741.Cherry-Pickup

741. Cherry Pickup

题目地址

https://leetcode.com/problems/cherry-pickup/

题目描述

In a N x N grid representing a field of cherries, each cell is one of three possible integers.

0 means the cell is empty, so you can pass through;
1 means the cell contains a cherry, that you can pick up and pass through;
-1 means the cell contains a thorn that blocks your way.

Your task is to collect maximum number of cherries possible by following the rules below:

Starting at the position (0, 0) and reaching (N-1, N-1) by moving right or down through valid path cells (cells with value 0 or 1);
After reaching (N-1, N-1), returning to (0, 0) by moving left or up through valid path cells;
When passing through a path cell containing a cherry, you pick it up and the cell becomes an empty cell (0);
If there is no valid path between (0, 0) and (N-1, N-1), then no cherries can be collected.

Example 1:

Input: grid =
[[0, 1, -1],
 [1, 0, -1],
 [1, 1,  1]]
Output: 5
Explanation: 
The player started at (0, 0) and went down, down, right right to reach (2, 2).
4 cherries were picked up during this single trip, and the matrix becomes [[0,1,-1],[0,0,-1],[0,0,0]].
Then, the player went left, up, up, left to return home, picking up one more cherry.
The total number of cherries picked up is 5, and this is the maximum possible.

Note:
grid is an N by N 2D array, with 1 <= N <= 50.
Each grid[i][j] is an integer in the set {-1, 0, 1}.
It is guaranteed that grid[0][0] and grid[N-1][N-1] are not -1.

代码

Approach #1 Dynamic Programming (Top Down)

https://www.cnblogs.com/grandyang/p/8215787.html

相当于同时从(N-1, N-1) => (0, 0) 出发两个人 (x1, y1) 和 (x2, y2)

x1 + x2 = y1 + y2

三维数组dp[x1][y1][x2] 有四种走法，取最大值:

dp(x1 + 1, y1, x2 + 1) dp(x1 + 1, y1, x2) dp(x1, y1 + 1, x2 + 1) dp(x1, y1 + 1, x2)

class Solution {
  int[][][] dp;
  int[][] grid;
  int N;
  public int cherryPickup(int[][] grid) {
        this.grid = grid;
    N = grid.length;
    dp = new int[N][N][N];
    for (int[][] layer: dp) {
      for (int[] row: layer) {
        Arrays.fill(row, Integer.MIN_VALUE);
      }
    }
    return Math.max(0, dp(0, 0, 0));
  }

  public int dp(int x1, int y1, int x2) {
    int y2 = x1 + y1 - x2;
    if (x1 == N || x2 == N || y1 == N || y2 == N ||
       grid[x1][y1] == -1 || grid[x2][y2] == -1) {
      return -9999; // 免去后面的ans判断<0 
    } else if (x1 == N - 1 && y1 == N - 1) {
      return grid[x1][y1];
    } else if (dp[x1][y1][x2] != Integer.MIN_VALUE) {
      return dp[x1][y1][x2];
    } else {
      int ans = Math.max(
        Math.max(dp(x1 + 1, y1, x2 + 1), dp(x1, y1 + 1, x2 + 1)),
       Math.max(dp(x1, y1 + 1, x2), dp(x1 + 1, y1, x2))
      );

      ans += grid[x1][y1];
      if (x1 != x2)     ans += grid[x2][y2];
      dp[x1][y1][x2] = ans;
      return ans;
    }
  }
}