Trapping Rain Water LT42

The above elevation map is represented by array [0,1,0,2,1,0,1,3,2,1,2,1]. In this case, 6 units of rain water (blue section) are being trapped. Thanks Marcos for contributing this image!

Example:

Input: [0,1,0,2,1,0,1,3,2,1,2,1]
Output: 6

1. Given n non-negative integers representing an elevation map where the width of each bar is 1, compute how much water it is able to trap after raining.

Brute force: for each element/bar, find the left boundary and right boundary, the level of the water trapped on this bar is Min(leftBoundary, rightBoundary) - height[i]. 

start from the bar, scan to the left to get the leftBoundary, scan to the right to get the rightBoundary

Time complexity: O(n2)

Space complexity: O(1)

class Solution {
    public int trap(int[] height) {
       int area = 0;
        
       for(int i = 0; i < height.length; ++i) {
          
           int leftBoundary = height[i];
           for(int j = 0; j <i; ++j) {
               leftBoundary = Math.max(leftBoundary, height[j]);
           }
           
           int rightBoundary = height[i];
           for(int j = i+1; j < height.length; ++j) {
               rightBoundary = Math.max(rightBoundary, height[j]);
           }
           
           area += Math.min(leftBoundary, rightBoundary) - height[i];
       }
        
       return area;
    }
}

1.a It is observed that the leftBoundary and rightBoundary has been recomputed multiple times, with dynamic programing, we could compute them once and store the result in the array.

leftBoundary[i]: maximum height starting from left and ending at i, leftBoundary[i] = Math.max(leftBoundary[i-1], height[i])

rightBoundary[j]: maximum height starting from right and ending at j, rightBoundary[j] = Math.max(rightBoundary[j+1], height[j])

Time Complexity: O(n)

Space Complexity: O(n)

public class SolutionLT42 {
    private int findMaxHeight(int[] height) {
        int result = 0;
        for(int i = 0; i < height.length; ++i) {
            if(height[i] > height[result]) {
                result = i;
            }
        }
        return result;
    }
    public int trap(int[] height) {
        if(height == null || height.length <= 1) return 0;

        int highestBar = findMaxHeight(height);

        int area = 0;
        int leftBoundary = 0;
        for(int i = 0; i < highestBar; ++i) {
            leftBoundary = Math.max(leftBoundary, height[i]);
            area += leftBoundary - height[i];
        }

        int rightBoundary = 0;
        for(int i = height.length - 1; i > highestBar; --i) {
            rightBoundary = Math.max(rightBoundary, height[i]);
            area += rightBoundary - height[i];
        }

        return area;
    }

    public static void main(String[] args) {
        SolutionLT42 subject = new SolutionLT42();
        int[] testData = {0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1};
        System.out.println(subject.trap(testData));
    }
}

1.4a Instead of doing two passes, we can start from the two ends to find out the boundary on the go with only 1 pass

if leftBounday <= rightBoundary, ++left, if height[left] < leftBoundary, area = leftBoundary - height[left]; otherwise leftBoundary = height[left]

else ++right, if height[right] < rightBoundary, area = rightBoundary - height[right]; otherwise rightBoundary = height[right]

public class SolutionLT42 {

    public int trap(int[] height) {
        if(height == null || height.length <= 1) return 0;



        int area = 0;
        int leftBoundary = height[0];
        int rightBoundary = height[height.length - 1];

        for(int left = 0, right = height.length - 1; left < right;) {
            if(leftBoundary <= rightBoundary) {
                ++left;
                leftBoundary = Math.max(leftBoundary, height[left]);
                area += leftBoundary - height[left];
            }
            else {
                --right;
                rightBoundary = Math.max(rightBoundary, height[right]);
                area += rightBoundary - height[right];
            }
        }
        return area;
    }

    public static void main(String[] args) {
        SolutionLT42 subject = new SolutionLT42();
        int[] testData = {0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1};
        System.out.println(subject.trap(testData));
    }
}

2. Use stack, in order to store the boundary for a bar, we need to store the index of height in decreasing order of height, hence the previous element before the current would be the leftBoundary, if height[i] <= height[leftBoundaryStack.peek()], leftBoundaryStack.push(i); otherwise, the current element would be the rightBoundary.

public class SolutionLT42 {

    public int trap(int[] height) {
        if(height == null || height.length <= 1) return 0;

        int area = 0;
        Deque<Integer> leftBoundaryStack = new LinkedList<>();

        for(int i = 0; i < height.length; ++i) {
            while(!leftBoundaryStack.isEmpty() && height[leftBoundaryStack.peek()] < height[i]) {
                int rightBoundary = height[i];
                int current = leftBoundaryStack.pop();
                if(leftBoundaryStack.isEmpty()) {
                    break;
                }
                int leftBoundary = height[leftBoundaryStack.peek()];
                area += (Math.min(leftBoundary, rightBoundary) - height[current]) * (i - leftBoundaryStack.peek() - 1);
            }
            leftBoundaryStack.push(i);
        }
        return area;
    }
}

Refactoring the above code, replace while loop with if, especially the ++i, interesting...

public class SolutionLT42 {

    public int trap(int[] height) {
        if(height == null || height.length <= 1) return 0;

        int area = 0;
        Deque<Integer> leftBoundaryStack = new LinkedList<>();

        for(int i = 0; i < height.length;) {
            if(leftBoundaryStack.isEmpty() || height[leftBoundaryStack.peek()] > height[i]) {
                leftBoundaryStack.push(i);
                ++i;
            }
            else {
                int current = leftBoundaryStack.pop();
                if(leftBoundaryStack.isEmpty()) {
                    continue;
                }
                int rightBoundary = height[i];
                int leftBoundary = height[leftBoundaryStack.peek()];
                int distance = i - leftBoundaryStack.peek() - 1;
                int boundedHeight = Math.min(leftBoundary, rightBoundary) - height[current];
                area += distance * boundedHeight;
            }
        }
        return area;
    }

}