Graph Theory in Networks

1
Graph Theory in Networks

• Lecture 3, 9/7/04
• EE 228A, Fall 2004
• Rajarshi Gupta
• University of California, Berkeley

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Plan for Graph Segment

• Lecture 3 Tue (Sep 7, 2004)
• Paths and Routing
• Cycles and Protection
• Matching and Switching
• Lecture 4 Thu (Sep 9, 2004)
• Coloring and Capacity
• Trees and Broadcast, Multicast
• Lecture 5 Tue (Sep 14, 2004)
• Complete example Capacity in Ad-Hoc Networks
• Lectures 9 10
• Student Presentations (have you signed up ?)

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Network looks like Graph !
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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

5
Routing Dijkstra

• Dijkstra, E.W., A Note on Two Problems in
  Connexion with Graphs, Numerische Mathematik,
  vol. 1, pp. 269-271, 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

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Dijkstra Review

• Initialize
• while
• end while

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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)

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Principle of Optimality

• Partial paths of SP are also shortest
• Proof
• Let Ps,,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 gt Cost p,j,q
• Then, Ps,,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

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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 5

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Optimality Counter-example
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Dijkstra Complexity

• Initialize
• while
• end while

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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(n2m)
• Can improve loopmin (using binary heap) to give
  O(nlognm)

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Other Routing Algorithms

• Bellman Ford
• single source shortest path, negative wts
• O(mn)
• Floyd Warshall
• all pairs shortest path
• O(n3)
• Johnson
• all pairs shortest path
• O(n(mnlogn)) obvious for ve edge weights
• All-pairs shortest path reqd to set up routing
  tables

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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

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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 (s1) to n
• next j
• Complexity
• We only check each edge once, so O(m)

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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

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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 ?

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P-Cycle
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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

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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

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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

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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

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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
  boys 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.

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Vertex/Node Cover

• Definition
• 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

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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)

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Eulers Formula

• For any convex polyhedron, V-EF2
• V Vertices
• E Edges
• F Faces
• Examples
• Tetrahedron V4, E6, F4
• Cube V8, E12, F6
• Octahedron V6, E12, F8
• Dodecahedron V20, E30, F12
• Icosahedron V12, E30, F20
• BuckyBall V60, E90, F32

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Proof of Eulers Formula

• Proof by induction
• If no edges, its an isolated vertex. So V1, E0,
  F1
• 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.

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Multicast Trees

• Application Want to send packets from a source
  (e.g. internet radio station) to many hosts
• Sending n copies of the stream is wasteful
• Solution Form tree rooted at source that spans
  all member nodes

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Distributed Algorithm

• Need to know address of group
• First Join
• First member sends route request
• Request will be forwarded all the way to source
• Confirmation flows back to member
• All routers in path will add entry (mc_addr
  port)
• Subsequent Join
• New member send route request
• Request forwarded to router that has mc_addr
  entry
• Send confirmation from this point on
• Augment route entry (mc_addr port1, port2)
• Think Optimal multicast tree

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Multicast Algorithm (contd)

• Leave/Pruning
• Node leaving sends leave message to router
• Router removes port from route entry
• If last port entry deleted, send prune message to
  parent
• Multi-source multicast tree
• Less optimal packets between two branches travel
  long distance
• But need to minimize state in routers

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Lessons from Todays Lecture

• Path algorithms useful for routing
• Prove correctness
• Principle of Optimality allows distributedness
• Matching useful in Switching etc
• Key Analyses
• Correctness
• Optimality
• Complexity

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Papers to read

• Coloring (one of the two)
• H. Luo, S, Lu, and V. Bhargavan, A New Model for
  Packet Scheduling in Multihop Wireless Networks,
  Proceedings ACM Mobicom 2000, pp.76-86.
• 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, 55-64.
• 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 2309-2326,
  December 1998.

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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

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Words of Wisdom (1)

• Graph Theory is an old field
• Konigsberg Bridge problem Euler 1736
• Lots of famous people
• 18th and 19th Century Euler, Hamilton, Gauss
• 20th Century Erdos, Lovasz, Harary
• It is a very deep field. Takes years to get to
  cutting edge
• Replete in terminology names for everything !
• hole, cut, tree, forest, cycle, walk, degree,
  girth, face, cover, heredity, clique,
  independence, animal

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Words of Wisdom (2)

• Searching for results in Graph Theory is tricky
• E.g. Clique Algorithms yielded nice list of
  referrences 150 pages long !!!
• Need to have some understanding of Graph Theory
  to be able to utilize its results
• Take IEOR 266 Network Flows and Graphs