Problem Description

You are tasked with finding the number of complete connected components in an undirected graph. The graph has n vertices, each identified by a unique number from 0 to n - 1. An array edges contains pairs of integer indices representing the edges that connect vertices in the graph.

A connected component is a set of vertices in which each pair of vertices is connected by some path. Importantly, no vertex in a connected component connects to any vertex outside that component in the graph.

A complete connected component (also known as a clique) is a special kind of connected component where there is a direct edge between every pair of nodes within the component.

So, the objective is to count how many complete connected components there are in the given graph.

Flowchart Walkthrough

First, let's determine the appropriate algorithm using the Flowchart. Here's a step-by-step analysis:

Is it a graph?

• Yes: We need to consider if pairs of employees meet conditions (completeness) and form components which implies a graph where nodes represent employees, and edges represent the meet condition.

Is it a tree?

• No: The graph is likely not a tree as we're asked to count complete components where any node can have connections with multiple other nodes, making it general and potentially cyclical.

Is the problem related to directed acyclic graphs (DAGs)?

• No: The description does not suggest anything about directionality or acyclic conditions, as we're focused on completeness in components.

Is the problem related to shortest paths?

• No: The problem is centered on counting the number of complete components, not finding paths between nodes.

Does the problem involve connectivity?

• Yes: The problem is specifically about finding and counting complete components, which is directly related to how nodes are connected.

Is the graph weighted?

• No: All the information provided indicates that the conditions are binary (yes/no - meets/does not meet), not involving weights on the connections.

Conclusion: Based on the flowchart, DFS (Depth-First Search) is an apt choice for an unweighted connectivity problem to explore and count complete components in the graph, as it efficiently handles the visiting of all nodes in each connected component.

Intuition

To solve this problem, we can use Depth-First Search (DFS). DFS allows us to explore each vertex and its neighbors recursively. The solution employs DFS to count vertices (x) and edges (y) in each connected component. Since a complete graph with n vertices should have exactly n * (n - 1) / 2 edges, this fact can be used to verify if a connected component is complete.

Here's how the algorithm works step by step:

1. Create a graph representation using a list where each index represents a node and the values are lists of connected nodes.
2. Initialize a list of boolean values to track visited nodes during DFS to ensure nodes are not counted more than once.
3. Iterate through each node. If a node is unvisited, perform DFS starting from that node to determine the size of the connected component.
4. During DFS, count the number of nodes (x) and the number of edges (y). Note that since the graph is undirected, each edge will be counted twice - once for each endpoint.
5. After each DFS call, check if the number of edges matches the formula x * (x - 1) (which is double x * (x - 1) / 2 since each edge is counted twice). If it does, then we have found a complete connected component.
6. Increment a counter for complete connected components any time the check in step 5 passes.
7. After completing the iteration through all nodes, return the counter value.

This solution is efficient because it only requires a single pass through the graph, using DFS to explore the connected components and immediately checking which ones are complete.

Learn more about Depth-First Search, Breadth-First Search and Graph patterns.

Solution Approach

The solution uses the following algorithms, data structures, and patterns:

Depth-First Search (DFS): This algorithm allows us to explore all nodes in a connected component of the graph. We define a recursive dfs function that takes a starting vertex i and explores every connected vertex, returning the total count of vertices and edges in the connected component.

Adjacency List: The graph is represented as an adjacency list using a defaultdict(list) from Python's collections library. This data structure is efficient for storing sparse graphs and allows us to quickly access all neighbors of a graph node.

Visited List: To keep track of which nodes have been visited during the DFS exploration, we maintain a list vis of boolean values. this avoids counting the same node more than once and ensures we don't enter into an infinite loop by revisiting the same nodes.

Here's how the implementation follows the algorithm:

1. Initialize the graph as an adjacency list (g) from the edges array, where for each edge [a, b] we add b to the list of adjacent vertices of a and vice versa since the graph is undirected.

