AcWing 1027. 方格取数(数字三角形线性DP)

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#动态规划

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本文探讨了四维动态规划算法的未优化版本及其在三维空间中的优化实现，通过对比两种方法，详细阐述了如何在保持算法核心思想不变的前提下，通过减少状态空间维度来提高算法效率。

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算法思想：

四维未优化版


import java.util.*;
class Main{
    static int n = 0, N = 15;
    static int[][] nums = new int[N][N];
    static int[][][][] max = new int[N][N][N][N];
    public static void main(String[] args)throws Exception{
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        while(true){
            int a = sc.nextInt();
            int b = sc.nextInt();
            int w = sc.nextInt();
            if(a == 0 && b == 0 && w == 0)break;
            nums[a][b] = w;
        }
            for(int i1 = 1; i1 <= n; ++i1)
                for(int j1 = 1; j1 <= n; ++j1)
                    for(int i2 = 1; i2 <= n; ++i2){
                        for(int j2 = 1; j2 <= n; ++j2){
                            int x = max[i1][j1][i2][j2];
                            if(i1 == i2 && j1 == j2){
                                x = Math.max(x, max[i1 - 1][j1][i2 - 1][j2] + nums[i1][j1]);
                                x = Math.max(x, max[i1 - 1][j1][i2][j2 - 1] + nums[i1][j1]);
                                x = Math.max(x, max[i1][j1 - 1][i2 - 1][j2] + nums[i1][j1]);
                                x = Math.max(x, max[i1][j1 - 1][i2][j2 - 1] + nums[i1][j1]);
                            }else{
                                x = Math.max(x, max[i1 - 1][j1][i2 - 1][j2] + nums[i1][j1] + nums[i2][j2]);
                                x = Math.max(x, max[i1 - 1][j1][i2][j2 - 1] + nums[i1][j1] + nums[i2][j2]);
                                x = Math.max(x, max[i1][j1 - 1][i2 - 1][j2] + nums[i1][j1] + nums[i2][j2]);
                                x = Math.max(x, max[i1][j1 - 1][i2][j2 - 1] + nums[i1][j1] + nums[i2][j2]);
                            }
                            max[i1][j1][i2][j2] = x; 
                        }
                    }
        System.out.print(max[n][n][n][n]);
        
    }
}

三维优化版


import java.util.*;
class Main{
    static int n = 0, N = 15;
    static int[][] nums = new int[N][N];
    static int[][][] max = new int[N + N][N][N];
    public static void main(String[] args)throws Exception{
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        while(true){
            int a = sc.nextInt();
            int b = sc.nextInt();
            int w = sc.nextInt();
            if(a == 0 && b == 0 && w == 0)break;
            nums[a][b] = w;
        }
        for(int k = 2; k <= n + n; ++k)
            for(int i1 = 1; i1 <= n; ++i1)
                for(int i2 = 1; i2 <= n; ++i2){
                    int j1 = k - i1, j2 = k - i2;
                    if(j1 >= 1 && j1 <= n && j2 >= 1 && j2 <= n){
                        int t = nums[i1][j1];
                        if(i1 != i2)t += nums[i2][j2];
                        int x = max[k][i1][i2];
                        x = Math.max(x, max[k - 1][i1 - 1][i2 - 1] + t);
                        x = Math.max(x, max[k - 1][i1][i2 - 1] + t);
                        x = Math.max(x, max[k - 1][i1 - 1][i2] + t);
                        x = Math.max(x, max[k - 1][i1][i2] + t);
                        max[k][i1][i2] = x;
                    }
                }
        System.out.print(max[n + n][n][n]);
        
    }
}


import java.util.*;
class Main{
    static int n = 0, N = 20;
    static int[][] nums = new int[N][N];
    static int[][][] max = new int[N + N][N][N];
    public static void main(String[] args)throws Exception{
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        while(true){
            int a = sc.nextInt();
            int b = sc.nextInt();
            int w = sc.nextInt();
            if(a == 0 && b == 0 && w == 0)break;
            nums[a][b] = w;
        }
         for(int k = 2; k <= n + n; ++k)
            for(int i = 1; i < k; ++i)
                for(int j = 1; j < k; ++j){
                        int t = nums[i][k - i];
                        if(i != j)t += nums[j][k - j];
                        int x = max[k][i][j];
                        x = Math.max(x, max[k - 1][i - 1][j - 1]);
                        x = Math.max(x, max[k - 1][i][j - 1]);
                        x = Math.max(x, max[k - 1][i - 1][j]);
                        x = Math.max(x, max[k - 1][i][j]);
                        max[k][i][j] = x + t;
                }
        System.out.print(max[n + n][n][n]);
        
    }
}