Efficiently Finding Shortest Paths in Unweighted Graphs: A BFS Approach

Introduction

In the realm of graph theory, determining the shortest path from a start vertex to all other vertices in an unweighted graph is a fundamental and frequent task. This is central to various applications, from navigation systems to network routing and social network analysis. This article delves into the intricacies and practical implementation of finding the shortest path in an unweighted graph using the Breadth-First Search (BFS) algorithm. We will explore how to effectively represent the graph and execute the BFS to achieve the desired results within optimal time complexity.

Problem Statement

Given an unweighted graph ( g ) and a start vertex ( S ), the objective is to find the shortest path from vertex ( S ) to all other nodes in the graph. The term 'shortest path' in an unweighted graph implies the path with the minimum number of edges, which is equivalent to the path with the minimum length. This is a common use case in many real-world scenarios, such as computer networks, where the cost is minimal and thus routed through the least number of nodes.

Graph Representation

To represent the graph efficiently, we will utilize an adjacency list, which is an essential data structure in graph theory. The adjacency list is implemented using an STL vector of vectors in C . The adjacency list storage format is intuitive and highly scalable, making it suitable for both dense and sparse graphs.

Below is the C code snippet for creating the adjacency list:

#include iostream
#include vector
using namespace std;
void addEdge(vectorvectorint adjList, int src, int dest) {
    adjList[src].push_back(dest);
    adjList[dest].push_back(src);
}
int main() {
    int V  5; // Number of vertices
    vectorvectorint adjList(V);
    addEdge(adjList, 0, 1);
    addEdge(adjList, 0, 4);
    addEdge(adjList, 1, 2);
    addEdge(adjList, 1, 3);
    addEdge(adjList, 1, 4);
    addEdge(adjList, 2, 3);
    // Display the adjacency list
    for (int i  0; i 
Breadth-First Search (BFS)
Breadth-First Search, often used for traversing or searching tree or graph data structures, is a perfect choice for this problem. BFS explores the graph level by level, ensuring that all nodes at a certain level are visited before moving on to the next. This ensures that the shortest path from the source vertex is found first, as paths with fewer edges are visited earlier in the search.
Below is the C   code snippet for performing BFS to find the shortest path from the source vertex ( S ).
#include iostream
#include queue
#include vector
using namespace std;
void bfs(vectorvectorint adjList, int source, vectorint dist) {
    vectorbool visited((), false);
    queueint q;
    q.push(source);
    dist[source]  0;
    visited[source]  true;
    while (!q.empty()) {
        int u  ();
        q.pop();
        for (auto x : adjList[u]) {
            if (!visited[x]) {
                q.push(x);
                dist[x]  dist[u]   1;
                visited[x]  true;
            }
        }
    }
}
int main() {
    int V  5; // Number of vertices
    vectorvectorint adjList(V);
    // Add edges as previously defined
    vectorint dist(V, -1); // Initialize distances with -1
    bfs(adjList, 0, dist); // Source vertex is 0
    // Display the shortest distances
    for (int i  0; i 
Time Complexity Analysis
The time complexity for BFS in this context is ( O(V   E) ), where ( V ) is the number of vertices and ( E ) is the number of edges. This complexity makes BFS an efficient algorithm for finding shortest paths in unweighted graphs. The reason behind this complexity is straightforward: we visit each vertex and each edge exactly once. The space complexity is ( O(V) ) due to the storage requirements for the distance array and the queue.
Conclusion
In conclusion, finding the shortest path in an unweighted graph is a critical task in many computational problems. By utilizing the adjacency list representation and the BFS algorithm, we can efficiently determine the shortest path from a source to all other vertices in the graph. This approach not only ensures the accuracy of the path but also handles large graphs without significant performance degradation.
Keywords
    unweighted graph
    shortest path
    BFS
    adjacency list