Reaching Points

Problem Description

Given four integers sx, sy, tx, and ty, return true if it is possible to convert the point (sx, sy) to the point (tx, ty) through some operations, or false otherwise. The allowed operation on some point (x, y) is to convert it to either (x, x + y) or (x + y, y).

Key Insights

• The operations allowed can be thought of as adding the coordinates together, which leads to larger values over time.
• The problem can be reversed: instead of trying to reach (tx, ty) from (sx, sy), we can check if we can reduce (tx, ty) back to (sx, sy) using the reverse operations.
• The reverse operations are:
  • If tx > ty, then tx can be reduced to tx - ty until tx is less than or equal to ty.
  • If ty > tx, then ty can be reduced to ty - tx until ty is less than or equal to tx.
• The process continues until tx and ty are equal to or less than sx and sy.
• The final check is whether tx and ty can both equal sx and sy by considering the modulo operation.

Space and Time Complexity

Time Complexity: O(log(max(tx, ty)))
Space Complexity: O(1)

Solution

To determine if it is possible to reach the target point (tx, ty) from the starting point (sx, sy), we can reverse the operations. We repeatedly apply the reverse operations until we either reach or overshoot the starting point. The algorithm effectively checks if we can adjust tx and ty through subtraction to match sx and sy. If at any point tx or ty becomes less than sx or sy and they are not equal, we return false.

Code Solutions