Compute Minimum L-Moves on Infinite Grid

On an infinite 2D grid, a chess-like piece moves in L-shapes: from (x, y) it can go to (x±1, y± 2) or (x±2, y± 1). Given integer target coordinates (tx, ty) with |tx|, |ty| ≤ 1e9, implement a function minLMoves(tx, ty) that returns the minimum number of moves needed to reach (tx, ty) starting from (0, 0). Exploit symmetry to optimize your solution and handle edge cases carefully. Explain your algorithm, prove correctness intuitively, and analyze time and space complexity.

What a Strong Answer Covers

• A clear algorithm with the right data structures and enough pseudocode or code-level detail to implement it.
• A correctness argument that explains why the algorithm covers all required cases.
• Time and space complexity, plus at least one alternative approach when relevant.
• Focused tests for normal cases, edge cases, and failure modes.

Follow-up Questions

• How would the approach change if the input were streaming or too large for memory?
• What invariants would you assert in production code?
• Which tests would catch off-by-one, duplicate, or tie-breaking bugs?

Minimum L-Moves on an Infinite Grid

On an infinite 2D grid, a chess-knight-like piece moves in L-shapes: from (x, y) it can jump to (x±1, y±2) or (x±2, y±1) — eight possible moves in total. Starting from the origin (0, 0), return the minimum number of moves needed to reach the target cell (tx, ty).

The target may have very large coordinates (|tx|, |ty| up to 1e9), so a plain breadth-first search over the grid is far too slow and memory-hungry. You must exploit the problem's symmetry and derive a closed-form (O(1)) answer.

Implement minLMoves(tx, ty) returning that minimum move count.

Symmetry to exploit:

• The eight L-moves are symmetric across both axes, so the answer depends only on (|tx|, |ty|). Reduce to the first quadrant by taking absolute values.
• The answer is also symmetric in its two arguments, so you may further assume x ≥ y.

Key idea: after reducing to x ≥ y ≥ 0, let delta = x - y. Each move changes the coordinate sum's parity and lets you 'spend' the gap efficiently. Two small corner cases, (1, 0) and (2, 2), break the general pattern and must be special-cased (their true answers are 3 and 4, not what the bulk formula yields).

Constraints

• -1e9 <= tx, ty <= 1e9
• The piece starts at the origin (0, 0).
• Eight legal moves from (x, y): (x±1, y±2) and (x±2, y±1).
• A move count always exists (the grid is infinite and the piece can reach any cell).
• A BFS solution is correct in principle but will TLE / run out of memory for |tx|, |ty| near 1e9 — an O(1) closed form is required.

Examples

Example 1

Input: (0, 0)
Output: 0

Explanation: Already at the origin; no moves needed.

Example 2

Input: (1, 1)
Output: 2

Explanation: (0,0) -> (2,-1) -> (1,1): two L-moves reach the near-diagonal cell.

Hints

1. The eight moves are symmetric across both the x- and y-axes, so the answer is unchanged by the signs of tx and ty. Replace them with their absolute values immediately.
2. The move set is also symmetric in swapping the two coordinates, so without loss of generality assume x >= y after taking absolute values.
3. Brute-force BFS on small targets (say all x,y in 0..12) to build a ground-truth table, then look for a pattern in terms of delta = x - y and y.
4. Two cells refuse to fit the bulk pattern: (1, 0) needs 3 moves and (2, 2) needs 4. Special-case them before applying the general formula.
5. Branch on whether y exceeds delta = x - y: when it does you correct toward the diagonal in chunks of 3, otherwise you advance along one axis in chunks of 4 — each correction adds 2 'extra' moves.

Solution

Solution Alignment

The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests.

Problem Restatement

On an infinite 2D grid, a chess-like piece moves in L-shapes: from (x, y) it can go to (x±1, y± 2) or (x±2, y± 1). Given integer target coordinates (tx, ty) with |tx|, |ty| ≤ 1e9, implement a function minLMoves(tx, ty) that returns the minimum number of moves needed to reach (tx, ty) starting from (0, 0). Exploit symmetry to optimize your solution and handle edge cases carefully. Explain your algorithm, prove correctness intuitively, and analyze time and space complexity.

Recommended Approach

Represent each cell as a state. Use BFS for minimum-distance propagation, DFS with memoization for longest monotonic paths, and careful boundary checks for simulation. Store obstacles or visited cells in sets when the grid is sparse.

Correctness

The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call.

Complexity

A full grid traversal is O(mn) time and O(mn) space in the worst case. Sparse simulation is O(number_of_commands) plus obstacle storage.

Edge Cases and Tests

Empty grid, one row/column, blocked start or target, boundaries, repeated visits, and tie-breaking among equal-distance cells.