LeetCode-in-Java
2709. Greatest Common Divisor Traversal
Hard
You are given a 0-indexed integer array nums, and you are allowed to traverse between its indices. You can traverse between index i and index j, i != j, if and only if gcd(nums[i], nums[j]) > 1, where gcd is the greatest common divisor.
Your task is to determine if for every pair of indices i and j in nums, where i < j, there exists a sequence of traversals that can take us from i to j.
Return true if it is possible to traverse between all such pairs of indices, or false otherwise.
Example 1:
Input: nums = [2,3,6]
Output: true
Explanation: In this example, there are 3 possible pairs of indices: (0, 1), (0, 2), and (1, 2). To go from index 0 to index 1, we can use the sequence of traversals 0 -> 2 -> 1, where we move from index 0 to index 2 because gcd(nums[0], nums[2]) = gcd(2, 6) = 2 > 1, and then move from index 2 to index 1 because gcd(nums[2], nums[1]) = gcd(6, 3) = 3 > 1. To go from index 0 to index 2, we can just go directly because gcd(nums[0], nums[2]) = gcd(2, 6) = 2 > 1. Likewise, to go from index 1 to index 2, we can just go directly because gcd(nums[1], nums[2]) = gcd(3, 6) = 3 > 1.
Example 2:
Input: nums = [3,9,5]
Output: false
Explanation: No sequence of traversals can take us from index 0 to index 2 in this example. So, we return false.
Example 3:
Input: nums = [4,3,12,8]
Output: true
Explanation: There are 6 possible pairs of indices to traverse between: (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), and (2, 3). A valid sequence of traversals exists for each pair, so we return true.
Constraints:
1 <= nums.length <= 1051 <= nums[i] <= 105
Solution
import java.util.HashMap;
import java.util.Map;
public class Solution {
private Map<Integer, Integer> map = null;
private int[] set;
private int findParent(int u) {
if (u == set[u]) {
return u;
}
set[u] = findParent(set[u]);
return set[u];
}
private void union(int a, int b) {
int p1 = findParent(a);
int p2 = findParent(b);
if (p1 != p2) {
set[b] = p1;
}
set[p2] = p1;
}
private void solve(int n, int index) {
if (n % 2 == 0) {
int x = map.getOrDefault(2, -1);
if (x != -1) {
union(x, index);
}
while (n % 2 == 0) {
n /= 2;
}
map.put(2, index);
}
int sqrt = (int) Math.sqrt(n);
for (int i = 3; i <= sqrt; i++) {
if (n % i == 0) {
int x = map.getOrDefault(i, -1);
if (x != -1) {
union(x, index);
}
while (n % i == 0) {
n /= i;
}
map.put(i, index);
}
}
if (n > 2) {
int x = map.getOrDefault(n, -1);
if (x != -1) {
union(x, index);
}
map.put(n, index);
}
}
public boolean canTraverseAllPairs(int[] nums) {
set = new int[nums.length];
map = new HashMap<>();
for (int i = 0; i < nums.length; i++) {
set[i] = i;
}
for (int i = 0; i < nums.length; i++) {
solve(nums[i], i);
}
int p = findParent(0);
for (int i = 0; i < nums.length; i++) {
if (p != findParent(i)) {
return false;
}
}
return true;
}
}