1. Graph Theory in Networks Lecture 5, 9/14/04 EE 228A, Fall 2004 Rajarshi Gupta University of California, Berkeley

2. Rajarshi Gupta 2 Lecture 5, 9/14/04 EE 228A, Fall 2004 Announcements Pass-NoPass requirements Only class presentations (~2 per student) No Projects Register Want to get student involvement 2 presentations in semester is small cost for knowledge ;-) Presentations Evaluation by Instructor, Peers Also evaluated on Evaluations Graph – next week. Need to sign up today.

3. Rajarshi Gupta 3 Lecture 5, 9/14/04 EE 228A, Fall 2004 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 7 & 8 Student Presentations (have you signed up ?)

4. Rajarshi Gupta 4 Lecture 5, 9/14/04 EE 228A, Fall 2004 Goal Support quality of service for flows over ad-hoc networks Collaborators: John Musacchio Zhanfeng Jia Prof. Jean Walrand

5. Rajarshi Gupta 5 Lecture 5, 9/14/04 EE 228A, Fall 2004 Ad-Hoc Networks No base station Multi-hop transmissions Distributed and dynamic operations

6. Rajarshi Gupta 6 Lecture 5, 9/14/04 EE 228A, Fall 2004 Application Scenarios Disaster ReliefConvention Center

7. Rajarshi Gupta 7 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Interference-based QoS Routing

8. Rajarshi Gupta 8 Lecture 5, 9/14/04 EE 228A, Fall 2004 QoS for Flows Want to support flows with quality (bandwidth) requirements Aspects of the problem Maximum capacity in a network Feasibility of a given set of flows Available capacity once flows are assigned Routing a given set of flows

9. Rajarshi Gupta 9 Lecture 5, 9/14/04 EE 228A, Fall 2004 Random vs Arbitrary Network Capacity of ad-hoc networks Random/homogenous topology, traffic matrix Asymptotic bounds on capacity Our Approach Arbitrary topology, traffic matrix Graph theoretic model Feasibility of given set of flows Distributed, localized and dynamic algorithm Gupta+Kumar (2000), Grossglauser+Tse (2002), El Gamal et. al. (2003)

10. Rajarshi Gupta 10 Lecture 5, 9/14/04 EE 228A, Fall 2004 What’s the problem with ad-hoc networks ? Ans: Interference In wired networks, all links may be used simultaneously In Ad-Hoc networks, neighboring links interfere Interference Range (Ix) > Transmission Range (Tx)

11. Rajarshi Gupta 11 Lecture 5, 9/14/04 EE 228A, Fall 2004 Representing a Link by its Center Approximate the interference of a link by a circle centered at mid-point S Since Ix > Tx, the extra area is small Interference range of S D Interference range of D L Interference range of link L

12. Rajarshi Gupta 12 Lecture 5, 9/14/04 EE 228A, Fall 2004 Conflict Graph (CG) Every link in G is represented by a node in CG Edge in CG if the two links interfere

13. Rajarshi Gupta 13 Lecture 5, 9/14/04 EE 228A, Fall 2004 Single Link: F 1 <= C Two Links: F 1 + F 2 <= C Three Links: F 1 + F 2 <= C and F 2 + F 3 <= C Constraints on Conflict Graph L3L3 L2L2 L1L1 Interference Radius L1L1 L2L2 L3L3 Conflict Graph:

14. Rajarshi Gupta 14 Lecture 5, 9/14/04 EE 228A, Fall 2004 Independent Set Solution Identify All Maximal Independent Sets {L 1, L 3 } L1L1 L2L2 L3L3 L4L4 L5L5, {L 1, L 4 } {L 2, L 4 }, {L 2, L 5 }, {L 3, L 5 } Write Constraints such that Only one Independent Set “on” at a time QoS requirements met for flow at each link “A New Model for Packet Scheduling in Multihop Wireless Networks”, H. Luo, S. Lu, and V. Bhargavan, ACM Mobicom 2000. Construct Conflict Graph

15. Rajarshi Gupta 15 Lecture 5, 9/14/04 EE 228A, Fall 2004 Issues with Independent Sets Shown to be necessary and sufficient for existence of global feasible schedule But scales poorly Need centralized information Finding all maximal independent sets is exponential Takes 10’s of minutes for simple graph (<100 links) Want distributed and sufficient constraints that can be computed quickly in a large network "Impact of Interference on Multi-hop Wireless Network Performance”, K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, ACM Mobicom 2003.

