Number of Valid Move Combinations On Chessboard — C++

// Time:  O(n^p) = O(1), n is the max number of possible moves for each piece, and n is at most 29
//                     , p is the number of pieces, and p is at most 4
// Space: O(1)

class Solution {
public:
    int countCombinations(vector<string>& pieces, vector<vector<int>>& positions) {
        vector<vector<int>> lookup(8, vector<int>(8));
        return backtracking(pieces, positions, 0, &lookup);
    }

private:
    int backtracking(const vector<string>& pieces, const vector<vector<int>>& positions, int i, vector<vector<int>> *lookup) {
        static const unordered_map<string, vector<pair<int, int>>> directions = {
            {"rook", {{0, 1}, {1, 0}, {0, -1}, {-1, 0}}},
            {"bishop", {{1, 1}, {1, -1}, {-1, 1}, {-1, -1}}},
            {"queen", {{0, 1}, {1, 0}, {0, -1}, {-1, 0}, {1, 1}, {1, -1}, {-1, 1}, {-1, -1}}}
        };
        static const int all_mask = (1 << 7) - 1;  // at most 7 seconds in 8x8 board

        if (i == size(pieces)) {
            return 1;
        }
        int result = 0;
        int r = positions[i][0] - 1, c = positions[i][1] - 1;
        int mask = all_mask;
        if (!((*lookup)[r][c] & mask)) {
            (*lookup)[r][c] += mask;  // stopped at (r, c)
            result += backtracking(pieces, positions, i + 1, lookup);
            (*lookup)[r][c] -= mask;
        }
        for (const auto& [dr, dc] : directions.at(pieces[i])) {
            int bit = 1, nr = r + dr, nc = c + dc;
            int mask = all_mask;  // (mask&bit == 1): (log2(bit)+1)th second is occupied
            for (; 0 <= nr && nr < 8 && 0 <= nc && nc < 8 && !((*lookup)[nr][nc] & bit); bit <<= 1, nr += dr, nc += dc) {
                (*lookup)[nr][nc] += bit;
                mask -= bit;
                if (!((*lookup)[nr][nc] & mask)) {  // stopped at (nr, nc)
                    (*lookup)[nr][nc] += mask;
                    result += backtracking(pieces, positions, i + 1, lookup);
                    (*lookup)[nr][nc] -= mask;
                }
            }
            while (bit >> 1) {
                bit >>= 1, nr -= dr, nc -= dc;
                (*lookup)[nr][nc] -= bit;
            }
        }
        return result;
    }
};