# 6.3: Graph Traversal

In this section we present two algorithms for exploring a graph, starting at one of its vertices, $$\mathtt{i}$$, and finding all vertices that are reachable from $$\mathtt{i}$$. 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.

## $$\PageIndex{1}$$ Breadth-First Search

The bread-first-search algorithm starts at a vertex $$\mathtt{i}$$ and visits, first the neighbours of $$\mathtt{i}$$, then the neighbours of the neighbours of $$\mathtt{i}$$, then the neighbours of the neighbours of the neighbours of $$\mathtt{i}$$, and so on.

This algorithm is a generalization of the breadth-first traversal algorithm for binary trees (Open Data Structures, Section 6.1.2), and is very similar; it uses a queue, $$\mathtt{q}$$, that initially contains only $$\mathtt{i}$$. It then repeatedly extracts an element from $$\mathtt{q}$$ and adds its neighbours to $$\mathtt{q}$$, provided that these neighbours have never been in $$\mathtt{q}$$ before. 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 $$\mathtt{q}$$ more than once. It does this by using an auxiliary boolean array, $$\mathtt{seen}$$, that tracks which vertices have already been discovered.

    void bfs(Graph g, int r) {
boolean[] seen = new boolean[g.nVertices()];
Queue<Integer> q = new SLList<Integer>();
seen[r] = true;
while (!q.isEmpty()) {
int i = q.remove();
for (Integer j : g.outEdges(i)) {
if (!seen[j]) {
seen[j] = true;
}
}
}
}


An example of running $$\mathtt{bfs(g,0)}$$ on the graph from Figure 6.1 is shown in Figure $$\PageIndex{1}$$. Different executions are possible, depending on the ordering of the adjacency lists; Figure $$\PageIndex{1}$$ uses the adjacency lists in Figure 6.2.1. Figure $$\PageIndex{1}$$: An example of bread-first-search starting at node 0. Nodes are labelled with the order in which they are added to $$\mathtt{q}$$. Edges that result in nodes being added to $$\mathtt{q}$$ are drawn in black, other edges are drawn in grey.

Analyzing the running-time of the $$\mathtt{bfs(g,i)}$$ routine is fairly straightforward. The use of the $$\mathtt{seen}$$ array ensures that no vertex is added to $$\mathtt{q}$$ more than once. Adding (and later removing) each vertex from $$\mathtt{q}$$ takes constant time per vertex for a total of $$O(\mathtt{n})$$ 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 $$G$$ is processed at most once. This processing, which is done in the inner loop takes constant time per iteration, for a total of $$O(\mathtt{m})$$ time. Therefore, the entire algorithm runs in $$O(\mathtt{n}+\mathtt{m})$$ time.

The following theorem summarizes the performance of the $$\mathtt{bfs(g,r)}$$ algorithm.

Theorem $$\PageIndex{1}$$

When given as input a Graph, $$\mathtt{g}$$, that is implemented using the AdjacencyLists data structure, the $$\mathtt{bfs(g,r)}$$ algorithm runs in $$O(\mathtt{n}+\mathtt{m})$$ time.

A breadth-first traversal has some very special properties. Calling $$\mathtt{bfs(g,r)}$$ will eventually enqueue (and eventually dequeue) every vertex $$\mathtt{j}$$ such that there is a directed path from $$\mathtt{r}$$ to $$\mathtt{j}$$. Moreover, the vertices at distance 0 from $$\mathtt{r}$$ ( $$\mathtt{r}$$ itself) will enter $$\mathtt{q}$$ before the vertices at distance 1, which will enter $$\mathtt{q}$$ before the vertices at distance 2, and so on. Thus, the $$\mathtt{bfs(g,r)}$$ method visits vertices in increasing order of distance from $$\mathtt{r}$$ and vertices that cannot be reached from $$\mathtt{r}$$ 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 $$\mathtt{r}$$ to every other vertex, we use a variant of $$\mathtt{bfs(g,r)}$$ that uses an auxilliary array, $$\mathtt{p}$$, of length $$\mathtt{n}$$. When a new vertex $$\mathtt{j}$$ is added to $$\mathtt{q}$$, we set $$\mathtt{p[j]=i}$$. In this way, $$\mathtt{p[j]}$$ becomes the second last node on a shortest path from $$\mathtt{r}$$ to $$\mathtt{j}$$. Repeating this, by taking $$\mathtt{p[p[j]}$$, $$\mathtt{p[p[p[j]]]}$$, and so on we can reconstruct the (reversal of) a shortest path from $$\mathtt{r}$$ to $$\mathtt{j}$$.

