Check if Point is Reachable (Leetcode) Added problem number 233

dsa-solutions/lc-solutions/2500 - 3000/2543 - Check if Point is Reachable.md

@@ -0,0 +1,121 @@
+---
+id: check-if-point-is-reachable
+title: Check if Point is Reachable
+sidebar_label: 2543 Check if Point is Reachable
+tags:
+- Number Theory
+- Java
+- gcd
+- Math
+description: "This document provides a solution where we need to reach the point (targetX, targetY) using a finite number of steps."
+---
+
+## Problem
+
+There exists an infinitely large grid. You are currently at point $(1, 1)$, and you need to reach the point $(targetX, targetY)$ using a finite number of steps.
+
+In one Step, you can move from point $(x, y)$ to any one of the following points:
+
+- $(x, y - x)$
+
+- $(x - y, y)$
+
+- $(2 * x, y)$
+
+- $(x, 2 * y)$
+
+Given two integers $targetX$ and $targetY$ representing the X-coordinate and Y-coordinate of your final position, return $true$ if you can reach the point from $(1, 1)$ using some number of steps, and $false$ otherwise.
+
+### Examples
+
+**Example 1:**
+
+```
+Input: targetX = 6, targetY = 9
+
+Output: false
+
+Explanation: It is impossible to reach (6,9) from (1,1) using any sequence of moves, so false is returned.
+
+```
+
+**Example 2:**
+
+```
+Input: targetX = 4, targetY = 7
+
+Output: true
+
+Explanation: You can follow the path (1,1) -> (1,2) -> (1,4) -> (1,8) -> (1,7) -> (2,7) -> (4,7).
+
+```
+
+### Constraints
+
+- `1 <= targetX, targetY <= 10^9`
+
+---
+
+## Approach
+
+To solve the problem, we need to understand the nature of the allowed moves:
+
+1. $(x, y - x)$
+
+2. $(x - y, y)$
+
+3. $(2 * x, y)$
+
+4. $(x, 2 * y)$
+
+These moves suggest that if we can reach a certain point **'(x, y)'**, we can generate other points by specific transformations. By analyzing the moves:
+
+  - The first two moves involve subtraction, reducing one coordinate.
+
+  - The last two moves involve multiplication by $2$.
+
+We can reverse the logic to check whether we can reach $(1, 1)$ from $(targetX, targetY)$. This reversal involves:
+
+ - Checking if $targetX$ or $targetY$ can be divided by $2$ (to reverse the multiplication).
+
+ - Checking if one coordinate can be reduced by subtracting the other.
+
+By reversing the operations, we trace the problem back to whether $(targetX, targetY)$ can be reduced to $(1, 1)$.
+
+The key insight here is that we only need to check if **'gcd(targetX, targetY)'** is a power of $2$. This is because if both $targetX$ and $targetY$ share a common factor other than $2$, it won't be possible to reach $(1, 1)$.
+
+
+## Solution for Check if Point is Reachable
+
+The given problem involves reaching a target point $(targetX, targetY)$ from the starting point $(1, 1)$ using a set of allowed moves. The challenge is to determine whether it's possible to reach the target using these moves.
+
+#### Code in Java
+
+ ```java
+class Solution {
+  public boolean isReachable(int targetX, int targetY) {
+    return Integer.bitCount(gcd(targetX, targetY)) == 1;
+  }
+
+  private int gcd(int a, int b) {
+    return b == 0 ? a : gcd(b, a % b);
+  }
+}
+
+```
+
+### Complexity Analysis
+
+#### Time Complexity: $O(log(min(targetX, targetY)))$
+
+> **Reason**: Calculating the gcd of two numbers takes $O(log(min(targetX, targetY)))$ time using the Euclidean algorithm.
+
+#### Space Complexity: $O(1)$
+
+> **Reason**: The space complexity is $O(1)$ since we are using a constant amount of extra space for computation (ignoring the space used by the recursive stack of the gcd function, which is logarithmic in terms of the depth of the recursion).
+
+# References
+
+- **LeetCode Problem:** [Check if Point is Reachable](https://leetcode.com/problems/check-if-point-is-reachable/description/)
+- **Solution Link:** [Check if Point is Reachable Solution on LeetCode](https://leetcode.com/problems/check-if-point-is-reachable/solutions/)
+- **Authors LeetCode Profile:** [Vivek Vardhan](https://leetcode.com/u/vivekvardhan43862/)