1510. Stone Game IV

1510. Stone Game IV (Hard)

Alice and Bob take turns playing a game, with Alice starting first.

Initially, there are n stones in a pile.  On each player's turn, that player makes a move consisting of removing any non-zero square number of stones in the pile.

Also, if a player cannot make a move, he/she loses the game.

Given a positive integer n. Return True if and only if Alice wins the game otherwise return False, assuming both players play optimally.

 

Example 1:

Input: n = 1
Output: true
Explanation: Alice can remove 1 stone winning the game because Bob doesn't have any moves.

Example 2:

Input: n = 2
Output: false
Explanation: Alice can only remove 1 stone, after that Bob removes the last one winning the game (2 -> 1 -> 0).

Example 3:

Input: n = 4
Output: true
Explanation: n is already a perfect square, Alice can win with one move, removing 4 stones (4 -> 0).

Example 4:

Input: n = 7
Output: false
Explanation: Alice can't win the game if Bob plays optimally.
If Alice starts removing 4 stones, Bob will remove 1 stone then Alice should remove only 1 stone and finally Bob removes the last one (7 -> 3 -> 2 -> 1 -> 0). 
If Alice starts removing 1 stone, Bob will remove 4 stones then Alice only can remove 1 stone and finally Bob removes the last one (7 -> 6 -> 2 -> 1 -> 0).

Example 5:

Input: n = 17
Output: false
Explanation: Alice can't win the game if Bob plays optimally.

 

Constraints:

• 1 <= n <= 10^5

Related Topics:
Dynamic Programming

Solution 1. Bottom-up DP

Let dp[i] be whether Alice and win starting with i stones.

dp[i] = true    // If any dp[i - j * j] is false
      = false   // otherwise
      where 1 <= j * j <= i  -- the stone taken by Bob.

// OJ: https://leetcode.com/problems/stone-game-iv/
// Author: github.com/lzl124631x
// Time: O(N * sqrt(N))
// Space: O(N)
class Solution {
public:
    bool winnerSquareGame(int n) {
        vector<bool> dp(n + 1);
        for (int i = 1; i <= n; ++i) {
            for (int j = 1; j * j <= i && !dp[i]; ++j) dp[i] = !dp[i - j * j];
        }
        return dp[n];
    }
};

Or use static variable to save computation

// OJ: https://leetcode.com/problems/stone-game-iv/
// Author: github.com/lzl124631x
// Time: O(N * sqrt(N))
// Space: O(N)
class Solution {
public:
    bool winnerSquareGame(int n) {
        static vector<bool> dp(1);
        for (int i = dp.size(); i <= n; ++i) {
            dp.push_back(false);
            for (int j = 1; j * j <= i && !dp[i]; ++j) dp[i] = !dp[i - j * j];
        }
        return dp[n];
    }
};

Solution 2. Top-down DP

// OJ: https://leetcode.com/problems/stone-game-iv/
// Author: github.com/lzl124631x
// Time: O(N * sqrt(N))
// Space: O(N)
class Solution {
    vector<int> dp; // -1 unvisited, 0 lose, 1 win
    bool dfs(int n) {
        if (n == 0) return false;
        if (dp[n] != -1) return dp[n];
        for (int i = 1; i * i <= n && dp[n] != 1; ++i) dp[n] = !dfs(n - i * i);
        return dp[n];
    }
public:
    bool winnerSquareGame(int n) {
        dp.assign(n + 1, -1);
        return dfs(n);
    }
};