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Figure 42 A physical model of a graph
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Figure 43 Breadth- rst search procedure bfs(G, s) Input: Graph G = (V, E), directed or undirected; vertex s V Output: For all vertices u reachable from s, dist(u) is set to the distance from s to u for all u V : dist(u) = dist(s) = 0 Q = [s] (queue containing just s) while Q is not empty: u = eject(Q) for all edges (u, v) E: if dist(v) = : inject(Q, v) dist(v) = dist(u) + 1
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42 Breadth- rst search
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In Figure 42, the lifting of s partitions the graph into layers: s itself, the nodes at distance 1 from it, the nodes at distance 2 from it, and so on A convenient way to compute distances from s to the other vertices is to proceed layer by layer Once we have picked out the nodes at distance 0, 1, 2, , d, the ones at d + 1 are easily determined: they are precisely the as-yetunseen nodes that are adjacent to the layer at distance d This suggests an iterative algorithm in which two layers are active at any given time: some layer d, which has been fully identi ed, and d + 1, which is being discovered by scanning the neighbors of layer d Breadth- rst search (BFS) directly implements this simple reasoning (Figure 43) Initially the queue Q consists only of s, the one node at distance 0 And for each subsequent distance d = 1, 2, 3, , there is a point in time at which Q contains all the nodes at distance d and nothing else As these nodes are processed (ejected off the front of the queue), their as-yet-unseen neighbors are injected into the end of the queue Let s try out this algorithm on our earlier example (Figure 41) to con rm that it does the 106
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Figure 44 The result of breadth- rst search on the graph of Figure 41 Order of visitation S A C D E B Queue contents after processing node [S] [A C D E] [C D E B] [D E B] [E B] [B] []
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right thing If S is the starting point and the nodes are ordered alphabetically, they get visited in the sequence shown in Figure 44 The breadth- rst search tree, on the right, contains the edges through which each node is initially discovered Unlike the DFS tree we saw earlier, it has the property that all its paths from S are the shortest possible It is therefore a shortestpath tree Correctness and ef ciency We have developed the basic intuition behind breadth- rst search In order to check that the algorithm works correctly, we need to make sure that it faithfully executes this intuition What we expect, precisely, is that For each d = 0, 1, 2, , there is a moment at which (1) all nodes at distance d from s have their distances correctly set; (2) all other nodes have their distances set to ; and (3) the queue contains exactly the nodes at distance d
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This has been phrased with an inductive argument in mind We have already discussed both the base case and the inductive step Can you ll in the details The overall running time of this algorithm is linear, O(|V | + |E|), for exactly the same reasons as depth- rst search Each vertex is put on the queue exactly once, when it is rst encountered, so there are 2 |V | queue operations The rest of the work is done in the algorithm s innermost loop Over the course of execution, this loop looks at each edge once (in directed graphs) or twice (in undirected graphs), and therefore takes O(|E|) time Now that we have both BFS and DFS before us: how do their exploration styles compare Depth- rst search makes deep incursions into a graph, retreating only when it runs out of new nodes to visit This strategy gives it the wonderful, subtle, and extremely useful properties we saw in the 3 But it also means that DFS can end up taking a long and convoluted route to a vertex that is actually very close by, as in Figure 41 Breadth- rst search makes sure to visit vertices in increasing order of their distance from the starting point This is a broader, shallower search, rather like the propagation of a wave upon water And it is achieved using almost exactly the same code as DFS but with a queue in place of a stack 107
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