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).
Example 1:
Input: sx = 1, sy = 1, tx = 3, ty = 5 Output: true Explanation: One series of moves that transforms the starting point to the target is: (1, 1) -> (1, 2) (1, 2) -> (3, 2) (3, 2) -> (3, 5)
Example 2:
Input: sx = 1, sy = 1, tx = 2, ty = 2 Output: false
Example 3:
Input: sx = 1, sy = 1, tx = 1, ty = 1 Output: true
Constraints:
1 <= sx, sy, tx, ty <= 109In #780 Reaching Points, you are given a starting point (sx, sy) and a target point (tx, ty). From any point, you can transform it to (x + y, y) or (x, x + y). Instead of simulating forward moves, which can grow exponentially, the key insight is to work backwards from the target.
By reversing the operations, you reduce the larger coordinate using the smaller one. This resembles the logic of the Euclidean algorithm. Instead of subtracting repeatedly, you can apply a modulo operation to jump multiple steps at once, significantly improving efficiency.
While reducing the target coordinates, ensure they do not fall below the starting coordinates. Special handling is needed when one coordinate matches the starting value, as the remaining difference must be divisible by the other coordinate.
This reverse greedy approach avoids brute force exploration and runs in logarithmic time relative to the target values, making it suitable for very large inputs.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Reverse Greedy with Modulo (Euclidean-style reduction) | O(log(max(tx, ty))) | O(1) |
NeetCode
The goal could be efficiently approached by reversing the operations. Instead of trying to reach (tx, ty) from (sx, sy), we attempt to determine whether (sx, sy) can be reached from (tx, ty) using reverse operations. This is achieved by repeatedly applying the inverse operations: (x, x+y) can be reverted to (x, y), and (x+y, y) can be reverted to (x, y). The main idea is to use modulus operations when x != y.
Time Complexity: O(log(max(tx, ty)))
Space Complexity: O(1)
1#include <stdbool.h>
2bool reachingPoints(int sx, int sy, int tx, int ty) {
3 while (tx > sx && ty > sy &&This solution uses a loop to apply the reverse operations until tx or ty is reduced to sx or sy. It checks if further reduction is possible using modulus operations, ensuring the feasibility of reaching (sx, sy) based on the remainder. The logic ensures that one either complements the other by reducing one coordinate using the other. This approach is efficient due to its use of modulus operation, providing a quick reduction of the problem space.
Another strategy involves recursive backtracking, where the function makes recursive calls to simulate both directions (x + y, y) and (x, x + y) to reach the target point (tx, ty) from the start point (sx, sy). Although less efficient compared to the reverse operation method due to its depth, it's an introductory way to explore the possibilities.
Time Complexity: Exponential in the worst case
Space Complexity: Recursive stack size
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Yes, problems like Reaching Points are common in FAANG-style interviews because they test mathematical intuition and optimization. Candidates are expected to recognize patterns and convert brute force approaches into efficient ones.
The optimal approach works backwards from the target point instead of simulating forward moves. By repeatedly reducing the larger coordinate using modulo with the smaller one, the process mimics the Euclidean algorithm and avoids exponential growth.
Using modulo allows us to skip multiple subtraction steps at once when reducing the larger coordinate. This optimization dramatically improves efficiency and ensures the algorithm runs in logarithmic time.
The problem relies on mathematical reasoning, greedy reduction, and understanding the Euclidean algorithm. Recognizing that reversing the operations is more efficient than forward simulation is the key insight.
The backtracking approach uses recursion to simulate both possible moves at each step, verifying if any combination of moves can transition the start point into the target point. However, this implementation will face efficiency issues with larger inputs due to recursive depth and repeated evaluations.