0.7 Graphs  (Page 12/21)

Pic.15 Animated example of a breadth-first search

Algorithm (informal)

1. Put the ending node (the root node) in the queue.
2. Pull a node from the beginning of the queue and examine it.
   • If the searched element is found in this node, quit the search and return a result.
   • Otherwise push all the (so-far-unexamined) successors (the direct child nodes) of this node into the end of the queue, if there are any.
3. If the queue is empty, every node on the graph has been examined -- quit the search and return "not found".
4. Repeat from Step 2.

C implementation

Algorithm of Breadth-first search：

void BFS(VLink G[], int v) {

int w;

VISIT(v); /*visit vertex v*/

visited[v] = 1; /*mark v as visited : 1 */

ADDQ(Q,v);

while(!QMPTYQ(Q)) {

v = DELQ(Q); /*Dequeue v*/

w = FIRSTADJ(G,v); /*Find first neighbor, return -1 if no neighbor*/

while(w != -1) {

if(visited[w] == 0) {

VISIT(w); /*visit vertex v*/

ADDQ(Q,w); /*Enqueue current visited vertext w*/

visited[w] = 1; /*mark w as visited*/

}

W = NEXTADJ(G,v); /*Find next neighbor, return -1 if no neighbor*/

}

}

}

Main Algorithm of apply Breadth-first search to graph G=(V,E)：

void TRAVEL_BFS(VLink G[], int visited[], int n) {

int i;

for(i = 0; i<n; i ++) {

visited[i] = 0; /* Mark initial value as 0 */

}

for(i = 0; i<n; i ++)

if(visited[i] == 0)

BFS(G,i);

}

C++ implementation

This is the implementation of the above informal algorithm, where the "so-far-unexamined" is handled by the parent array. For actual C++ applications, see the Boost Graph Library.

Suppose we have a struct:

struct Vertex {

...

std::vector<int>out;

...

};

and an array of vertices: (the algorithm will use the indexes of this array, to handle the vertices)

std::vector<Vertex>graph(vertices);

the algorithm starts from start and returns true if there is a directed path from start to end:

bool BFS(const std::vector<Vertex>&graph, int start, int end) {

std::queue<int>next;

std::map<int,int>parent;

parent[start] = -1;

next.push(start);

while (!next.empty()) {

int u = next.front();

next.pop();

// Here is the point where you can examine the u th vertex of graph

// For example:

if (u == end) return true;

for (std::vector<int>::const_iterator j = graph[u].out.begin(); j != graph[u].out.end(); ++j) {

// Look through neighbors.

int v = *j;

if (parent.count(v) == 0) {

// If v is unvisited.

parent[v] = u;

next.push(v);

}

}

}

return false;

}

it also stores the parents of each node, from which you can get the path.

Features

• Space Complexity

Since all nodes discovered so far have to be saved, the space complexity of breadth-first search is O(|V| + |E|) where |V| is the number of nodes and |E| the number of edges in the graph. Note: another way of saying this is that it is O(BM) where B is the maximum branching factor and M is the maximum path length of the tree. This immense demand for space is the reason why breadth-first search is impractical for larger problems.

• Time Complexity

Since in the worst case breadth-first search has to consider all paths to all possible nodes the time complexity of breadth-first search is O(|V| + |E|) where |V| is the number of nodes and |E| the number of edges in the graph. The best case of this search is o(1). It occurs when the node is found at first time.