16. Rajarshi Gupta 16 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Interference-based QoS Routing

17. Rajarshi Gupta 17 Lecture 5, 9/14/04 EE 228A, Fall 2004 Single Link: F 1 <= C Two Links: F 1 + F 2 <= C Three Links: F 1 + F 2 <= C and F 2 + F 3 <= C Constraints on Conflict Graph L3L3 L2L2 L1L1 Interference Radius L1L1 L2L2 L3L3 Conflict Graph: Alternatively: F 1 + F 2 + F 3 <= C

18. Rajarshi Gupta 18 Lecture 5, 9/14/04 EE 228A, Fall 2004 Row Constraints At Node 2: F 2 + F 1 <= C At Node 1: F 1 + F 2 + F 3 + F 4 + F 5 <= C Proved to be sufficient for existence of feasible schedule Often too pessimistic F 2 = F 3 = F 4 = F 5 = C possible Row constraints allow only F 2 = F 3 = F 4 = F 5 = C/4 Each row in the Conflict Graph incidence matrix yields a constraint

19. Rajarshi Gupta 19 Lecture 5, 9/14/04 EE 228A, Fall 2004 Sufficiency of Row Constraints: Proof Assume each weight F i is integral (else take ) where T is number of slots Transform CG  CG F Replace each node i with K i fully connected nodes Color this graph Each node will be scheduled for requisite number of slots Neighboring nodes will be scheduled for disjoint slots Need to achieve coloring in T colors/slots Greedy algorithm Color each node with smallest available color Can always find such a color since sum of colors of all neighbors (row constraints) < T

20. Rajarshi Gupta 20 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Interference-based QoS Routing

21. Rajarshi Gupta 21 Lecture 5, 9/14/04 EE 228A, Fall 2004 Cliques Observe Cliques in CG are local structures (IS are global) Only one node in a clique may be active at once Maximal Cliques: ABC, BCEF, CDF Definitions Clique = Complete Subgraph Maximal Clique = Clique not a subset of any other

22. Rajarshi Gupta 22 Lecture 5, 9/14/04 EE 228A, Fall 2004 Clique Constraints Identify All Maximal Cliques {L 1, L 2 }, {L 1, L 5 }, {L 2, L 3 }, {L 3, L 4 }, {L 4, L 5 } Write Constraints Only one member of a Clique can be on at once F 1 + F 2 <= C, F 1 + F 5 <= C,... Necessary conditions for a feasible schedule [MSR 2003] L1L1 L2L2 L3L3 L4L4 L5L5 Clique

23. Rajarshi Gupta 23 Lecture 5, 9/14/04 EE 228A, Fall 2004 Insufficiency of Clique Constraints But, clique constraints are not sufficient F 1 =F 2 =F 3 =F 4 =F 5 = C/2 satisfy clique constraints But, we see that only 2 of 5 nodes may be on at once F 1 =F 2 =F 3 =F 4 =F 5 = 2C/5 is the max possible allocation Sufficient only for ‘Perfect Graphs’ L1L1 L2L2 L3L3 L4L4 L5L5

24. Rajarshi Gupta 24 Lecture 5, 9/14/04 EE 228A, Fall 2004 Unit Disk Graph (UDG) Need to introduce unit disk graph UDG = Graph in which two nodes have an edge between them if and only if their distance is less than 1 When we represent links by their mid-point, the CG is an UDG

25. Rajarshi Gupta 25 Lecture 5, 9/14/04 EE 228A, Fall 2004 Sufficiency using Cliques: Proof Equivalent weighted coloring problem Transform CG  CG F (as with Row Constraints) R eplace each node i by clique of size F i Color CG f with fewest colors Observe Schedule of a clique = color allocation for nodes in it Capacity of a clique = total number of colors used (T) Chromatic number Clique number is the largest clique in CG F

26. Rajarshi Gupta 26 Lecture 5, 9/14/04 EE 228A, Fall 2004 Duplication Recall Duplication Lemma Take vertex v in G. Add v’ s.t. v’ is neighbor to every neighbor of v. If G is perfect, then G’ is perfect Alternative version: Holds also if we connect v to v’ So by replacing a node by a clique, we do not change perfection of graph Also, duplication does not change UDG nature of graph

27. Rajarshi Gupta 27 Lecture 5, 9/14/04 EE 228A, Fall 2004 Imperfection Ratio is the ratio between the weighted Chromatic and Clique numbers Supremum over all weight (flow) vectors Bounded when the underlying graph is UDG Feasible schedule exists if scaled clique constraints are satisfied on a conflict graph Scale capacity of each link by So, Imperfection Ratio “Graph Imperfection I”, S. Gerke and C. McDiarmid, Journal of Combinatorial Theory, Series B, vol. 83 (2001), pp. 58-78.

