1. Graph Theory in Networks
Lecture 2, 9/2/04
EE 228A, Fall 2004
Rajarshi Gupta
University of California, Berkeley

2. Plan for Graph Segment Lecture 2 – Thu (Sep 2, 2004)
Paths and Routing
Cycles and Protection
Matching and Switching
Lecture 3 – Tue (Sep 7, 2004)
Coloring and Capacity
Trees and Broadcast, Multicast
Lecture 4 – Thu (Sep 9, 2004)
Complete example: Capacity in Ad-Hoc Networks
Lectures 8 & 9 – (Sep 23 & 28, 2004)
Student Presentations (have you signed up ?)
Lecture 2, 9/2/04

3. Network looks like Graph !
Lecture 2, 9/2/04

4. Using Graph Theory in Networking
Modeling
Appropriate underlying graph
Captures required behavior
Warning: slight difference entirely changes results
Analysis
Proofs/results from relevant graph theoretic technique
Reality Check
Does in make sense ?
Perhaps ignored some important feature
Lecture 2, 9/2/04

5. Routing: Dijkstra Given a graph G = (V, E) Applications
Dijkstra, E.W., “A Note on Two Problems in Connexion with Graphs,” Numerische Mathematik, vol. 1, pp , 1959.
Given a graph G = (V, E)
Finds weighted shortest path from a single source to all other nodes
Works with non-negative edge weights
Applications
OSPF routing (edge weight = 1)
Mapquest (edge weight = traversal time)
Many, many, many more …
Lecture 2, 9/2/04

6. Dijkstra Review
Initialize
while
end while
Lecture 2, 9/2/04

7. Proof of Correctness Need to show that this is correct !
Thm: Label v(i) is the shortest path cost to i
Lemma: Labels in P are the shortest path involving nodes in P. Then when P = V, the theorem holds
Proof of Lemma
Suppose we have a shorter path using node in T
Let be two nodes in path, with in between
But if
Then we would have picked t before q (Contradicton)
Lecture 2, 9/2/04

8. Principle of Optimality
Partial paths of SP are also shortest
Proof:
Let P={s,…,p,…i,…q,…d} be the SP from s to d
Suppose {p,…i,…q} as above is not SP from p to q i.e. Cost {p,…i,…q} > Cost {p,…j,…q}
Then, P’={s,…,p,…j,…q,…d} has lesser cost than P
So P is not the SP (contradiction)
Allows computing partial path and extending hop by hop
Lecture 2, 9/2/04

9. General Principle of Optimality
In general, hope that partial paths of “optimal” paths are also optimal. E.g.
Minimum Delay Paths
Widest path
If true, amenable to distributed algorithms
If not, often NP-hard 
E.g. Shortest Widest Path
Both types of cases found in networking
Will cover in Lecture 4
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10. Optimality Counter-example
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11. Dijkstra Complexity
Initialize
while
end while
Lecture 2, 9/2/04

12. Dijkstra Complexity (contd)
True metric of algorithm usefulness
n = # of nodes, m = # of edges
Dijkstra
Main loop runs n times
Finding min takes O(n)
Then, updating values is O(m)
So, total algo is O(n2+m)
Can improve loop+min (using binary heap) to give O(nlogn+m)
Lecture 2, 9/2/04

13. Other Routing Algorithms
Bellman Ford
single source shortest path, negative wts
O(mn)
Floyd Warshall
all pairs shortest path
O(n3)
Johnson
O(n(m+nlogn)) obvious for +ve edge weights
All-pairs shortest path reqd to set up routing tables
Lecture 2, 9/2/04

14. Directed Acyclic Graph (DAG)
Generalization of the tree concept to directed graphs
Special Features
Cannot have cycles
Always exists one node with in-degree 0 (Quick Proof)
Allows topological ordering of nodes
Applications
Tournaments, Games
Scheduling problems
Lecture 2, 9/2/04

15. SP in DAG
Due to lack of cycles, topological ordering of nodes (check up)
Can be done in O(m)
Correctness
Since no cycles, we have checked every possible path over the for loop
Algorithm
Initialize
for j = (s+1) to n
next j
Complexity
We only check each edge once, so O(m)
Lecture 2, 9/2/04

16. Protection Paths In many situations, we need two paths
Normal Path
Protection Path
Types of protection
Path Protection
Span Protection
Link Protection
Type of protection paths
Node disjoint
Link Disjoint
Lecture 2, 9/2/04

17. Min Cost Protection Paths
Common Algorithm
Find Shortest Path 
Make G’ = (V, E\)
Find shortest path ’ in G’
Advantages
Computationally simple
Two Dijkstra/BF
Shortest Normal Path, which is used much
Backup path used rarely
Problems
Cost (+’) may not be optimal
 may block off all possible ’
Lecture 2, 9/2/04

18. P-Cycle
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19. P-Cycle (contd) P-Cycle Algorithm
Want to find set of elementary cycles
Minimizes total capacity on these cycles
All working links must be protected
Integer programming solution
Followup: Given two nodes i,j, find min cost cycle in G that covers i and j
Lecture 2, 9/2/04

