#1728 - Cat and Mouse II

Hard • View on LeetCode • View on GitHub

Array Dynamic Programming Math Matrix Graph Memoization Topological Sort Game Theory

Problem Description

A game is played by a cat and a mouse named Cat and Mouse.

The environment is represented by a grid of size rows x cols, where each element is a wall, floor, player (Cat, Mouse), or food.

Players are represented by the characters 'C'(Cat),'M'(Mouse).
Floors are represented by the character '.' and can be walked on.
Walls are represented by the character '#' and cannot be walked on.
Food is represented by the character 'F' and can be walked on.
There is only one of each character 'C', 'M', and 'F' in grid.

Mouse and Cat play according to the following rules:

Mouse moves first, then they take turns to move.
During each turn, Cat and Mouse can jump in one of the four directions (left, right, up, down). They cannot jump over the wall nor outside of the grid.
catJump, mouseJump are the maximum lengths Cat and Mouse can jump at a time, respectively. Cat and Mouse can jump less than the maximum length.
Staying in the same position is allowed.
Mouse can jump over Cat.

The game can end in 4 ways:

If Cat occupies the same position as Mouse, Cat wins.
If Cat reaches the food first, Cat wins.
If Mouse reaches the food first, Mouse wins.
If Mouse cannot get to the food within 1000 turns, Cat wins.

Given a rows x cols matrix grid and two integers catJump and mouseJump, return true if Mouse can win the game if both Cat and Mouse play optimally, otherwise return false.

Solution

/**
 * @param {string[]} grid
 * @param {number} catJump
 * @param {number} mouseJump
 * @return {boolean}
 */
var canMouseWin = function(grid, catJump, mouseJump) {
  if (typeof grid === 'string') grid = [grid];
  const rows = grid.length;
  const cols = grid[0].length;
  const dirs = [[1, 0], [-1, 0], [0, 1], [0, -1]];
  let mouse;
  let cat;
  let food;
  let available = 0;

  for (let i = 0; i < rows; i++) {
    for (let j = 0; j < cols; j++) {
      const cell = grid[i][j];
      if (cell !== '#') available++;
      if (cell === 'M') mouse = [i, j];
      else if (cell === 'C') cat = [i, j];
      else if (cell === 'F') food = [i, j];
    }
  }

  const memo = new Map();

  return canWin(0, mouse, cat);

  function getKey(turn, [mr, mc], [cr, cc]) {
    return `${turn},${mr},${mc},${cr},${cc}`;
  }

  function isValid(r, c) {
    return r >= 0 && r < rows && c >= 0 && c < cols && grid[r][c] !== '#';
  }

  function canWin(turn, mousePos, catPos) {
    if (turn >= available * 2) return false;

    const key = getKey(turn, mousePos, catPos);
    if (memo.has(key)) return memo.get(key);

    const isMouseTurn = turn % 2 === 0;
    const [r, c] = isMouseTurn ? mousePos : catPos;
    const maxJump = isMouseTurn ? mouseJump : catJump;

    for (const [dr, dc] of dirs) {
      for (let jump = 0; jump <= maxJump; jump++) {
        const nr = r + dr * jump;
        const nc = c + dc * jump;
        if (!isValid(nr, nc)) break;

        if (isMouseTurn) {
          if (grid[nr][nc] === 'F' || canWin(turn + 1, [nr, nc], catPos)) {
            memo.set(key, true);
            return true;
          }
        } else {
          if (grid[nr][nc] === 'F' || (nr === mousePos[0] && nc === mousePos[1])
              || !canWin(turn + 1, mousePos, [nr, nc])) {
            memo.set(key, false);
            return false;
          }
        }
      }
    }

    memo.set(key, !isMouseTurn);
    return !isMouseTurn;
  }
};