Minimum Moves to Reach Target in Grid

In this problem, the goal is to calculate the fewest moves required to reach a target point from a starting point on a grid. The solution focuses on the mathematical relationships between the coordinates, and by working backward from the target, you can optimize the number of moves needed.

Problem Statement

You are given two points (sx, sy) and (tx, ty) on an infinitely large 2D grid. The task is to find the minimum number of moves to reach from the starting point (sx, sy) to the target point (tx, ty). You can move by either moving horizontally or vertically, but each move involves changing both coordinates, where m = max(x, y).

To solve this, consider working backward from the target (tx, ty) to the start (sx, sy), undoing one move at each step. The problem presents a challenging math-driven strategy to minimize the total number of moves required.

Examples

Example 1

Input: sx = 1, sy = 2, tx = 5, ty = 4

Output: 2

The optimal path is: Thus, the minimum number of moves to reach (5, 4) is 2.

Example 2

Input: sx = 0, sy = 1, tx = 2, ty = 3

Output: 3

The optimal path is: Thus, the minimum number of moves to reach (2, 3) is 3.

Example 3

Input: sx = 1, sy = 1, tx = 2, ty = 2

Output: -1

Constraints

• 0 <= sx <= tx <= 109
• 0 <= sy <= ty <= 109

Solution Approach

Working Backwards

Start by reversing the move process from the target point (tx, ty) to the start point (sx, sy). Each time you reverse a move, you reduce the larger coordinate until both are closer to the starting coordinates. This method reduces the complexity of exploring all possible moves.

Maximizing the Distance in Each Step

When working backward, in each step, the larger of the two coordinates (tx, ty) must be reduced by either subtracting from it or taking advantage of their difference to undo one of the moves, thus maximizing progress in each iteration.

Identifying Impossible Cases

If at any point the target cannot be reduced to the starting position using this method, the problem indicates that it is impossible to reach the target from the start. Specifically, situations where sx, sy cannot be reached from tx, ty by this strategy.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the approach chosen for solving the problem. Working backward with coordinate reduction typically leads to a logarithmic time complexity, as the process reduces one coordinate at each step.

What Interviewers Usually Probe

• Candidate effectively demonstrates an understanding of working backward through coordinates.
• Candidate identifies situations where the solution is impossible.
• Candidate uses the max(x, y) relationship accurately to optimize the solution.

Common Pitfalls or Variants

Common pitfalls

• Failing to recognize impossible scenarios where coordinates cannot be reduced.
• Incorrectly implementing the working-backward strategy by missing necessary coordinate adjustments.
• Not fully understanding the significance of maximizing the distance reduction in each move.

Follow-up variants

• Handling variations where the starting point is beyond the target point.
• Optimizing for large grid sizes with constraints.
• Adapting the strategy when there are multiple ways to move toward the target.

FAQ

What is the minimum number of moves to reach the target in the Minimum Moves to Reach Target in Grid problem?

The minimum number of moves is calculated by working backward from the target point, reducing the larger coordinate in each step.

How can I approach the Minimum Moves to Reach Target in Grid problem effectively?

Use a math-driven strategy by working backwards from the target point, considering the relationship between the coordinates and maximizing distance reductions.

What are some common mistakes in solving the Minimum Moves to Reach Target in Grid problem?

Common mistakes include failing to detect impossible cases or misunderstanding the significance of the max(x, y) relationship in moves.

What happens if it's impossible to reach the target in the Minimum Moves to Reach Target in Grid problem?

In such cases, the solution will return -1 to indicate that reaching the target is not possible based on the rules of movement.

What is the time complexity of the solution for the Minimum Moves to Reach Target in Grid problem?

The time complexity typically follows a logarithmic pattern, as each move reduces one coordinate towards the starting point.