1. Graphs
CS 2606

2. Topics Graph notation revisited More graph problems Undirected graph
Weighted/labeled graph
More graph problems
Euler Tour
Hamiltonian cycle
Max Clique
Traveling salesman
Graphs
Slide 2

3. Undirected Graph Graph G=(V,E):
V={a,b,c,d}, E={(a,b),(b,c),(b,d),(a,d)} a b c d
Graphs
Slide 3

4. Directed Graph Edge set is a set of ordered pairs Graph G=(V,E):
V={a,b,c,d}, E={(a,b),(b,c),(b,d),(d,b),(d,a)} a b c d
Graphs
Slide 4

5. Weighted Graph Graph G=(V,E,w):
V={a,b,c,d,e}, E={(a,b),(a,d),(b,c),(b,d),(b,e)}, w: function that maps vertex-pairs to values (can be stored in adjacency matrix)
w(a,b) = w(b,a) = 20 w(a,d) = w(d,a) = 65 w(b,c) = w(c,b) = 32 w(b,d) = w(d,b) = 12 w(b,e) = w(e,b) = 35
for all other vertex pairs: w(x,y) = 0 if x = y else, w(x,y) = infinity a 20 35 e b 12 32 c 65 d
Graphs
Slide 5

6. Graph problems Euler tour Hamiltonian cycle Max Clique
Find a sequence of vertices that traverse all edges in the graph exactly once
Hamiltonian cycle
Find a simple cycle that connects all vertices in the graph
Max Clique
Find the largest subset of vertices that are all pair-wise adjacent (connected by an edge)
Graphs
Slide 6

7. Graph problems 2 Shortest Paths Minimum Cost Spanning Tree
Section 11.4 of textbook
Minimum Cost Spanning Tree
Section 11.5 of textbook
Traveling Salesman
Given a weighted graph G, find the shortest (minimum-length) cycle that contains all vertices in G
Graphs
Slide 7

8. Euler Tour
Euler Tour/Cycle: sequence of vertices that traverse all edges exactly once c d a b f e
Euler tour: c,d,f,e,a,b,e,c
Graphs
Slide 8

9. f,c,d and e have odd degree
Euler Tour (2)
Some graphs do not have euler tours
specifically those with vertices whose degree (# of neighbors) is odd c a b d e g f
f,c,d and e have odd degree
Graphs
Slide 9

10. Euler Tour (3)
There is a straightforward algorithm that obtains an euler tour of a graph, if one such tour exists
Algorithm sketch
Start with any vertex; traverse edges by repeatedly visiting neighbors without repeating an edge
Repeat process, combining the partial tours as you go, until all edges are exhausted
Graphs
Slide 10

11. hamiltonian cycle: a,b,c,d,f,e,(then back to a)
Hamiltonian cycle: simple cycle containing all vertices w c a b z d x y f e
no hamiltonian cycle
hamiltonian cycle: a,b,c,d,f,e,(then back to a)
Graphs
Slide 11

12. Hamiltonian Cycle (2)
Exhaustive (inefficient) algorithm that finds a hamiltonian cycle if one such cycle exists:
Consider all permutations of the vertices
Return the permutation that forms a cycle
An O( n! n ) algorithm
Hamiltonian cycle is an example of an NP-complete (intractable) problem
No algorithm that performs better (than the exhaustive algorithm) is known
Graphs
Slide 12

13. Max Clique
Clique: subset of vertices that are all connected by an edge
Clique examples:
{a,b,e} {e,g} {b,c,d,f}
Max-clique: {b,c,d,f} c a b d e f g h
Graphs
Slide 13

14. Max Clique (2) Exhaustive algorithm for Max Clique:
Consider all subsets of vertices
Determine if the subset is a clique
Return the clique with the largest size
An O( 2n n2 ) algorithm
Max Clique is also NP-Complete
Graphs
Slide 14

15. w,x,y,z,w – length: 34 y,w,x,z,y – length: 36 z,w,y,x,z – length: 28
Traveling Salesman
Find shortest tour of all vertices (may impose that the salesman end where he started) w
Sample tours
w,x,y,z,w – length: 34 y,w,x,z,y – length: 36 z,w,y,x,z – length: 28 4 10 14 y 3 7 z x 11
Graphs
Slide 15

16. Traveling Salesman (2) O( n! n ) algorithm:
Consider all permutations
Compute resulting length for each permutation
Return the permutation that yields the shortest length
Traveling salesman is also NP-complete
Graphs
Slide 16

17. More on NP-Completeness
See chapter 15 of the textbook
Graphs
Slide 17