# 1627. Graph Connectivity With Threshold

## Description

We have n cities labeled from 1 to n. Two different cities with labels x and y are directly connected by a bidirectional road if and only if x and y share a common divisor strictly greater than some threshold. More formally, cities with labels x and y have a road between them if there exists an integer z such that all of the following are true:

• x % z == 0,
• y % z == 0, and
• z > threshold.

Given the two integers, n and threshold, and an array of queries, you must determine for each queries[i] = [ai, bi] if cities ai and bi are connected directly or indirectly. (i.e. there is some path between them).

Return an array answer, where answer.length == queries.length and answer[i] is true if for the ith query, there is a path between ai and bi, or answer[i] is false if there is no path.

Example 1:

Input: n = 6, threshold = 2, queries = [[1,4],[2,5],[3,6]]
Output: [false,false,true]
Explanation: The divisors for each number:
1:   1
2:   1, 2
3:   1, 3
4:   1, 2, 4
5:   1, 5
6:   1, 2, 3, 6
Using the underlined divisors above the threshold, only cities 3 and 6 share a common divisor, so they are the
only ones directly connected. The result of each query:
[1,4]   1 is not connected to 4
[2,5]   2 is not connected to 5
[3,6]   3 is connected to 6 through path 3--6


Example 2:

Input: n = 6, threshold = 0, queries = [[4,5],[3,4],[3,2],[2,6],[1,3]]
Output: [true,true,true,true,true]
Explanation: The divisors for each number are the same as the previous example. However, since the threshold is 0,
all divisors can be used. Since all numbers share 1 as a divisor, all cities are connected.


Example 3:

Input: n = 5, threshold = 1, queries = [[4,5],[4,5],[3,2],[2,3],[3,4]]
Output: [false,false,false,false,false]
Explanation: Only cities 2 and 4 share a common divisor 2 which is strictly greater than the threshold 1, so they are the only ones directly connected.
Please notice that there can be multiple queries for the same pair of nodes [x, y], and that the query [x, y] is equivalent to the query [y, x].


Constraints:

• 2 <= n <= 104
• 0 <= threshold <= n
• 1 <= queries.length <= 105
• queries[i].length == 2
• 1 <= ai, bi <= cities
• ai != bi

## Solutions

Solution 1: Union-Find

We can enumerate $z$ and its multiples, and use union-find to connect them. In this way, for each query $[a, b]$, we only need to determine whether $a$ and $b$ are in the same connected component.

The time complexity is $O(n \times \log n \times (\alpha(n) + q))$, and the space complexity is $O(n)$. Here, $n$ and $q$ are the number of nodes and queries, respectively, and $\alpha$ is the inverse function of the Ackermann function.

• class UnionFind {
private int[] p;
private int[] size;

public UnionFind(int n) {
p = new int[n];
size = new int[n];
for (int i = 0; i < n; ++i) {
p[i] = i;
size[i] = 1;
}
}

public int find(int x) {
if (p[x] != x) {
p[x] = find(p[x]);
}
return p[x];
}

public boolean union(int a, int b) {
int pa = find(a), pb = find(b);
if (pa == pb) {
return false;
}
if (size[pa] > size[pb]) {
p[pb] = pa;
size[pa] += size[pb];
} else {
p[pa] = pb;
size[pb] += size[pa];
}
return true;
}
}

class Solution {
public List<Boolean> areConnected(int n, int threshold, int[][] queries) {
UnionFind uf = new UnionFind(n + 1);
for (int a = threshold + 1; a <= n; ++a) {
for (int b = a + a; b <= n; b += a) {
uf.union(a, b);
}
}
List<Boolean> ans = new ArrayList<>();
for (var q : queries) {
}
return ans;
}
}

