class Solution {
    private Map<String, Integer> memo = new HashMap<>();
    
    public int minKnightMoves(int x, int y) {
        return dfs(Math.abs(x), Math.abs(y));
    }
    
    private int dfs(int x, int y) {
        String memoKey = x + "|" + y;
        if (memo.containsKey(memoKey)) {
            return memo.get(memoKey);
        }
        
        if (x + y == 0) {
            // Base case 1: Because x, y is positive, so when x + y == 0, both x and y is 0
            return 0;
        } else if (x + y == 2) {
            // Base case 2: If x + y == 2, Knight can reach destination in 2 steps
            return 2;
        } else {
            int ans = 1 +  Math.min(dfs(Math.abs(x - 1), Math.abs(y - 2)),
                                    dfs(Math.abs(x - 2), Math.abs(y - 1)));
            memo.put(memoKey, ans);
            return ans;
        }
    }
}