1197. Minimum Knight Moves - Detailed Explanation

Problem Statement

Description:
On an infinite chessboard, a knight starts at the origin (0, 0) and moves according to the standard knight moves in chess (an “L” shape). Given target coordinates (x, y), return the minimum number of moves required for the knight to reach (x, y) from (0, 0).

A knight moves in an “L” shape: two squares in one direction and one square perpendicular to that direction. For example, from (0, 0), possible moves include (2, 1), (1, 2), (-1, 2), (-2, 1), and their symmetric counterparts.

Examples:

• Example 1:

  • Input: x = 2, y = 1
  • Output: 1
  • Explanation:
    The knight can move directly from (0, 0) to (2, 1) in one move.

• Example 2:

  • Input: x = 5, y = 5
  • Output: 4
  • Explanation:
    One possible sequence of moves is:
    (0, 0) → (2, 1) → (4, 2) → (3, 4) → (5, 5)
    Total moves = 4.

• Example 3:

  • Input: x = 0, y = 0
  • Output: 0
  • Explanation:
    The knight is already at the target.

Constraints:

• (-300 \le x, y \le 300)
• The answer is guaranteed to exist.

Hints

• Hint 1:
  Due to symmetry, note that the minimum moves to reach (x, y) is the same as reaching (abs(x), abs(y)). You can work in the first quadrant (where both coordinates are non-negative) to simplify your solution.

• Hint 2:
  A straightforward approach is to perform a Breadth-First Search (BFS) starting from (0, 0) while keeping track of visited positions.

• Hint 3:
  There exists a mathematical observation that helps compute the answer in O(1) time by handling a couple of special cases and then using a formula based on the relative values of x and y. Explore both the BFS simulation and the mathematical approach.

Approaches

BFS (Breadth-First Search) Approach

Idea:

• Simulation:
  Start at (0, 0) and use BFS to explore all reachable positions. Each move follows one of the eight possible knight moves.
• Symmetry and Pruning:
  Convert the target to (abs(x), abs(y)) and restrict your search to the first quadrant. Additionally, you can limit the search space by recognizing that if the knight has overshot far beyond the target, those moves are unlikely to lead to an optimal solution.
• Visited Set:
  Keep track of visited coordinates to avoid repeated work.

Steps:

1. Initialize a queue with the starting position (0, 0) and move count 0.

2. Loop:
   For each position, generate all 8 possible moves.

3. Check:
   If a move reaches (abs(x), abs(y)), return the move count + 1.

4. Boundaries:
   Since the board is infinite, use symmetry and reasonable bounds (e.g., positions where both coordinates are not “too negative”) to limit the search.

5. Continue until the target is found.

Time Complexity:

In the worst case, BFS may explore many nodes, but symmetry and bounds reduce the practical search space.

Space Complexity:

O(n) for the queue and visited set (n being the number of positions visited).

Mathematical / Greedy Approach

Idea:

• Observation:
  Because of symmetry and the “L” shaped movement, the problem has a closed-form solution for many cases.
• Steps:

  1. Convert (x, y) to (a, b) where a = abs(x) and b = abs(y), and assume a >= b.

  2. Handle special cases:

     • If (a, b) == (1, 0) → answer is 3 moves.
     • If (a, b) == (2, 2) → answer is 4 moves.

  3. Otherwise, use the formula:
     [ \text{moves} = \max\left(\left\lceil \frac{a+1}{2} \right\rceil, \left\lceil \frac{a+b+2}{3} \right\rceil\right) ] Then adjust for parity—if the parity of the sum of moves and (a + b) differ, increment the moves until they match.

• Benefits:
  The mathematical solution runs in O(1) time.

Note:

While the mathematical approach is very efficient, it requires careful handling of corner cases.

Code Implementations

Python Code (Using BFS Approach)