DIGRAPHS
wake up
1
A typical student day
2
cs16 meditation
3
eat
4
work
7
play
8
cs16 program
9
make cookies
for cs16 HTA
10
sleep
Digraphs
5
more cs16
6
cxhextris
11
dream of cs16
1
What’s a Digraph?
a) A small burrowing animal with long sharp
teeth and a unquenchable lust for the blood of
computer science majors
b) A distressed graph
c) A directed graph
Each edge goes in one direction
Edge (a,b) goes from a to b, but not b to a
a
b
You’re saying, “Yo, how about an example of
how we might be enlightened by the use of
digraphs!!” − Well, if you insist. . .
Digraphs
2
Applications
Maps: digraphs handle one-way streets
(especially helpful in Providence)
Tunnel O’ Doom
D’Angelo’s!
Bookstore
Thomas J. Watson Jr.
Center for
Information
143
Technology
Store
24
Angell
SciLi
Waterman
Thayer
Brook
ME!
Digraphs
3
Another Application
Scheduling: edge (a,b) means task a must be
completed before b can be started
cs15
cs16
cs22
cs126
cs127
cs141
cs32
cs167
cs31
Old programmers never die they just fall into black holes
Digraphs
4
DAG’s
dag: (noun) dÂ-g
1. Di-Acyl-Glycerol − My favorite snack!
2.“man’s best friend”
person’s
3. directed acyclic graph
Say What?!
directed graph with no directed cycles
a
b
a
c
DAG
Digraphs
c
e
d
b
e
d
not a DAG
5
Depth-First Search
Same algorithm as for undirected graphs
On a connected digraph, may yield
unconnected DFS trees (i.e., a DFS forest)
a
b
c
f
e
d
a
e
f
b
d
c
Digraphs
6
Reachability
DFS tree rooted at v: vertices reachable from
v via directed paths
a
b
c
f
e
d
e
b
c
b
a
c
d
f
a
d
Digraphs
7
Strongly Connected Digraphs
Each vertex can reach all other vertices
a
g
c
d
f
Digraphs
e
b
8
Strongly Connected
Components
a
g
c
d
e
b
f
{a,c,g}
{f,d,e,b}
Digraphs
9
Transitive Closure
Digraph G* is obtained from G using the rule:
If there is a directed path in G from a to b,
then add the edge (a,b) to G*
G
Digraphs
G*
10
Computing the Transitive
Closure
We can perform DFS starting at each vertex
Time: O(n(n+m))
Alternatively ... Floyd-Warshall Algorithm:
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
Digraphs
11
Example
BOS
ORD
v7
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
BOS
ORD
v7
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Digraphs
12
Floyd-Warshall Algorithm
• this algorithms assumes that methods areAdjacent
and insertDirectedEdge take O(1) time (e.g.,
adjacency matrix structure)
Algorithm FloydWarshall(G)
let v1 ... vn be an arbitrary ordering of the vertices
G0 = G
for k = 1 to n do
// consider all possible routing vertices vk
Gk = Gk-1
for each (i, j = 1, ..., n) (i != j) (i, j != k) do
// for each pair of vertices vi and vj
if Gk-1.areAdjacent(vi,vk) and
Gk-1.areAdjacent(vk,vj) then
Gk.insertDirectedEdge(vi,vj,null)
return G0
• digraph Gk is the subdigraph of the transitive closure
of G induced by paths with intermediate vertices in
the set { v1, ..., vk }
• running time: O(n3)
Digraphs
13
Example
• digraph G
BOS
ORD
v7
v4
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Digraphs
14
Example
• digraph G*
BOS
v4
v7
ORD
JFK
v2
v6
SFO
DFW
LAX
v1
v3
MIA
v5
Digraphs
15
Topological Sorting
For each edge (u,v), vertex u is visited before
vertex v
1
A typical student day
wake up
2
cs16 meditation
3
eat
4
work
7
play
8
cs16 program
9
make cookies
for cs16 HTA
10
sleep
Digraphs
5
more cs16
6
cxhextris
11
dream of cs16
16
Topological Sorting
Topological sorting may not be
unique
A
B
C
ABCD
or
ACBD
D
− You make the call!
Digraphs
17
Topological Sorting
Labels are increasing along a directed path
A digraph has a topological sorting if and
only if it is acyclic (i.e., a dag)
1
A
2
3
B
C
4
D
Digraphs
5
E
18
Algorithm for Topological
Sorting
method TopologicalSort
if there are more vertices
let v be a source;
// a vertex w/o incoming edges
label and remove v;
TopologicalSort;
A
B
C
D
Digraphs
E
19
Algorithm (continued)
Simulate deletion of sources using indegree
counters
TopSort(Vertex v);
label v;
foreach edge(v,w)
indeg(w) = indeg(w) − 1;
if indeg(w) = 0
TopSort(w);
1. Compute indeg(v) for all vertices
2. Foreach vertex v do
if v not labeled and indeg(v) = 0
then TopSort(v)
Digraphs
20
Example
? 0
? 0
A
B
? 1
C
D
? 3
E
F
? 2
? 1
G
H
? 2
? 1
I
? 3
Digraphs
21
Reverse Topological Sorting
RevTopSort(Vertex v)
mark v;
foreach edge(v,w)
if v not marked
RevTopSort(w);
label v;
A
B
C
D
Digraphs
E
22