BreadthFirstSearch

Breadth-first search

DFS is called "depth first" because it descends into the graph as quickly as possible.

Breadth-search first, in contrast, explores nodes by following a "wave front". It explores in order of their distance from the start vertex.

BFS(graph G, vertex v)

   FIFO_Queue Q;
   Array<int> distance(-1);
   
   distance[v] = 0;
   insert vertex v into the queue Q.

   while (Q is not empty)
      take the next vertex W off the front of the queue Q
      for each neighbor X of W do
         if (distance[X] == -1) then
            distance[X] = distance[W] + 1
            add vertex X to the tail of the queue Q
end

(example here)

Upper bounds on its running time:

• O(n²) because the outer loop executes at most n times, and each time the inner loop executes, it executes at most n times.
• O(n+m) because the time can be counted as ∑_{vertices w} 1+degree(w) = n+2m . Here n is the number of vertices and m is the number of edges.

O(n+m) is linear in the input size, because encoding the graph takes Θ(n+m) space.

Algorithms that run in time linear in the input size are often the best possible, because for many problems, any algorithm must at least examine the entire input.

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References

• GoodrichAndTomassia section 6.3.3.