Part4 graph algorithms

Part IV
Graph Algorithms

Saturday, April 30, 2011 1

1

Undirected Graphs

Undirected graph. G = (V, E)
V = nodes.
!

E = edges between pairs of nodes.
!

Captures pairwise relationship between objects.
!

Graph size parameters: n = |V|, m = |E|.
!

V={1,2,3,4,5,6,7,8}
V = { 1, 2, 3, 4, 5, 6, 7, 8 }
E = {E={(1,2),(1,3),(2,3),...,(7,8)}3-8, 4-5, 5-6 }
1-2, 1-3, 2-3, 2-4, 2-5, 3-5, 3-7,
n=8
n=8
m = 11
m=11

2

Directed Graphs

hs Directed graph. G = (V, E)
! Edge (u, v) goes from node u to node v.

Ex. Web graph - hyperlink points from one web page to a
! Directedness of graph is crucial.
! Modern web search engines exploit hyperlink structur
pages by importance.

Graph Representation: Adjacency Matrix

Adjacency matrix. n-by-n matrix with Auv = 1 if (u, v) is an edge.
!Two representations of each edge.
!Space proportional to n2.
!Checking if (u, v) is an edge takes !(1) time.
!Identifying all edges takes !(n2) time.
1 if (i, j) ∈ E,
A = (aij ), where aij =
0 otherwise.
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 1 0 0 0
3 1 1 0 0 1 0 1 1
4 0 1 0 1 1 0 0 0
5 0 1 1 1 0 1 0 0
6 0 0 0 0 1 0 0 0
7 0 0 1 0 0 0 0 1
8 0 0 1 0 0 0 1 0

7 8


Paths and Connectivity

Def. A path in an undirected graph G = (V, E) is a sequence P of nodes
v1, v2, …, vk-1, vk with the property that each consecutive pair vi, vi+1 is
joined by an edge in E.

Def. A path is simple if all nodes are distinct.

Def. An undirected graph is connected if for every pair of nodes u and
v, there is a path between u and v.

10


Breadth First Search

tion. Explore outward from s in all possible directions, adding
layer at a time.

s L1 L2 L
ithm. n-1

s }.
neighbors of L0.
nodes that do not belong to L0 or L1, and that have an edge
de in L1.
ll nodes that do not belong to an earlier layer, and that have
e to a node in Li.

For each i, Li consists of all nodes at distance exactly i
here is a path from s to t iff t appears in some layer.

Saturday, April 30, 2011 18 10


t T be a BFS tree of G = (V, E), and let (x, y) be an edge of
evel of x and y differ by at most 1.
s=1
√ 1. insert the start node into queue

a queue
1

L0

L1

L2


evel of x and y differ by at mostremove a node from queue
2. 1.
s=1
√ 3. insert neighbors of it into the queue

√ √

1

2 3
L0

L1

L2


2. 1.
s=1

√ √

3
√ √
2

3 4 5
L0

L1

L2


2. 1.
s=1

√ √

4 5
√ √
3
√

4 5 6
L0

L1

L2

94
96

Figure 3.2 Exploring a graph is rather like navigating themaze. Figure 3.2.
begin here Figure 3.4 The result of explore(A) on a graph of

L A
K
D A B E B D F
B

E F G

G H C F I J C
I
J C E H

J
K L

Figure 3.5 Depth-ﬁrst search. I A
procedure dfs(G)

for all v ∈ V :
visited(v) = false

What parts of the graph are reachable from a given vertex?
for all v ∈ V :
Saturday, April 30, 2011 if not visited(v): explore(v) 20

➙

➙ ➙


Def. A graph is strongly connected if every pair of nodes i
reachable.

Lemma. Let s be any node. G is strongly connected iff eve
reachable from s, and s is reachable from every node.

Pf. ! Follows from definition.
Pf. Path from u to v: concatenate u-s path with s-v path
Path from v to u: concatenate v-s path with s-u pat

ok if paths over

s u

v


Strong Connectivity: Algorithm

Theorem. Can determine if G is strongly connected in O(m + n) time.
Pf.
! Pick any node s.
! Run BFS from s in G. reverse orientation of every edge in G

! Run BFS from s in Grev.
! Return true iff all nodes reached in both BFS executions.
! Correctness follows immediately from previous lemma. !

strongly connected not strongly connected

37


strongly connected not strongly connected

37

Directed Acyclic Graphs

Def. An DAG is a directed graph that contains no directed cycles. Pr

Ex. Precedence constraints: edge (vi, vj) means vi must precede vj. Ap
!

Def. A topological order of a directed graph G = (V, E) is an ordering !

of its nodes as v1, v2, …, vn so that for every edge (vi, vj) we have i j.

v2 v3

v6 v5 v4 v1 v2 v3 v4 v5 v6 v7

v7 v1

a DAG a topological ordering

39


O(n+m) with adjacent list


b
b ee

a SCC
d
d g
c
c h
h ff


b
b ee

a
d
d g
c
c h
h ff

more than one SCC


a SCC
another SCC

YES !!

The SCCs partition nodes of graph.


b
b ee

a
d
d g
c
c h
h ff


bb
ee C2
C1

a
a

d
d g
g
C4
c
c
h C3
h
f f


b
b ee C2
C1

a
d
d g C4
C3
c
c h
h ff


13/14 11/16 1/10 8/9

12/15 3/4 2/7 5/6


13/14 11/16 1/10 8/9

start here

12/15 3/4 2/7 5/6


13/14 11/16 1/10 8/9

12/15 3/4 2/7 5/6

start here


13/14 11/16 1/10 8/9

12/15 3/4 2/7 5/6


Part4 graph algorithms

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