28. Rajarshi Gupta 28 Lecture 5, 9/14/04 EE 228A, Fall 2004 Earlier results valid for CG that are UDG Variance in interference range Model interference range varying between [x,1] Then, need to scale the clique constraints by Obstructions in network Consider virtual CG V without obstructions Feasible schedule in CG V implies schedule in CG Satisfy scaled clique constraints in CG V Extensions to Realistic Networks

29. Rajarshi Gupta 29 Lecture 5, 9/14/04 EE 228A, Fall 2004 Constraint-based Algorithms Background Computation Local link state exchange (position, flows) Distributedly compute maximal cliques in CG Constraint-based approach Check sufficiency with row constraints Estimate capacity using scaled clique constraints Useful for Admission Control Clustering Routing

30. Rajarshi Gupta 30 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Interference-based QoS Routing

31. Rajarshi Gupta 31 Lecture 5, 9/14/04 EE 228A, Fall 2004 Computing Cliques General algorithms are centralized and exponential Propose computationally simple heuristic approximation (for ad-hoc networks) Key observations for an interference CG All links sharing cliques with this link must lie within a circle of radius Ix (interference range) All links that lie within a circle of diameter Ix must form a clique Harary+Ross (1957), Bierstone (1960s), Augustson et. al. (1970), Bron+Kerbosch (1973)

32. Rajarshi Gupta 32 Lecture 5, 9/14/04 EE 228A, Fall 2004 Heuristic Clique Algorithm Use a disk of radius Ix/2 to scan a disk of radius Ix around link Each position of scanning disk generates a clique Heuristically shrink set of cliques Only remember previous clique Check containment Can further shrink to set of maximal cliques Brute force check against all existing cliques

33. Rajarshi Gupta 33 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Interference-based QoS Routing

34. Rajarshi Gupta 34 Lecture 5, 9/14/04 EE 228A, Fall 2004 X position in km Y position in km 0 kbps1000 kbps500 kbps Choose Source Choose DestinationClick on bar to choose flow rateRouting…

35. Rajarshi Gupta 35 Lecture 5, 9/14/04 EE 228A, Fall 2004 0 kbps1000 kbps500 kbps Choose Next SourceChoose DestinationClick on bar to choose flow rateRouting…

36. Rajarshi Gupta 36 Lecture 5, 9/14/04 EE 228A, Fall 2004

37. Rajarshi Gupta 37 Lecture 5, 9/14/04 EE 228A, Fall 2004 0 kbps1000 kbps500 kbps Choose Next Source Choose DestinationClick on bar to choose flow rate Flow Rejected. Insufficient Resources

38. Rajarshi Gupta 38 Lecture 5, 9/14/04 EE 228A, Fall 2004 Overview Introduction and Motivation QoS in Ad-Hoc Networks Model and Related Work Row Constraints Clique Constraints Computing Cliques Implementation of Algorithms Simulations of 802.11b Interference-based QoS Routing

39. Rajarshi Gupta 39 Lecture 5, 9/14/04 EE 228A, Fall 2004 Shortest Path Methods ?? 1-3 is widest path from node 1 to 3 Consider path from 1 to 5 Path 1-3-4-5: F A +F D +F E <=C, so f<=C/3 Path 1-2-3-4-5: F B +F C <=C, F C +F D <=C, F D +F E <=C, so f<=C/2 Violates Bellman’s principle of optimality Does not conform to distributed algorithm extending path hop by hop Distributed algorithm unlikely to be optimal Work with distributed heuristic algorithms

40. Rajarshi Gupta 40 Lecture 5, 9/14/04 EE 228A, Fall 2004 Ad-Hoc Shortest Widest Path Recall Lec 2: distributed SWP is sub-optimal Solution At each node, remember every possible combination of path length and width Exponential algoritm :-( Approximation Remember a few sets of optimal paths ASWP (remembers only best set) 2-ASWP (remembers two)  -ASWP (optimal solution)

41. Rajarshi Gupta 41 Lecture 5, 9/14/04 EE 228A, Fall 2004 SWP Tradeoffs Width vs Resource utilization Denote width of a path as the max flow possible on that path When introducing a new flow, clearly width  -ASWP  4-ASWP  2-ASWP  ASWP  SP But consider resources utilized by path. Then,  -ASWP  4-ASWP  2-ASWP  ASWP  SP  -ASWP may not be best in the long run

42. Rajarshi Gupta 42 Lecture 5, 9/14/04 EE 228A, Fall 2004 SWP Tradeoffs (contd) Short Paths Take least resources Tend to crowd middle of network Wide Paths Use up too much resources Computation intensive Turns out (simulations) that ASWP is typically good enough to provide long term benefits

43. Rajarshi Gupta 43 Lecture 5, 9/14/04 EE 228A, Fall 2004 Lessons from this lecture Important to model critical phenomenon as appropriate graph (CG) Map physical behavior to graph feature Utilize graph theory and results – Cliques, IS Opens up many other related avenues, e.g. routing (ASWP)

44. Rajarshi Gupta 44 Lecture 5, 9/14/04 EE 228A, Fall 2004 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

45. Rajarshi Gupta 45 Lecture 5, 9/14/04 EE 228A, Fall 2004 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.