Graph Traversals
Graph Traversals
• For solving most problems on graphs
– Need to systematically visit all the vertices and edges of a graph
• Two major traversals
– Breadth-First Search (BFS)
– Depth-First Search(DFS)
BFS
• Starts at some source vertex s
• Discover every vertex that is reachable from s
• Also produces a BFS tree with root s and including all
reachable vertices
• Discovers vertices in increasing order of distance from s
– Distance between v and s is the minimum number of edges on a
path from s to v
• i.e. discovers vertices in a series of layers
BFS : vertex colors stored in color[]
• Initially all undiscovered: white
• When first discovered: gray
– They represent the frontier of vertices between discovered
and undiscovered
– Frontier vertices stored in a queue
– Visits vertices across the entire breadth of this frontier
• When processed: black
Additional info stored (some applications of BFS
need this info)
• pred[u]: points to the vertex which first discovered u
• d[u]: the distance from s to u
BFS(G=(V,E),s)
for each (u in V)
color[u] = white
d[u] = infinity
pred[u] = NIL
color[s] = gray
d[s] = 0
Q.enqueue(s)
while (Q is not empty) do
u = Q.dequeue()
for each (v in Adj[u]) do
if (color(v] == white) then
color[v] = gray //discovered but not yet processed
d[v] = d[u] + 1
pred[v] = u
Q.enqueue(v) //added to the frontier
color[u] = black //processed
Example
identified vertex
visited vertex
unexplored edge
discovery edge
cross edge
A
A
A
B
E
A
B
D
F
A
C
E
C
D
F
Chapter 12: Graphs
B
C
E
7
D
F
Example (cont.)
A
B
A
C
E
D
B
F
E
A
B
D
F
A
C
E
C
D
F
Chapter 12: Graphs
B
C
E
8
D
F
Example (cont.)
A
B
L2
A
C
E
D
B
C
F
E
C
E
F
A
A
B
D
D
F
Chapter 12: Graphs
B
C
E
9
D
F
Analysis
• Each vertex is enqued once and dequeued once : Θ(n)
• Each adjacency list is traversed once:
 deg( u )   ( m )
u V
• Total: Θ(n+m)
BFS Tree
• predecessor pointers after BFS is completed define an
inverted tree
– Reverse edges: BFS tree rooted at s
– The edges of the BFS tree are called : discovery (tree)
edges
– Remaining graph edges are called: cross edges
BFS and shortest paths
• Theorem: Let G=(V,E) be a directed or undirected graph,
and suppose BFS is run on G starting from vertex s.
During its execution BFS discovers every vertex v in V
that is reachable from s. Let δ(s,v) denote the number of
edges on the shortest path form s to v. Upon termination
of BFS, d[v] = δ(s,v) for all v in V.
DFS
• Start at a source vertex s
• Search as far into the graph as possible and backtrack
when there is no new vertices to discover
– recursive
color[u] and pred[u] as before
• color[u]
– Undiscovered: white
– Discovered but not finished processing: gray
– Finished: black
• pred[u]
– Pointer to the vertex that first discovered u
Time stamps,
• We store two time stamps:
– d[u]: the time vertex u is first discovered (discovery time)
– f[u]: the time we finish processing vertex u (finish time)
DFS(G)
for each (u in V) do
color[u] = white
pred[u] = NIL
time = 0
for each (u in V) do
if (color[u] == white)
DFS-VISIT(u)
DFS-VISIT invoked by
each vertex exactly once.
Θ(V+E)
DFS-VISIT(u)
//vertex u is just discovered
color[u] = gray
time = time +1
d[u] = time
for each (v in Adj[u]) do
//explore neighbor v
if (color[v] == white) then
//v is discovered by u
pred[v] = u
DFS-VISIT(v)
color[u] = black // u is finished
f[u] = time = time+ 1
DFS Forest
• Tree edges: inverse of pred[] pointers
– Recursion tree, where the edge
(u,v) arises when processing a
vertex u, we call DFS-VISIT(v)
Remaining graph edges are classified as:
• Back edges: (u, v) where v is an
ancestor of u in the DFS forest (self
loops are back edges)
• Forward edges: (u, v) where v is a
proper descendent of u in the DFS
forest
• Cross edges: (u,v) where u and v are
not ancestors or descendents of one
another.
s
A
B
D
If DFS run on an undirected graph
• No difference between back and forward edges: all
called back edges
• No cross edges: can you see why?
Parenthesis Theorem
• In any DFS of a graph G= (V,E), for any two vertices u and
v, exactly one of the following three conditions holds:
– The intervals [d[u],f[u]] and [d[v],f[v]] are entirely disjoint and
neither u nor v is a descendent of the other in the DFS forest
– The interval [d[u],f[u]] is contained within interval [d[v],f[v]]
and u is a descendent of v in the DFS forest
– The interval [d[v],f[v]] is contained within interval [d[u],f[u]]
and v is a descendent of u in the DFS forest
How can we use DFS to determine whether a
graph contains any cycles?