Minimum Knight Moves — C++

// Time:  O(1)
// Space: O(1)

class Solution {
private:
    template <typename T>
    struct PairHash {
        size_t operator()(const pair<T, T>& p) const {
            size_t seed = 0;
            seed ^= std::hash<T>{}(p.first)  + 0x9e3779b9 + (seed<<6) + (seed>>2);
            seed ^= std::hash<T>{}(p.second) + 0x9e3779b9 + (seed<<6) + (seed>>2);
            return seed;
        }
    };

public:
    int minKnightMoves(int x, int y) {
        // we can observe from:
        // [0]
        // [3, 2]
        // [2,(1),4]
        // [3, 2, 3, 2]
        // [2, 3,(2) 3, 4]
        // [3, 4, 3, 4, 3, 4]
        // [4, 3, 4,(3),4, 5, 4]
        // [5, 4, 5, 4, 5, 4, 5, 6]
        // [4, 5, 4, 5,(4),5, 6, 5, 6]
        // [5, 6, 5, 6, 5, 6, 5, 6, 7, 6]
        // [6, 5, 6, 5, 6,(5),6, 7, 6, 7, 8]
        // [7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8]
        // [6, 7, 6, 7, 6, 7,(6),7, 8, 7, 8, 9, 8]
        // [7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 10]
        // [8, 7, 8, 7, 8, 7, 8,(7),8, 9, 8, 9, 10, 9, 10]
        // [9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 11, 10]

        x = abs(x), y = abs(y);
        if (x < y) {
            swap(x, y);
        }
        static const unordered_map<pair<int, int>, int, PairHash<int>> lookup =
            {{{1, 0}, 3}, {{2, 2}, 4}};  // special cases
        if (lookup.count({x, y})) {
            return lookup.at({x, y});
        }
        const auto& k = x - y;
        if (y > k) {
            // if 2y > x, every period 3 of y (or k) with fixed another is increased by 2 (or 1)
            // and start from (2k, k) with (k) when y = k (diagonal line)
            // ex. (0, 0) ~ (12, 12) ~ ... : 0 => 2,4(special case),2 => 4,4,4 => 6,6,6 => 8,8,8 => ...
            // ex. (2, 1) ~ (14, 13) ~ ... : 1 => 3,3,3 => 5,5,5 => 7,7,7 => 9,9,9 => ...
            return k + 2 * ((y - k - 1) / 3 + 1);
        }
        // if 2y <= x, every period 4 of k (or y) with fixed another is increased by 2
        // and start from (2k, k) with (k) when y = k (vertical line)
        // ex. (0, 0) ~ (11, 0) ~ ... : 0,3(special case),2,3 => 2,3,4,5 => 4,5,6,7 => ...
        // ex. (2, 1) ~ (13, 1) ~ ... : 1,2,3,4 => 3,4,5,6 => 5,6,7,8 => ...
        return k - 2 * ((k - y) / 4);
    }
};

// Time:  O(n^2)
// Space: O(n^2)
class Solution2 {
public:
    int minKnightMoves(int x, int y) {
        return dp(x, y);
    }

private:
    template <typename T>
    struct PairHash {
        size_t operator()(const pair<T, T>& p) const {
            size_t seed = 0;
            seed ^= std::hash<T>{}(p.first)  + 0x9e3779b9 + (seed<<6) + (seed>>2);
            seed ^= std::hash<T>{}(p.second) + 0x9e3779b9 + (seed<<6) + (seed>>2);
            return seed;
        }
    };

    int dp(int x, int y) {
        static unordered_map<pair<int, int>, int, PairHash<int>> lookup =
            {{{0, 0}, 0}, {{1, 1}, 2}, {{1, 0}, 3}};  // special cases
        x = abs(x), y = abs(y);
        if (x < y) {
            swap(x, y);
        }
        if (!lookup.count({x, y})) {  // greedy, smaller x, y is always better if not special cases
            lookup[{x, y}] = min(dp(x - 1, y - 2), dp(x - 2, y - 1)) + 1;
        }
        return lookup[{x, y}];
    }
};