In this section we present two algorithms for exploring a graph, starting at one of its vertices, , and finding all vertices that are reachable from . Both of these algorithms are best suited to graphs represented using an adjacency list representation. Therefore, when analyzing these algorithms we will assume that the underlying representation is an AdjacencyLists.
The bread-first-search algorithm starts at a vertex and visits, first the neighbours of , then the neighbours of the neighbours of , then the neighbours of the neighbours of the neighbours of , and so on.
This algorithm is a generalization of the breadth-first traversal
algorithm for binary trees (Section 6.1.2), and is
very similar; it uses a queue,
, that initially contains only
It then repeatedly extracts an element from
and adds its neighbours
, provided that these neighbours have never been in
The only major difference between the breadth-first-search algorithm
for graphs and the one for trees is that the algorithm for graphs has
to ensure that it does not add the same vertex to
more than once.
It does this by using an auxiliary boolean array,
, that tracks
which vertices have already been discovered.
An example of running on the graph from Figure 12.1 is shown in Figure 12.4. Different executions are possible, depending on the ordering of the adjacency lists; Figure 12.4 uses the adjacency lists in Figure 12.3.
Analyzing the running-time of the routine is fairly straightforward. The use of the array ensures that no vertex is added to more than once. Adding (and later removing) each vertex from takes constant time per vertex for a total of time. Since each vertex is processed by the inner loop at most once, each adjacency list is processed at most once, so each edge of is processed at most once. This processing, which is done in the inner loop takes constant time per iteration, for a total of time. Therefore, the entire algorithm runs in time.
The following theorem summarizes the performance of the algorithm.
A breadth-first traversal has some very special properties. Calling will eventually enqueue (and eventually dequeue) every vertex such that there is a directed path from to . Moreover, the vertices at distance 0 from ( itself) will enter before the vertices at distance 1, which will enter before the vertices at distance 2, and so on. Thus, the method visits vertices in increasing order of distance from and vertices that cannot be reached from are never visited at all.
A particularly useful application of the breadth-first-search algorithm is, therefore, in computing shortest paths. To compute the shortest path from to every other vertex, we use a variant of that uses an auxilliary array, , of length . When a new vertex is added to , we set . In this way, becomes the second last node on a shortest path from to . Repeating this, by taking , , and so on we can reconstruct the (reversal of) a shortest path from to .
The depth-first-search algorithm is similar to the standard algorithm for traversing binary trees; it first fully explores one subtree before returning to the current node and then exploring the other subtree. Another way to think of depth-first-search is by saying that it is similar to breadth-first search except that it uses a stack instead of a queue.
During the execution of the depth-first-search algorithm, each vertex,
, is assigned a colour,
if we have never seen
the vertex before,
if we are currently visiting that vertex,
if we are done visiting that vertex. The easiest way to
think of depth-first-search is as a recursive algorithm. It starts by
. When visiting a vertex
, we first mark
Next, we scan
's adjacency list and recursively visit any white vertex
we find in this list. Finally, we are done processing
, so we colour
black and return.
An example of the execution of this algorithm is shown in Figure 12.5.
Although depth-first-search may best be thought of as a recursive
algorithm, recursion is not the best way to implement it. Indeed, the code
given above will fail for many large graphs by causing a stack overflow.
An alternative implementation is to replace the recursion stack with an
. The following implementation does just that:
In the preceding code, when the next vertex, , is processed, is coloured and then replaced, on the stack, with its adjacent vertices. During the next iteration, one of these vertices will be visited.
Not surprisingly, the running times of and are the same as that of :
As with the breadth-first-search algorithm, there is an underlying tree associated with each execution of depth-first-search. When a node goes from to , this is because was called recursively while processing some node . (In the case of algorithm, is one of the nodes that replaced on the stack.) If we think of as the parent of , then we obtain a tree rooted at . In Figure 12.5, this tree is a path from vertex 0 to vertex 11.
An important property of the depth-first-search algorithm is the following: Suppose that when node is coloured , there exists a path from to some other node that uses only white vertices. Then will be coloured first then before is coloured . (This can be proven by contradiction, by considering any path from to .)
One application of this property is the detection of cycles. Refer to Figure 12.6. Consider some cycle, , that can be reached from . Let be the first node of that is coloured , and let be the node that precedes on the cycle . Then, by the above property, will be coloured and the edge will be considered by the algorithm while is still . Thus, the algorithm can conclude that there is a path, , from to in the depth-first-search tree and the edge exists. Therefore, is also a cycle.