leetcode动态规划笔记二---矩阵型DP

动态规划

题目分类

一维dp

矩阵型DP

• Unique Paths II : 矩阵型DP，求所有方法总数
• Minimum Path Sum:矩阵型，求最大最小值
• Triangle : 矩阵型，求最大最小值
• Maximum Square ：矩阵型，求最大最小值
• Range Sum Query 2D - Immutable : 求和

Unique Paths II : 矩阵型DP，求所有方法总数

方法一：自顶向下
递归法,time limited

class Solution {
    int helper(int[][] g, int x, int y){
        int top = 0, left = 0;
        if(g[x][y] == 1) return 0;
        
        if(x == 0 && y == 0) return 1;
        
        if(x > 0){
            top = helper(g, x - 1, y);
        }
        
        if(y > 0){
            left = helper(g, x, y - 1);
        }
        
        return top + left;
    }
    public int uniquePathsWithObstacles(int[][] obstacleGrid) {
        return helper(obstacleGrid, obstacleGrid.length - 1, obstacleGrid[0].length -1);
    }
}

方法一：自顶向下
递归法 + memo ac

class Solution {
    Map<Long, Integer> m = new HashMap<>();
    
    int helper(int[][] g, int x, int y){
        int top = 0, left = 0;
        if(g[x][y] == 1) return 0;
        
        if(x == 0 && y == 0) return 1;
        
        long key = ((long)x) << 32 | y; // long key = x << 32 | y; 错误
        
        if(m.containsKey(key)) return m.get(key);
        
        if(x > 0){
            top = helper(g, x - 1, y);
        }
        
        if(y > 0){
            left = helper(g, x, y - 1);
        }
        
        m.put(key, top + left);
        return top + left;
    }
    public int uniquePathsWithObstacles(int[][] obstacleGrid) {
        return helper(obstacleGrid, obstacleGrid.length - 1, obstacleGrid[0].length -1);
    }
}

方法三：自底向上
递推 + memo
这里用了二维memo，其实状态转移中，当前step只与上一step有关系，只用一维memo就可以了。

class Solution {
    public int uniquePathsWithObstacles(int[][] g) {
        if(g.length == 0) return 0;
        
        int[][] ps = new int[g.length][g[0].length];
        boolean obstacle = false;
        
        for(int i = 0; i < g.length; i ++){
            if(g[i][0] == 1){
                obstacle = true;
            }
            ps[i][0] = obstacle ? 0 : 1;
        }
        
        obstacle = false;
        for(int i = 0; i < g[0].length; i ++){
            if(g[0][i] == 1){
                obstacle = true;
            }
            ps[0][i] = obstacle ? 0 : 1;
        }        
        
        for(int i = 1; i < g.length; i++){
            
            for(int j = 1; j < g[0].length; j++){
                if(g[i][j] == 1){
                    ps[i][j] = 0;
                }        
                else{
                    ps[i][j] = ps[i-1][j] + ps[i][j-1];
                }
            }
        }
        
        return ps[g.length-1][g[0].length-1];
    }
}

Minimum Path Sum:矩阵型，求最大最小值

方法一：自底向上 + 二维memo

class Solution {
    public int minPathSum(int[][] g) {
        if (g.length == 0) return 0;
        
        int[][] sum = new int[g.length][g[0].length];
        sum[0][0] = g[0][0];
        
        for(int i = 1; i < g.length; i++){
            sum[i][0] = g[i][0] + sum[i-1][0];
        }
        
        for(int i = 1; i < g[0].length; i++){
            sum[0][i] = g[0][i] + sum[0][i-1];
        }        
        
        for(int i = 1; i < g.length; i ++){
            
            for(int j = 1; j < g[0].length; j++){
                sum[i][j] = g[i][j] + Math.min(sum[i-1][j], sum[i][j-1]);
            }
        }
        
        return sum[g.length-1][g[0].length-1];
    }
}

方法二：自底向上 + 一维memo

class Solution {
    public int minPathSum(int[][] g) {
        if (g.length == 0) return 0;
        
        int[] sum = new int[g[0].length];
        
        sum[0] = g[0][0];
        for(int i = 1; i < g[0].length; i++){
            sum[i] = g[0][i] + sum[i-1];
        }        
        
        for(int i = 1; i < g.length; i ++){
            sum[0] = sum[0] + g[i][0];
            
            for(int j = 1; j < g[0].length; j++){
                sum[j] = g[i][j] + Math.min(sum[j], sum[j-1]);
            }
        }
        
