Breadth-first search traversal of a graph using the algorithm given in CLRS book.

BFS is one of the ways to traverse a graph. It is named so because it expands the frontier between discovered and undiscovered vertices uniformly across the breadth of the frontier. What it means is that the algorithm first discovers all the vertices connected to “u” at a distance of k before discovering the vertices at a distance of k+1 from u. The algorithm given in CLRS uses the concept of “colour” to check if a vertex is discovered fully or partially or undiscovered. It also keeps a track of the distance a vertex u is from the source s.

BFS(G,s) 1 for each vertex u in G.V - {s} 2 u.color = white 3 u.d = INF 4 u.p = NIL 5 s.color = green 6 s.d = 0 7 s.p = NIL 8 Q = NULL 9 ENQUEUE(Q,s) 10 while Q != NULL 11 u = DEQUEUE(Q) 12 for each v in G.Adj[u] 13 if v.color == white 14 v.color = green 15 v.d = u.d + 1 16 v.p = u 17 ENQUEUE(Q,v) 18 u.color = dark_green

It produces a “breadth-first tree” with root s that contains all reachable vertices. Let’s take a simple directed graph and see how BFS traverses it.

// CPP program to implement BFS as per CLRS // algorithm. #include <bits/stdc++.h> using namespace std; // Declaring the vectors to store color, distance // and parent vector<string> colour; vector<int> d; vector<int> p; /* This function adds an edge to the graph. It is an undirected graph. So edges are added for both the nodes. */ void addEdge(vector <int> g[], int u, int v) { g[u].push_back(v); g[v].push_back(u); } /* This function does the Breadth First Search*/ void BFSSingleSource(vector <int> g[], int s) { // The Queue used for the BFS operation queue<int> q; // Pushing the root node inside the queue q.push(s); /* Distance of root node is 0 & colour is gray as it is visited partially now */ d[s] = 0; colour[s] = "green"; /* Loop to traverse the graph. Traversal will happen traverse until the queue is not empty.*/ while (!q.empty()) { /* Extracting the front element(node) and poping it out of queue. */ int u = q.front(); q.pop(); cout << u << " "; /* This loop traverses all the child nodes of u */ for (auto i = g[u].begin(); i != g[u].end(); i++) { /* If the colour is white then the said node is not traversed. */ if (colour[*i] == "white") { colour[*i] = "green"; d[*i] = d[u] + 1; p[*i] = u; /* Pushing the node inside queue to traverse its children. */ q.push(*i); } } /* Now the node u is completely traversed and colour is changed to black. */ colour[u] = "dark_green"; } } void BFSFull(vector <int> g[], int n) { /* Initially all nodes are not traversed. Therefore, the colour is white. */ colour.assign(n, "white"); d.assign(n, 0); p.assign(n, -1); // Calling BFSSingleSource() for all white // vertices. for (int i = 0; i < n; i++) if (colour[i] == "white") BFSSingleSource(g, 0); } // Driver Function int main() { // Graph with 7 nodes and 6 edges. int n = 7; // The Graph vector vector <int> g[n]; addEdge(g, 0, 1); addEdge(g, 0, 2); addEdge(g, 1, 3); addEdge(g, 1, 4); addEdge(g, 2, 5); addEdge(g, 2, 6); BFSFull(g, n); return 0; }

Output:

0 1 2 3 4 5 6

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