• class UnionFind {
public:
UnionFind(int n) {
p = vector<int>(n);
size = vector<int>(n, 1);
iota(p.begin(), p.end(), 0);
}

bool unite(int a, int b) {
int pa = find(a), pb = find(b);
if (pa == pb) {
return false;
}
if (size[pa] > size[pb]) {
p[pb] = pa;
size[pa] += size[pb];
} else {
p[pa] = pb;
size[pb] += size[pa];
}
return true;
}

int find(int x) {
if (p[x] != x) {
p[x] = find(p[x]);
}
return p[x];
}

private:
vector<int> p, size;
};

class Solution {
public:
vector<bool> areConnected(int n, int threshold, vector<vector<int>>& queries) {
UnionFind uf(n + 1);
for (int a = threshold + 1; a <= n; ++a) {
for (int b = a + a; b <= n; b += a) {
uf.unite(a, b);
}
}
vector<bool> ans;
for (auto& q : queries) {
ans.push_back(uf.find(q[0]) == uf.find(q[1]));
}
return ans;
}
};

• class UnionFind:
def __init__(self, n):
self.p = list(range(n))
self.size = [1] * n

def find(self, x):
if self.p[x] != x:
self.p[x] = self.find(self.p[x])
return self.p[x]

def union(self, a, b):
pa, pb = self.find(a), self.find(b)
if pa == pb:
return False
if self.size[pa] > self.size[pb]:
self.p[pb] = pa
self.size[pa] += self.size[pb]
else:
self.p[pa] = pb
self.size[pb] += self.size[pa]
return True

class Solution:
def areConnected(
self, n: int, threshold: int, queries: List[List[int]]
) -> List[bool]:
uf = UnionFind(n + 1)
for a in range(threshold + 1, n + 1):
for b in range(a + a, n + 1, a):
uf.union(a, b)
return [uf.find(a) == uf.find(b) for a, b in queries]


• type unionFind struct {
p, size []int
}

func newUnionFind(n int) *unionFind {
p := make([]int, n)
size := make([]int, n)
for i := range p {
p[i] = i
size[i] = 1
}
return &unionFind{p, size}
}

func (uf *unionFind) find(x int) int {
if uf.p[x] != x {
uf.p[x] = uf.find(uf.p[x])
}
return uf.p[x]
}

func (uf *unionFind) union(a, b int) bool {
pa, pb := uf.find(a), uf.find(b)
if pa == pb {
return false
}
if uf.size[pa] > uf.size[pb] {
uf.p[pb] = pa
uf.size[pa] += uf.size[pb]
} else {
uf.p[pa] = pb
uf.size[pb] += uf.size[pa]
}
return true
}

func areConnected(n int, threshold int, queries [][]int) []bool {
uf := newUnionFind(n + 1)
for a := threshold + 1; a <= n; a++ {
for b := a + a; b <= n; b += a {
uf.union(a, b)
}
}
ans := make([]bool, len(queries))
for i, q := range queries {
ans[i] = uf.find(q[0]) == uf.find(q[1])
}
return ans
}

• class UnionFind {
p: number[];
size: number[];
constructor(n: number) {
this.p = Array(n)
.fill(0)
.map((_, i) => i);
this.size = Array(n).fill(1);
}

find(x: number): number {
if (this.p[x] !== x) {
this.p[x] = this.find(this.p[x]);
}
return this.p[x];
}

union(a: number, b: number): boolean {
const [pa, pb] = [this.find(a), this.find(b)];
if (pa === pb) {
return false;
}
if (this.size[pa] > this.size[pb]) {
this.p[pb] = pa;
this.size[pa] += this.size[pb];
} else {
this.p[pa] = pb;
this.size[pb] += this.size[pa];
}
return true;
}
}

function areConnected(n: number, threshold: number, queries: number[][]): boolean[] {
const uf = new UnionFind(n + 1);
for (let a = threshold + 1; a <= n; ++a) {
for (let b = a * 2; b <= n; b += a) {
uf.union(a, b);
}
}
return queries.map(([a, b]) => uf.find(a) === uf.find(b));
}