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1627-graph-connectivity-with-threshold.adoc

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1627. Graph Connectivity With Threshold

{leetcode}/problems/graph-connectivity-with-threshold/[LeetCode - 1627. Graph Connectivity With Threshold ^]

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] = [a<sub>i</sub>, b<sub>i</sub>] if cities a<sub>i</sub> and b<sub>i</sub> 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 `a<sub>i</sub>` and `b<sub>i</sub>`, or `answer[i]` is `false` if there is no path._

Example 1: <img alt="" src="https://assets.leetcode.com/uploads/2020/10/09/ex1.jpg" style="width: 382px; height: 181px;" />

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: <img alt="" src="https://assets.leetcode.com/uploads/2020/10/10/tmp.jpg" style="width: 532px; height: 302px;" />

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: <img alt="" src="https://assets.leetcode.com/uploads/2020/10/17/ex3.jpg" style="width: 282px; height: 282px;" />

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 ⇐ a<sub>i</sub>, b<sub>i</sub> ⇐ cities

  • a<sub>i</sub> != b<sub>i</sub>

思路分析

一刷
link:{sourcedir}/_1627_GraphConnectivityWithThreshold.java[role=include]

参考资料