Graph Theory Chapter on Paths and Cycles

Paths and cycles
Shortest path problem
Traveling saleman problem
Other notions of graphs

Discrete mathematics I
Graph theory

Huynh Tuong Nguyen & Huynh Viet Linh

Faculty of Computer Science and Engineering
University of Technology, Ho Chi Minh City.
{htnguyen;vietlinh}@cse.hcmut.edu.vn

8 mai 2010

Discrete Math 1 N. Huynh Tuong, V.L. Huynh Chapter 7 - Paths and Cycles 1/54

Paths and cycles
Traveling saleman problem

1 Paths and cycles

2 Shortest path problem

3 Traveling saleman problem

4 Other notions of graphs


Paths and cycles
Shortest path problem Paths and cycles
Traveling saleman problem Connectivity

Paths and cycles

A path of length n from u to v , in an undirected graph is a sequence of
edges e1 , e2 , . . ., en such that f (e1 ) = (x0 , x1 ), f (e2 ) = (x1 , x2 ), . . .,
f (en ) = (xn−1 , xn ), where xi ∈ V and x0 = u, xn = v .
Simple graph case : the path is denoted by its vertex sequence x0 , x1 , . . .,
xn .
Path is a circuit (or cycle) if it begins and ends at the same vertex (i.e. if
u = v ).
Path or circuit is simple if it does not contain the same edge more than
once.


Paths and cycles

Paths and cycles

b
a

c
d

e f

Simple path

Cycle

Paths and cycles

Connectedness in undirected graph

An undirected graph is called connected if there is a path between every
pair of dictinct vertices of the graph.

∀u, v ∈ V , u = v , ∃ a path between u and v

Connected undirected graph ⇒ ∃ a simple path between every pair of
dictinct vertices.

b
a

c
d

e f


Paths and cycles

Connectedness in directed graph

b
a

c
d

e f

Cut edge - cut bridge

Cut vertex - articulation point


Paths and cycles

Connectedness in directed graph

An directed graph is strongly connected if there is a path a path between
any two vertices in the graph (for both directions).
An directed graph is weakly connected if there is a path between any two
vertices in the underlying undirected graph.

a b a b

c c

e d e d
Strongly connected Weakly connected


Paths and cycles

Question

How to prove that there is a path from u to v in a directed graph G which
traverses at most a given constant number of intermediate vertices.


Paths and cycles

Question

How to prove that there is a path from u to v in a directed graph G which
traverses at most a given constant number of intermediate vertices.

Application
Consider four cities V1 , V2 , V3 , V4 in a country where the air traﬃc is still very
reduced : there is only a through ﬂight from V1 to V2 and V4 , from V2 to V3 ,
from V3 to V1 and V4 , from V4 to V2 .


Paths and cycles

Question

 
0 1 0 1
 0 0 1 0 
M=
 
1 0 0 1 
0 1 0 0


Paths and cycles

Question

   
0 1 0 1 0 1 1 0
 0 0 1 0  2 = 1
 0 0 1 
M=
  M 
1 0 0 1   0 2 0 1 
0 1 0 0 0 0 1 0


Paths and cycles

Question

     
0 1 0 1 0 1 1 0 1 0 1 1
 0 0 1 0  2 = 1
 0 0 1  3 = 0
 2 1 0 
M=
  M  M 
1 0 0 1   0 2 0 1   0 1 2 0 
0 1 0 0 0 0 1 0 1 0 0 1


Paths and cycles

Couting paths between vertices

Let G be a graph with adjacency matrix A with respect to the
ordering v1 , v2 , . . ., vn (with directed or undirected edges, with
multiple edges and loops allowed).
The number of diﬀerent paths of length r from vi to vj , where
r is a positive integer, equals the (i, j)th entry of Ar .


