1627 - Graph Connectivity With Threshold.

Hard

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 <code>queriesi = a_i, b_i</code> if cities <code>a_i</code> and <code>b_i</code> 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 <code>i^th</code> query, there is a path between <code>a_i</code> and <code>b_i</code>, 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:

• <code>2 <= n <= 10⁴</code>

• 0 <= threshold <= n

• <code>1 <= queries.length <= 10⁵</code>

• queries[i].length == 2

• <code>1 <= a_i, b_i<= cities</code>

• <code>a_i != b_i</code>