2. Create a visited list vis of boolean values, initialized to False, to mark vertices as visited.

3. For each vertex i from 0 to n-1, check if it has not been visited:

   • If not visited, call the dfs function on vertex i. This will traverse all vertices in the same connected component as i, marking them as visited and counting the vertices and edges.
   • The dfs function works as follows:
     • Mark the current vertex i as visited.
     • Initialize local variables x (vertex count) to 1 and y (edge count) to the length of the adjacency list at i since every adjacent vertex represents an edge.
     • Iterate over each neighbor j of vertex i. If j is not visited, call dfs recursively and update x and y with the counts from the component connected through j.
     • Return the count of vertices x and the double count of edges y.

4. After the recursive DFS, compare the number of vertices a and the double count of edges b using the formula a * (a - 1). A complete component must have exactly a * (a - 1) / 2 edges, but since each edge is counted twice, we check for a * (a - 1) without dividing by 2. If the component is complete, increment the ans.

5. Finally, when all vertices are processed, return ans, which represents the count of complete connected components.

The solution effectively utilizes the dfs function for graph traversal and the combinatorial property of complete graphs to detect complete connected components in a single scan.

Example Walkthrough

Let's assume we have n = 5 vertices, and we are given the following edges array, which defines an undirected graph:

edges = [[0, 1], [1, 2], [3, 4]]

The graph can be visualized as two separate connected components:

0 -- 1 -- 2

3 -- 4

Here's a step-by-step illustration of applying the solution approach:

1. First, we create an adjacency list g for the given graph. This list will look like:

g = {
  0: [1],
  1: [0, 2],
  2: [1],
  3: [4],
  4: [3]
}

2. We initialize a visited list vis = [False, False, False, False, False] to keep track of visited nodes.

3. We start iterating through each vertex. When iterating over vertex 0, since it's not visited, we call the dfs function.

4. The dfs function now starts on 0, marking it as visited (vis[0] = True) and setting x = 1 and y = len(g[0]) = 1.

5. Within DFS on vertex 0, we move to its neighbor 1 which is unvisited, and call DFS on 1, marking it visited and adding its own vertices and edges. We get x = 2 and y = 3 (since it's connected to 0 and 2).

6. Continuing the DFS, we visit vertex 2. No new vertices are discovered, but this adds to the edge count: y += len(g[2]) = 1.

7. The DFS call on vertex 0 ends, returning x = 3 and y = 4.

8. We then check if x * (x - 1) == y which would mean it's a complete connected component. 3 * (3 - 1) = 6 which does not equal 4, so this component is not complete.

9. We update the visited list for all vertices in the connected component containing 0, 1, and 2.

10. We continue the iteration to vertex 3, calling DFS since it's not visited.

11. Similar to the previous DFS, we mark 3 as visited, setting x = 1 and y = len(g[3]) = 1 since it's connected to 4.

12. On visiting neighbor 4, we update x = 2 and y = 2.

13. After the DFS on 3, we check if x * (x - 1) == y. 2 * (2 - 1) = 2 which equals y, so this is indeed a complete connected component.

14. We add 1 to our complete connected component count ans.

15. Finally, after checking all vertices, we find that there is 1 complete connected component in the graph.

In conclusion, after traversing this example graph, our algorithm would correctly report that there is 1 complete connected component.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code is O(N + E), where N represents the number of nodes and E represents the number of edges in the graph. The reasoning behind this assessment is as follows:

• Iterating over all edges to create the graph g yields O(E), as it is directly proportional to the number of edges.

• The dfs function visits each node exactly once due to the guard condition if not vis[i]:, which prevents revisiting. Each call to dfs also iterates over all neighbors of the node, which results in a total of O(E) for all dfs calls across all nodes because each edge will be considered exactly twice (once from each incident node).

• The for loop over n nodes is O(N), as it will attempt a dfs from each unvisited node.

The combination of creating the adjacency list and the DFS traversal then constitutes O(N + E).

Space Complexity

The space complexity of the code is O(N + E) as well. This assessment considers the following components that utilize memory:

• The adjacency list g can potentially hold all the edges, which has a space requirement of O(E).

• The visited list vis has a space requirement of O(N), since it stores a boolean for each node.

• The recursion stack for DFS can also grow up to O(N) in the case of a deep or unbalanced tree.

• The edges list itself occupies O(E).

Taking all these factors into account, the upper bound on space utilized by the algorithm is O(N + E), which accounts for all nodes and edges stored and processed.

Learn more about how to find time and space complexity quickly using problem constraints.