Directed Graphs
BOS
ORD
JFK
SFO
DFW
LAX
MIA
Directed Graphs
1
Outline and Reading (§6.4)
Reachability (§6.4.1)


Directed DFS
Strong connectivity
Transitive closure (§6.4.2)

The Floyd-Warshall Algorithm
Directed Acyclic Graphs (DAG’s) (§6.4.4)

Topological Sorting
Directed Graphs
2
Digraphs
A digraph is a graph
whose edges are all
directed



D
Short for “directed graph”
C
Applications

E
one-way streets
flights
task scheduling
B
A
Directed Graphs
3
E
Digraph Properties
D
C
B
A graph G=(V,E) such that

Each edge goes in one direction:
 Edge (a,b) goes from a to b, but not b to a.
A
If G is simple, m < n(n-1).
Directed Graphs
4
Digraph Application
Scheduling: edge (a,b) means task a must be
completed before b can be started
ics21
ics22
ics23
ics51
ics53
ics52
ics161
ics131
ics141
ics121
ics171
The good life
ics151
Directed Graphs
5
Directed DFS
We can specialize the
traversal algorithms (DFS
and BFS) to digraphs by
traversing edges only along
their direction
In the directed DFS
algorithm, we have four
types of edges




discovery edges
back edges
forward edges
cross edges
E
D
C
B
A directed DFS starting at a
vertex s determines the
vertices reachable from s
We say that u reaches v (and v
is reachable from u) if there is
a directed path from u to v
Directed Graphs
A
6
Directed DFS
Back edge (v,w)

w is an ancestor of v in the
E
tree of discovery edges
Forward edge (v,w)

v is an ancestor (but not the
parent) of w in the tree of
discovery edges
Cross edge (v,w)

w is in the same level as v
or in the next level in the
tree of discovery edges
Discovery edge (v,w)

v is the parent of w in the
tree of discovery edges
Directed Graphs
D
C
B
A
7
Reachability
DFS tree rooted at v: vertices reachable
from v via directed paths
E
E
D
C
A
D
C
F
A
E
B
D
C
A
Directed Graphs
F
B
8
Strong Connectivity
A digraph G is strongly connected if, for any two
vertices u and v of G, u reaches v and v reaches u.
Each vertex reaches all other vertices
a
g
c
d
e
b
f
Directed Graphs
9
Strong Connectivity
Algorithm
Pick a vertex v in G.
Perform a DFS from v in G.

G:

g
c
If there’s a w not visited, print “no”.
d
Let G’ be G with edges reversed.
Perform a DFS from v in G’.

a
e
b
f
If there’s a w not visited, print “no”.
Else, print “yes”.
a
G’:
g
c
d
Running time: O(n+m).
e
b
f
Directed Graphs
10
Strongly Connected
Components
A maximal subgraph such that each vertex can reach
all other vertices in the subgraph.
Can also be done in O(n+m) time using DFS, but is
more complicated (similar to biconnectivity).
a
g
c
d
e
{a,c,g}
{f,d,e,b}
b
f
Directed Graphs
11
Transitive Closure
Given a digraph G, the
transitive closure of G is the
digraph G* such that
 G* has the same vertices
as G
 if G has a directed path
from u to v (u  v), G*
has a directed edge from
u to v
The transitive closure
provides reachability
information about a digraph
Directed Graphs
D
E
B
C
G
A
D
E
B
C
A
G*
12
Computing the
Transitive Closure
We can perform
DFS starting at
each vertex

