maximum number of k divisible components leetcode 2872 python

certainly! the problem "maximum number of k divisible components" (leetcode 2872) revolves around finding the maximum number of components in a graph that can be formed such that the sizes of these components are divisible by a given integer \( k \).

problem description
you are given an undirected graph represented by its edges, and you need to determine the maximum number of connected components in the graph such that the size of each component is divisible by \( k \).

key concepts
1. **graph representation**: the graph can be represented using an adjacency list or an edge list.
2. **connected components**: a component in a graph is a subset of vertices such that there is a path between any two vertices in that subset.
3. **dfs/bfs**: depth-first search (dfs) or breadth-first search (bfs) can be used to explore the graph and find connected components.
4. **divisibility check**: for each connected component found, check if its size is divisible by \( k \).

steps to solve the problem
1. construct the graph using the edges provided.
2. use dfs or bfs to traverse the graph and find all connected components.
3. count the size of each component and check if it is divisible by \( k \).
4. maintain a count of the components that satisfy the divisibility condition.

python code example
here’s a python solution for the problem using dfs:

explanation of the code
1. **graph construction**: we use a `defaultdict` to store the adjacency list of the graph.
2. **dfs function**: the `dfs` function explores the graph to count the size of the connected component starting from a given node.
3. **component size counting**: for each unvisited node, we invoke the `dfs` function and check if the size of the component is divisible by \( k \).
4. **result**: finally, the function returns the count of components whose sizes are divisible by \( k \).

complexity analysis
- **time complexity**: \( o(v + e) \), where \( v \) is the number of vertices and \( e \) is the number of edg ...

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maximum number
k divisible
components
leetcode
python
graph theory
connected components
divisibility
algorithm challenges
code optimization
data structures
graph traversal
union find
problem solving
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