Is Count Connected Components in LCM Graph easy or hard?

The problem is classified as Hard because it requires combining graph connectivity, number theory, and Union Find optimization. The brute-force idea is straightforward, but recognizing how to connect numbers through shared multiples within the threshold is the challenging insight.

Count Connected Components in LCM Graph Python/Java solution

Most implementations use Union Find with path compression and union-by-rank. The algorithm iterates through multiples up to the threshold and unions indices that share valid LCM relationships. The same logic translates easily across Python, Java, C++, and Go.

How to solve Count Connected Components in LCM Graph in O(T log T)?

Use a Union Find structure and iterate through numbers up to the threshold. For each value x, union nodes corresponding to its multiples (2x, 3x, 4x, ...) that remain ≤ threshold. This groups numbers whose LCM stays within the limit without explicitly checking every pair.

What is the best approach for Count Connected Components in LCM Graph?

The most efficient approach uses Union Find combined with number theory. Instead of checking every pair of numbers, you connect values through shared multiples that do not exceed the threshold. This reduces the complexity to roughly O(T log T) and efficiently groups nodes into components.

Is Count Connected Components in LCM Graph asked at Google/Amazon/Meta?

Problems combining Union Find and number theory frequently appear in interviews at companies like Google, Amazon, and Meta. Variants involving LCM, GCD, or connectivity through divisibility are common because they test mathematical reasoning and graph modeling skills.

What data structure is used in Count Connected Components in LCM Graph?

The primary data structure is Union Find (Disjoint Set Union). It efficiently merges nodes that share a valid LCM relationship and allows quick checks of whether two nodes belong to the same component. Graph traversal with DFS or BFS can also be used in the brute-force approach.

What is the time complexity of Count Connected Components in LCM Graph?

The optimized Union Find solution runs in about O(T log T), where T is the threshold used for iterating through multiples. Each union and find operation is nearly constant due to path compression. A naive pairwise solution requires O(n² log M) time because it computes LCM for every pair.

Description Hints 2 Solution Video 35:09

3378. Count Connected Components in LCM Graph

Hard31.3% acceptance

You are given an array of integers nums of size n and a positive integer threshold.

There is a graph consisting of n nodes with the i^th node having a value of nums[i]. Two nodes i and j in the graph are connected via an undirected edge if lcm(nums[i], nums[j]) <= threshold.

Return the number of connected components in this graph.

A connected component is a subgraph of a graph in which there exists a path between any two vertices, and no vertex of the subgraph shares an edge with a vertex outside of the subgraph.

The term lcm(a, b) denotes the least common multiple of a and b.

Example 1:

Input: nums = [2,4,8,3,9], threshold = 5

Output: 4

Explanation:

The four connected components are (2, 4), (3), (8), (9).

Example 2:

Input: nums = [2,4,8,3,9,12], threshold = 10

Output: 2

Explanation:

The two connected components are (2, 3, 4, 8, 9), and (12).

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i] <= 10⁹
All elements of nums are unique.
1 <= threshold <= 2 * 10⁵

Topics

Hints open one at a time

Hint 1

Use DSU

Hint 2

Connect a number to all its multiples less than threshold

35:09

Still stuck?Read 2 approaches, or watch a walkthrough.

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1class Solution:
2    def countComponents(self, nums: List[int], threshold: int) -> int:
3        

Starter codePython 3Spaces: 4

Examples

Case 1Case 2

Input

[2,4,8,3,9]
5

Input

[2,4,8,3,9,12]
10

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