20. What are you Protecting ?
IP Layer
When modeling with a graph, remember
Present structure
Underlying behavior
In this case, unclear
Edge costs are IP layer costs
Protection guarantee is lower layer phenomenon
Perhaps need a different graph model
Optical Layer
Normal Path
Protection Path
Lecture 2, 9/2/04

21. Matching M  E s.t no two edges in M share a common end node
A maximum cardinality matching is matching with a maximum number of edges
Max/min weighted matching chooses edges with max weight
Of special interest are matchings in Bi-Partite Graphs
Lecture 2, 9/2/04

22. Matching in Bi-Partite Graph
Stable Marriage
Look at M men and N women, with partner preferences
For any pair Alice & Bob, one of following is true
Alice & Bob are married
Alice prefers her partner to Bob
Bob prefers his partner to Alice
No pair (A,B) and (C,D) such that (A,D) and (B,C) would both be preferred :-)
Switching !
N ports (input & output)
Effectively, N input and N output ports
Packets (w priority) are queued on input port
Each port ‘ranks’ its preferrence for output ports based on priority
Want optimal input-output match
Lecture 2, 9/2/04

23. Stable Marriage Problem
Stable Marriage always exists, for any preferences
Girl-oriented Algorithm
Girls successively make proposals to boys (engaged or not)
If a girl proposes to a boy who is free (not yet paired), the boy ‘accepts’ the proposal.
If the girl proposes to a boy who is already paired
If this proposal is a better proposal from the boy’s perspective, then the boy breaks the old proposal and accepts the new one.
If the proposal is not a better one, the boy rejects the proposal.
Key of Proof
Consider arbitrary g. If g prefers b’ to b, then b’ must have rejected g and been paired with some g’ whom he prefers to g. As choice of g is arbitrary, there are no unstable pairs.
Algorithm completes within N2 iterations and is ‘girl-optimal’.
Lecture 2, 9/2/04

24. Vertex Cover Definition Observe: Application
Set of nodes NC of G such that every edge of G has at least one node in NC
Observe:
‘Opposite’ of Matching
Max cardinality of matching is at most the min cardinality of node cover
Equal in a bi-partite graph
Application
Choose a min set of nodes in an ad-hoc network to exercise control over all the links
Lecture 2, 9/2/04

25. Min Cost Cycle Cover Problem: Cover all nodes with disjoint cycles
Want least total cost of these cycles
Formulate G’
Replicate V into U, W
Join ui to wj if (vi,vj)E
cost(ui,wj) = cost(vi,vj)
Observe
G’ is bipartite
Cycle Cover in G is perfect matching in G’
Therefore
Evaluate min cost perfect matching in G’
Polynomial time (bi-partite)
Lecture 2, 9/2/04

26. Euler’s Formula For any convex polyhedron, V-E+F=2 Examples
V = Vertices
E = Edges
F = Faces
Examples
Tetrahedron: V=4, E=6, F=4
Cube: V=8, E=12, F=6
Octahedron: V=6, E=12, F=8
Dodecahedron: V=20, E=30, F=12
Icosahedron: V=12, E=30, F=20
BuckyBall: V=60, E=90, F=32
Lecture 2, 9/2/04

27. Proof of Euler’s Formula
Proof by induction
If no edges, its an isolated vertex. So V=1, E=0, F=1
Else choose any edge
If it connects two vertices, contract it. This reduces V by 1 and E by 1
Else the edge must separate two faces (Jordan curve). Remove it. Reduces F by 1 and E by 1.
Lecture 2, 9/2/04

28. Lessons from Today’s Lecture
Path algorithms useful for routing
Prove correctness
Principle of Optimality allows distributedness
Matching useful in Switching etc
Key Analyses
Correctness
Optimality
Complexity
Lecture 2, 9/2/04

29. Papers to read Coloring (one of the two) Routing Matching
H. Luo, S, Lu, and V. Bhargavan, “A New Model for Packet Scheduling in Multihop Wireless Networks,” Proceedings ACM Mobicom 2000, pp
M. Kodialam, and T. Nandagopal, “Characterizing the Achievable Rates in Multihop Wireless Networks,” Proceedings ACM Mobicom 2003, San Diego, CA, September 2003.
Routing
M. Kodialam and T. Lakshman, “Minimum Interference Routing with Applications to MPLS Traffic Engineering,” Proceedings IEEE INFOCOM 2000.
S. Deering and D. Cheriton, "Multicast Routing in Internetworks and Extended LANs", SIGCOMM'88, Stanford, CA, Aug 1988,
Matching
Nick McKeown and Thomas E. Anderson, "A Quantitative Comparison of Scheduling Algorithms for Input-Queued Switches", Computer Networks and ISDN Systems, Vol 30, No 24, pp , December 1998.
Lecture 2, 9/2/04

30. References Graph Theory, by Frank Harary
Integer and Combinatorial Optimization, by G.L. Nemhauser and L.A. Wolsey
Network Flows: Theory, Algorithms and Applications by Ravindra K. Ahuja, Thomas L. Magnanti and James B. Orlin
Lecture 2, 9/2/04