Shortest Path
3/17/2005 2:08 AM
Outline and Reading (§6.4)
Reachability (§6.4.1)
Directed Graphs
„
BOS
„
ORD
Directed DFS
Strong connectivity
JFK
Transitive closure (§6.4.2)
SFO
LAX
„
DFW
The Floyd-Warshall Algorithm
MIA
Directed Acyclic Graphs (DAG’s) (§6.4.4)
„
Directed Graphs
Topological Sorting
1
Directed Graphs
2
E
Digraphs
Digraph Properties
C
A digraph is a graph
whose edges are all
directed
„
E
„
„
„
D
B
A
Directed Graphs
ics23
ics51
ics53
ics52
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
„
ics161
ics131
ics141
ics121
„
„
ics171
„
Directed Graphs
discovery edges
back edges
forward edges
cross edges
4
5
E
D
C
B
A directed DFS starting at a
vertex s determines the
vertices reachable from s
The good life
ics151
Directed Graphs
Directed DFS
Scheduling: edge (a,b) means task a must be
completed before b can be started
ics22
A
If G is simple, m < n(n-1).
If we keep in-edges and out-edges in separate
adjacency lists, we can perform listing of of
the sets of in-edges and out-edges in time
proportional to their size.
3
Digraph Application
ics21
Each edge goes in one direction:
Š Edge (a,b) goes from a to b, but not b to a.
C
one-way streets
flights
task scheduling
B
A graph G=(V,E) such that
Short for “directed graph”
Applications
„
D
Directed Graphs
A
6
1
Shortest Path
3/17/2005 2:08 AM
Strong Connectivity
Reachability
Each vertex can reach all other vertices
DFS tree rooted at v: vertices reachable
from v via directed paths
E
E
D
F
A
d
E
D
B
e
C
F
A
7
Directed Graphs
Strong Connectivity
Algorithm
a
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’.
„
e
a
g
c
a
G’:
g
c
d
Running time: O(n+m).
d
e
Directed Graphs
e
9
D
E
B
C
G
D
E
B
C
A
We can perform
DFS starting at
each vertex
„
A
{f,d,e,b}
Directed Graphs
Computing the
Transitive Closure
Transitive Closure
{a,c,g}
b
f
b
f
Directed Graphs
Maximal subgraphs 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).
b
f
If there’s a w not visited, print “no”.
Else, print “yes”.
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
8
Strongly Connected
Components
Pick a vertex v in G.
Perform a DFS from v in G.
„
b
f
B
Directed Graphs
„
g
c
C
C
A
a
D
10
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
G*
11
Directed Graphs
12
2
Shortest Path
3/17/2005 2:08 AM
Floyd-Warshall
Transitive Closure
Floyd-Warshall’s Algorithm
Idea #1: Number the vertices 1, 2, …, n.
Idea #2: Consider paths that use only
vertices numbered 1, 2, …, k, as
intermediate vertices:
Uses only vertices numbered 1,…,k
(add this edge if it’s not already in)
i
j
Uses only vertices
numbered 1,…,k-1
k
Uses only vertices
numbered 1,…,k-1
Directed Graphs
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
computed from Gk − 1
if Gk − 1.areAdjacent(vi, vk) ∧
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
Floyd-Warshall’s algorithm
numbers the vertices of G as
v1 , …, vn and computes a
series of digraphs G0, …, Gn
13
Floyd-Warshall Example
v7
Directed Graphs
14
Floyd-Warshall, Iteration 1
BOS
ORD
v4
ORD
JFK
v2
SFO
v1
v4
JFK
v2
v6
LAX
v6
SFO
DFW
LAX
v3
v1
MIA
DFW
v3
MIA
v5
v5
Directed Graphs
15
Floyd-Warshall, Iteration 2
v7
Directed Graphs
16
Floyd-Warshall, Iteration 3
BOS
ORD
v4
ORD
v2
v4
JFK
v2
v6
SFO
v1
v7
BOS
JFK
LAX
v7
BOS
v6
SFO
DFW
LAX
v3
v1
MIA
DFW
v3
MIA
v5
Directed Graphs
v5
17
Directed Graphs
18
3
Shortest Path
3/17/2005 2:08 AM
Floyd-Warshall, Iteration 4
v7
Floyd-Warshall, Iteration 5
v4
ORD
JFK
SFO
v1
JFK
v2
v6
LAX
v4
ORD
v2
v6
SFO
DFW
DFW
LAX
v3
v3
v1
MIA
MIA
v5
v5
Directed Graphs
19
Floyd-Warshall, Iteration 6
v7
Directed Graphs
20
Floyd-Warshall, Conclusion
BOS
JFK
JFK
v2
v6
SFO
v1
v4
ORD
v2
v7
BOS
v4
ORD
LAX
v7
BOS
BOS
v6
SFO
DFW
DFW
LAX
v3
v3
v1
MIA
MIA
v5
v5
Directed Graphs
21
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 a
task sequence that satisfies the
v2
precedence constraints
Theorem
A digraph admits a topological
v1
ordering if and only if it is a DAG
Directed Graphs
D
Topological Sorting
wake up
1
A typical student day
2
study computer sci.
C
A
DAG G
D
B
C
v4
E
22
Number vertices, so that (u,v) in E implies u < v
E
B
A
Directed Graphs
v5
4
7
play
nap
Topological
ordering of G
23
5
more c.s.
8
write c.s. program
9
make cookies
for professors
v3
3
eat
6
work out
10
sleep
11
dream about graphs
Directed Graphs
24
4
Shortest Path
3/17/2005 2:08 AM
Algorithm for Topological Sorting
Simulate the algorithm by using
depth-first search
Note: This algorithm is different than the
one in Goodrich-Tamassia
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
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)
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
Topological Sorting
Algorithm using DFS
O(n+m) time.
25
Topological Sorting Example
Directed Graphs
26
Topological Sorting Example
9
Directed Graphs
27
Topological Sorting Example
Directed Graphs
28
Topological Sorting Example
7
8
8
9
Directed Graphs
9
29
Directed Graphs
30
5
Shortest Path
3/17/2005 2:08 AM
Topological Sorting Example
Topological Sorting Example
6
6
7
5
7
8
8
9
9
Directed Graphs
31
Topological Sorting Example
Directed Graphs
32
Topological Sorting Example
3
4
6
4
6
5
7
5
7
8
8
9
9
Directed Graphs
33
Topological Sorting Example
Directed Graphs
34
Topological Sorting Example
2
2
3
1
3
4
6
4
6
5
7
5
7
8
8
9
Directed Graphs
9
35
Directed Graphs
36
6