        return sum[g[0].length-1];
    }
}

Triangle : 矩阵型，求最小值

方法一 ： 自定向下
递归 + memo
问题拆分、状态、状态转移方程、初始值，一个都不能少

class Solution {
    Map<Long, Integer> m = new HashMap<>();
    
    int helper(List<List<Integer>> tri, int row, int col){

        long key = (long)row << 32 | col;
        if(m.containsKey(key)){
            return m.get(key);
        }
        
        if ( row == 0 && col == 0 ){  //不能忽略此逻辑
            return tri.get(0).get(0);
        }
        
        int left = Integer.MAX_VALUE;
        int top = Integer.MAX_VALUE;
        if(row > 0){
            if(col < tri.get(row).size() - 1){
                top = helper(tri, row - 1, col);
            }

            if(col > 0){
                left = helper(tri, row - 1, col - 1);
            }
        }
        
        int val = Math.min(top, left) + tri.get(row).get(col);
        m.put(key, val);
        return val;
    }
    
    
    public int minimumTotal(List<List<Integer>> tri) {
        if(tri.size() == 0) return 0;
        
        int sum = Integer.MAX_VALUE;
        for(int j = 0; j < tri.get(tri.size()-1).size(); j++){
            sum = Math.min(sum, helper(tri, tri.size()-1, j));
        }

        return sum;
    }
}

方法二 ： 自底向上
递推 + memo，下面的方法还可以进一步优化

class Solution {
    public int minimumTotal(List<List<Integer>> tri) {
        if(tri.size() == 0) return 0;
        
        int[] memo = new int[tri.get(tri.size()-1).size()];
        memo[0] = tri.get(0).get(0);
        
        for(int i = 1; i < tri.size(); i ++){
            for(int j = tri.get(i).size() -1; j >= 0 ; j--){
                if(j == 0){
                    memo[j] = memo[j] + tri.get(i).get(j);
                }
                else if(j == tri.get(i).size() - 1){
                    memo[j] = memo[j-1] + tri.get(i).get(j);
                }
                else{
                    memo[j] = Math.min(memo[j-1], memo[j]) + tri.get(i).get(j);
                }
            }
        }
        
        int sum = Integer.MAX_VALUE;
        for(int i=0; i<memo.length; i++){
            sum = Math.min(sum, memo[i]);
        }
        
        return sum;
    }
}

Maximum Square ：矩阵型，求最大最小值

方法一：自底向上
递推 + memo
主要看斜对角线，当前位置能成为多大square，要看左上角位置有多大square。本地弯弯道道挺多。

class Solution {
    public int maximalSquare(char[][] m) {
        if(m.length == 0) return 0;
        
        int[][] memo = new int[m.length + 1][m[0].length + 1];
        
        for(int i = 0; i <= m.length; i++ ){
            memo[i][0] = 0;
        }
        
        for(int i = 0; i <= m[0].length; i++){
            memo[0][i] = 0;
        }
        
        int maxi = 0;
        
        for(int i = 0; i < m.length; i++){
            
            for(int j = 0; j < m[0].length; j++){
                if(m[i][j] != '1'){
                    memo[i+1][j+1] = 0;
                    continue;
                }
                
                if(memo[i][j] <= 0){
                    memo[i+1][j+1] = 1;
                }
                else {
                    int k = 1;
                    for(; k <= memo[i][j]; k++){
                        if(m[i-k][j] != '1' || m[i][j-k] != '1'){
                            break;
                        }
                    }
                    memo[i+1][j+1] = k;
                }
                //System.out.println("i" + i + " j" + j + " =" + memo[i+1][j+1]);
                maxi = Math.max(memo[i+1][j+1], maxi);
            }
        }
        
        return maxi * maxi;
    }
}

Range Sum Query 2D - Immutable

方法一 ： 自底向上， 递推 + memo

class NumMatrix {
    
    private int[][] memo;
    
    public NumMatrix(int[][] m) {
        if(m.length == 0 || m[0].length == 0) return;
        
        memo = new int[m.length][m[0].length + 1];
        
        for(int i = 0; i < m.length; i++){
            for(int j = 0; j < m[0].length; j++){
                memo[i][j+1] = memo[i][j] + m[i][j];
            }
        }
    }
    
    public int sumRegion(int row1, int col1, int row2, int col2) {
        int sum = 0;
        for(int i = row1; i <= row2; i++){
            sum += (memo[i][col2+1] - memo[i][col1]);
        }
        return sum;
    }
}