[C++] DP solution


  • 0
    C

    A few people have posted very nice solutions using DP. Here is another one.
    One feature is that I expand the grid by one cell in all four directions, by assigning the number of paths using 0 times of moves to be 1.

    class Solution {
    public:
        int findPaths(int m, int n, int N, int i, int j) {
            if (N == 0) return 0;
            long mod = 1000000007;
            vector<vector<vector<long>>> dp (m + 2, vector<vector<long>> (n + 2, vector<long> (N + 1, 0)));    // dp[k][l][p] number of paths for starting at (k, l) with p times
            long result = 0;
            for (int k = 1; k <= m; k ++) {
                dp[k][0][0] = 1;
                dp[k][n + 1][0] = 1;
            }
            for (int l = 1; l <= n; l++) {
                dp[0][l][0] = 1;
                dp[m + 1][l][0] = 1;
            }
            
            for (int p = 1; p <= N; p++) {
                for (int layer = 0; layer < min(m, n) / 2.0; layer ++) {
                    for (int k = layer + 1; k <= m - layer; k ++) {
                        for (int l = layer + 1; l <= n -layer; l++) {
                            dp[k][l][p] = dp[k - 1][l][p - 1] + dp[k + 1][l][p - 1] + dp[k][l - 1][p - 1] + dp[k][l + 1][p - 1];
                            dp[k][l][p] %= mod;
                        }
                    }
                }
            }
            for (int u = 1; u <= N; u++) {
                result += dp[i + 1][j + 1][u];
                result %= mod;
            }
            return result % mod;
        }
    };
    

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