If there's a way to get
from A to B and from
B to C, then there's a
way to get from A to C.
O(n(n+m))
Alternatively ... Use
dynamic programming:
the Floyd-Warshall
Algorithm
Directed Graphs
13
Floyd-Warshall
Transitive Closure
Idea #1: Number the vertices 1, 2, …, n.
Idea #2: Consider paths that use only
vertices numbered 1, 2, …, k, as
intermediate vertices:
i
Uses only vertices numbered 1,…,k
(add this edge if it’s not already in)
j
Uses only vertices
numbered 1,…,k-1
k
Uses only vertices
numbered 1,…,k-1
Directed Graphs
14
Floyd-Warshall’s Algorithm
Floyd-Warshall’s algorithm
numbers the vertices of G as
v1 , …, vn and computes a
series of digraphs G0, …, Gn
Algorithm FloydWarshall(G)
Input digraph G
Output transitive closure G* of G
i1
for all v  G.vertices()
 G0=G
denote v as vi
 Gk has a directed edge (vi, vj)
ii+1
if G has a directed path from
G0  G
vi to vj with intermediate
for k  1 to n do
vertices in the set {v1 , …, vk}
Gk  Gk - 1
We have that Gn = G*
for i  1 to n (i  k) do
for j  1 to n (j  i, k) do
In phase k, digraph Gk is
if Gk - 1.areAdjacent(vi, vk) 
computed from Gk - 1
Gk - 1.areAdjacent(vk, vj)
Running time: O(n3),
if Gk.areAdjacent(vi, vj)
assuming areAdjacent is O(1)
Gk.insertDirectedEdge(vi, vj , k)
(e.g., adjacency matrix)
return Gn
Directed Graphs
15
Floyd-Warshall Example
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
16
Floyd-Warshall, Iteration 1
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
17
Floyd-Warshall, Iteration 2
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
18
Floyd-Warshall, Iteration 3
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
19
Floyd-Warshall, Iteration 4
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
20
Floyd-Warshall, Iteration 5
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
21
Floyd-Warshall, Iteration 6
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
22
Floyd-Warshall, Conclusion
v7
BOS
ORD
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Directed Graphs
23
DAGs and Topological Ordering
A directed acyclic graph (DAG) is a
digraph that has no directed cycles
A topological ordering of a digraph
is a numbering
v1 , …, vn
of the vertices such that for every
edge (vi , vj), we have i < j
Example: in a task scheduling
digraph, a topological ordering is a
task sequence that satisfies the
precedence constraints
Theorem
A digraph has a topological
ordering if and only if it is a DAG
D
E
B
C
DAG G
A
v2
v1
Directed Graphs
D
B
C
A
v4
E
v5
v3
Topological
ordering of G
24
Topological Sorting
Number vertices, so that (u,v) in E implies u < v
wake up
1
A typical student day
3
2
study computer sci.
eat
4
7
play
nap
5
more c.s.
8
write c.s. program
9
make cookies
for professors
6
work out
10
sleep
11
dream about graphs
Directed Graphs
25
Algorithm for Topological Sorting
Note: This algorithm is different than the
one in Goodrich-Tamassia
Method TopologicalSort(G)
HG
// Temporary copy of G
n  G.numVertices()
while H is not empty do
Let v be a vertex with no outgoing edges
Label v  n
nn-1
Remove v from H
Running time: O(n + m). How…?
Directed Graphs
26
Topological Sorting
Algorithm using DFS
Simulate the algorithm by using
depth-first search
Algorithm topologicalDFS(G)
Input dag G
Output topological ordering of G
n  G.numVertices()
for all u  G.vertices()
setLabel(u, UNEXPLORED)
for all e  G.edges()
setLabel(e, UNEXPLORED)
for all v  G.vertices()
if getLabel(v) = UNEXPLORED
topologicalDFS(G, v)
O(n+m) time.
Algorithm topologicalDFS(G, v)
Input graph G and a start vertex v of G
Output labeling of the vertices of G
in the connected component of v
setLabel(v, VISITED)
for all e  G.incidentEdges(v)
if getLabel(e) = UNEXPLORED
w  opposite(v,e)
if getLabel(w) = UNEXPLORED
setLabel(e, DISCOVERY)
topologicalDFS(G, w)
else
{e is a forward or cross edge}
Label v with topological number n
nn-1
Directed Graphs
27
Topological Sorting Example
Directed Graphs
28
Topological Sorting Example
9
Directed Graphs
29
Topological Sorting Example
8
9
Directed Graphs
30
Topological Sorting Example
7
8
9
Directed Graphs
31
Topological Sorting Example
6
7
8
9
Directed Graphs
32
Topological Sorting Example
6
5
7
8
9
Directed Graphs
33
Topological Sorting Example
4
6
5
7
8
9
Directed Graphs
34
Topological Sorting Example
3
4
6
5
7
8
9
Directed Graphs
35
Topological Sorting Example
2
3
4
6
5
7
8
9
Directed Graphs
36
Topological Sorting Example
2
1
3
4
6
5
7
8
9
Directed Graphs
37