 * algorithms
 * Medium (42.86%)
 * Total Accepted:    344.3K
 * Total Submissions: 790.1K
 * Testcase Example:  '[1,8,6,2,5,4,8,3,7]'
 *
 * Given n non-negative integers a1, a2, ..., an , where each represents a
 * point at coordinate (i, ai). n vertical lines are drawn such that the two
 * endpoints of line i is at (i, ai) and (i, 0). Find two lines, which together
 * with x-axis forms a container, such that the container contains the most
 * water.
 * 
 * Note: You may not slant the container and n is at least 2.
 * 
 * 
 * 
 * 
 * 
 * The above vertical lines are represented by array [1,8,6,2,5,4,8,3,7]. In
 * this case, the max area of water (blue section) the container can contain is
 * 49. 
 * 
 * 
 * 
 * Example:
 * 
 * 
 * Input: [1,8,6,2,5,4,8,3,7]
 * Output: 49
 * 
 */
/**
 * @param {number[]} height
 * @return {number}
 */
var maxArea = function(height) {
    if (!height || height.length <= 1) return 0;
    
    // 双指针来进行优化
    // 时间复杂度是O(n)
    let leftPos = 0;
    let rightPos = height.length - 1;
    let max = 0;
    while(leftPos < rightPos) {
        
        const currentArea = Math.abs(leftPos - rightPos) * Math.min(height[leftPos] , height[rightPos]);
        if (currentArea > max) {
            max = currentArea;
        }
        // 更新小的
        if (height[leftPos] < height[rightPos]) {
            leftPos++;
        } else { // 如果相等就随便了
            rightPos--;
        }
    }

    return max;
};

C++ Code:

class Solution {
public:
    int maxArea(vector<int>& height) {
        auto ret = 0ul, leftPos = 0ul, rightPos = height.size() - 1;
        while( leftPos < rightPos)
        {
            ret = std::max(ret, std::min(height[leftPos], height[rightPos]) * (rightPos - leftPos));
            if (height[leftPos] < height[rightPos]) ++leftPos;
            else --rightPos;
        }
        return ret;
    }
};