## $$\PageIndex{2}$$ Depth-First Search

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, $$\mathtt{i}$$, is assigned a colour, $$\mathtt{c[i]}$$: $$\mathtt{white}$$ if we have never seen the vertex before, $$\mathtt{grey}$$ if we are currently visiting that vertex, and $$\mathtt{black}$$ if we are done visiting that vertex. The easiest way to think of depth-first-search is as a recursive algorithm. It starts by visiting $$\mathtt{r}$$. When visiting a vertex $$\mathtt{i}$$, we first mark $$\mathtt{i}$$ as $$\mathtt{grey}$$. Next, we scan $$\mathtt{i}$$'s adjacency list and recursively visit any white vertex we find in this list. Finally, we are done processing $$\mathtt{i}$$, so we colour $$\mathtt{i}$$ black and return.

    void dfs(Graph g, int r) {
byte[] c = new byte[g.nVertices()];
dfs(g, r, c);
}
void dfs(Graph g, int i, byte[] c) {
c[i] = grey;  // currently visiting i
for (Integer j : g.outEdges(i)) {
if (c[j] == white) {
c[j] = grey;
dfs(g, j, c);
}
}
c[i] = black; // done visiting i
}


An example of the execution of this algorithm is shown in Figure $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: An example of depth-first-search starting at node 0. Nodes are labelled with the order in which they are processed. Edges that result in a recursive call are drawn in black, other edges are drawn in $$\mathtt{grey}$$.

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 explicit stack, $$\mathtt{s}$$. The following implementation does just that:

    void dfs2(Graph g, int r) {
byte[] c = new byte[g.nVertices()];
Stack<Integer> s = new Stack<Integer>();
s.push(r);
while (!s.isEmpty()) {
int i = s.pop();
if (c[i] == white) {
c[i] = grey;
for (int j : g.outEdges(i))
s.push(j);
}
}
}


In the preceding code, when the next vertex, $$\mathtt{i}$$, is processed, $$\mathtt{i}$$ is coloured $$\mathtt{grey}$$ 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 $$\mathtt{dfs(g,r)}$$ and $$\mathtt{dfs2(g,r)}$$ are the same as that of $$\mathtt{bfs(g,r)}$$:

Theorem $$\PageIndex{2}$$

When given as input a Graph, $$\mathtt{g}$$, that is implemented using the AdjacencyLists data structure, the $$\mathtt{dfs(g,r)}$$ and $$\mathtt{dfs2(g,r)}$$ algorithms each run in $$O(\mathtt{n}+\mathtt{m})$$ time.

As with the breadth-first-search algorithm, there is an underlying tree associated with each execution of depth-first-search. When a node $$\mathtt{i}\neq \mathtt{r}$$ goes from $$\mathtt{white}$$ to $$\mathtt{grey}$$, this is because $$\mathtt{dfs(g,i,c)}$$ was called recursively while processing some node $$\mathtt{i'}$$. (In the case of $$\mathtt{dfs2(g,r)}$$ algorithm, $$\mathtt{i}$$ is one of the nodes that replaced $$\mathtt{i'}$$ on the stack.) If we think of $$\mathtt{i'}$$ as the parent of $$\mathtt{i}$$, then we obtain a tree rooted at $$\mathtt{r}$$. In Figure $$\PageIndex{2}$$, 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 $$\mathtt{i}$$ is coloured $$\mathtt{grey}$$, there exists a path from $$\mathtt{i}$$ to some other node $$\mathtt{j}$$ that uses only white vertices. Then $$\mathtt{j}$$ will be coloured first $$\mathtt{grey}$$ then $$\mathtt{black}$$ before $$\mathtt{i}$$ is coloured $$\mathtt{black}$$. (This can be proven by contradiction, by considering any path $$P$$ from $$\mathtt{i}$$ to $$\mathtt{j}$$.)

One application of this property is the detection of cycles. Refer to Figure $$\PageIndex{3}$$. Consider some cycle, $$C$$, that can be reached from $$\mathtt{r}$$. Let $$\mathtt{i}$$ be the first node of $$C$$ that is coloured $$\mathtt{grey}$$, and let $$\mathtt{j}$$ be the node that precedes $$\mathtt{i}$$ on the cycle $$C$$. Then, by the above property, $$\mathtt{j}$$ will be coloured $$\mathtt{grey}$$ and the edge $$\mathtt{(j,i)}$$ will be considered by the algorithm while $$\mathtt{i}$$ is still $$\mathtt{grey}$$. Thus, the algorithm can conclude that there is a path, $$P$$, from $$\mathtt{i}$$ to $$\mathtt{j}$$ in the depth-first-search tree and the edge $$\mathtt{(j,i)}$$ exists. Therefore, $$P$$ is also a cycle. Figure $$\PageIndex{3}$$: The depth-first-search algorithm can be used to detect cycles in $$G$$. The node $$\mathtt{j}$$ is coloured $$\mathtt{grey}$$ while $$\mathtt{i}$$ is still $$\mathtt{grey}$$. This implies that there is a path, $$P$$, from $$\mathtt{i}$$ to $$\mathtt{j}$$ in the depth-first-search tree, and the edge $$\mathtt{(j,i)}$$ implies that $$P$$ is also a cycle.