Paths and cycles Dijkstra’s algorithm
Shortest path problem Bellman-Moore algorithm
Traveling saleman problem Floyd-Warshall algorithm
Other notions of graphs Ford’s algorithm


Gbps
Lang Son Da Nang Ca Mau
100Mbps
1bps
10Kbps 50Mbps
Ha Noi Ho Chi Minh
1Gbps

100$
20$ 40$
100$
Ha Noi Ho Chi Minh
200$

800 km
200 km 400 km
800 km
Ha Noi Ho Chi Minh
1300 km




Given a graph G = (V , E ), and a weighted function w : E → Z, the length of
path p = e1 , e2 , . . . , ek is calculated by w (p) = k=1 w (ei ).
i




Given a graph G = (V , E ), and a weighted function w : E → Z, the length of
path p = e1 , e2 , . . . , ek is calculated by w (p) = k=1 w (ei ).
i

Given P a set of all path from s to t. Shortest path problem from s to t is
deﬁned by minp∈P w (p).



Problem

The problem is also sometimes called the single-pair shortest path problem, to
distinguish it from the following generalizations :
The single-source shortest path problem, in which we have to find
shortest paths from a source vertex v to all other vertices in the graph.
The single-destination shortest path problem, in which we have to find
shortest paths from all vertices in the graph to a single destination vertex
v . This can be reduced to the single-source shortest path problem by
reversing the edges in the graph.
The all-pairs shortest path problem, in which we have to find shortest
paths between every pair of vertices v , v ′ in the graph.
These generalizations have significantly more efficient algorithms than the
simplistic approach of running a single-pair shortest path algorithm on all
relevant pairs of vertices.



Algorithms

The most important algorithms
Dijkstra’s algorithm solves the single-pair, single-source, and
single-destination shortest path problems.
Bellman-Ford algorithm solves single source problem if edge weights may
be negative. It can be applied for directed graph.
A* search algorithm solves for single pair shortest path using heuristics to
try to speed up the search.
Floyd-Warshall algorithm solves all pairs shortest paths.
Johnson’s algorithm solves all pairs shortest paths, and may be faster
than Floyd-Warshall on sparse graphs.
Perturbation theory ﬁnds (at worst) the locally shortest path.
Additional algorithms and associated evaluations may be found in
[Cherkassky et al.].



Dijkstra’s algorithm [1959]
a 3 c a 3 c

4 2 4 2
3 3
s t s t
2 1 2 1
3 3
b d b d
a 3 c a 3 c

4 2 4 2
3 3
s t s t
2 1 2 1
3 3
b d b d
a 3 c

4 2
3
s t
2 1
3
b d



Dijkstra’s algorithm [1959]

This algorithm is based on the labelling.
Some notations :
k : k-th iteration
ℓk (u, v ) : minimal distance from u to v after k-th iteration
Sk : set of vertices with the smallest labels in each iteration.

Important formula
ℓk (a, v ) = min{ℓk−1 (a, v ), min(u,v )∈E {ℓk−1 (a, u) + w (u, v )}}



Dijkstra’s algorithm

Input : weighted simple graph G // G has vertex s = v0 , v1 , . . . , vn = t
Ourput : shortest path from s to t
ℓ(vi ) = ∞, i = 1, . . . , n ;
ℓ(s) = 0 ; S = ∅ ;
While (t ∈ S) do
u = vertex ∈ S having smallest ℓ(u) ;
S = S ∪ {u} ;
For each (vertex v ∈ S) do
If (ℓ(u) + w (u, v ) < ℓ(v )) then
ℓ(v ) = ℓ(u) + w (u, v ) ;
end
end
end



Dijkstra’s algorithm

Property
any G , any length ℓ(vi ) ≥ 0, ∀i ; one-to-all ; complexity O(|V |2 ).



Example



Example

i 1 2 3 4 5 6



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞
3 9 10 5 +∞



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞
3 9 10 5 +∞
9 10 5 6



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞
3 9 10 5 +∞
9 10 5 6
9 8 6



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞
3 9 10 5 +∞
9 10 5 6
9 8 6
9 8



Example

i 1 2 3 4 5 6
π(i) 0 3 9 +∞ +∞ +∞
3 9 10 5 +∞
9 10 5 6
9 8 6
9 8
9


Exercise


Graph Theory Chapter on Paths and Cycles

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