Graph Theory in Networks

Lecture 5, 9/14/04EE 228A, Fall 2004

Rajarshi Gupta

University of California, Berkeley

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

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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.

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

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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 7 & 8

Student Presentations (have you signed up ?)

Lecture 5, 9/14/04

EE 228A, Fall 2004

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Goal

Support quality of service for flows

over ad-hoc networks

Collaborators: John Musacchio Zhanfeng Jia Prof. Jean Walrand

Lecture 5, 9/14/04

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Ad-Hoc Networks

No base station Multi-hop transmissions Distributed and dynamic operations

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Application Scenarios

Disaster Relief

Convention Center

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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

Lecture 5, 9/14/04

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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

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

9 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)

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

10 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)

InterferenceRange

TransmissionRange

Station A

Station D

Station C

Station B

Link 2

Link 1

InterferenceRange

TransmissionRange

Station A

Station D

Station C

Station B

Link 2

Link 1

TransmissionRange

Station A

Station D

Station C

Station B

Link 2

Link 1

TransmissionRange

Station A Station BLink 1

TransmissionRange

Station A

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

11 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

Interferencerange of D

L

Interferencerange of link L

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Conflict Graph (CG)

Every link in G is represented by a node in CG

Edge in CG if the two links interfere

2

31

45

A

CB

E

Dinterference

E

CD

B

A

Connectivity Graph Conflict Graph

2

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A

CB

E

Dinterference

Connectivity Graph

2

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A

CB

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D

Connectivity Graph

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Single Link:F1 <= C

Two Links:F1 + F2 <= C

Three Links:F1 + F2 <= C

andF2 + F3 <= C

Constraints on Conflict Graph

L3

L2

L1

Interference Radius

L1

L2

L3

ConflictGraph:

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Independent Set Solution

Identify All Maximal Independent Sets

{L1, L3}

L1

L2

L3

L4L5

, {L1, L4}

{L2, L4} , {L2, L5} , {L3, L5}

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

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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.

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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

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Single Link:F1 <= C

Two Links:F1 + F2 <= C

Three Links:F1 + F2 <= C

andF2 + F3 <= C

Constraints on Conflict Graph

L3

L2

L1

Interference Radius

L1

L2

L3

ConflictGraph:

Alternatively:F1 + F2 + F3

<= C

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Row Constraints

At Node 2: F2 + F1 <= C At Node 1:

F1 + F2 + F3 + F4 + F5 <= C

L2

L4

L1 L3L5

Proved to be sufficient for existence of feasible schedule

Often too pessimistic F2 = F3 = F4 = F5 = C possible Row constraints allow only F2 = F3 = F4 = F5 = C/4

Each row in the Conflict Graph incidence matrix yields a constraint

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

19 Sufficiency of Row Constraints: Proof

Assume each weight Fi is integral (else take ) where T is number of slots

Transform CG CGF Replace each node i with Ki 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

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

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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

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Cliques

Observe Cliques in CG are local

structures (IS are global) Only one node in a clique

may be active at once

A

B C

E F

D

Maximal Cliques:

ABC, BCEF, CDF

Definitions Clique = Complete

Subgraph Maximal Clique =

Clique not a subset of any other

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Clique Constraints

Identify All Maximal Cliques {L1, L2}, {L1, L5} , {L2, L3}, {L3, L4}, {L4, L5}

Write Constraints Only one member of a Clique can be on at once

F1+ F2 <= C, F1+ F5 <= C, ...

Necessary conditions for a feasible schedule [MSR 2003]

L1

L2

L3

L4L5

Clique

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Insufficiency of Clique Constraints

But, clique constraints are not sufficient F1=F2=F3=F4=F5 = C/2 satisfy clique constraints But, we see that only 2 of 5 nodes may be on at once F1=F2=F3=F4=F5 = 2C/5 is the max possible allocation

Sufficient only for ‘Perfect Graphs’

L1

L2

L3

L4L5

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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

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

25 Sufficiency using Cliques: Proof

Equivalent weighted coloring problem Transform CG CGF (as with Row Constraints)

Replace each node i by clique of size Fi

Color CGf 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 CGF

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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

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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.

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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 CGV without obstructions Feasible schedule in CGV implies schedule in CG Satisfy scaled clique constraints in CGV

Extensions to Realistic Networks

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29 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

Lecture 5, 9/14/04

EE 228A, Fall 2004

Rajarshi Gupta

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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

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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)

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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

Lecture 5, 9/14/04

EE 228A, Fall 2004

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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

Lecture 5, 9/14/04

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Choose SourceChoose DestinationClick on bar to choose flow rateRouting…

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Flow 1 from 32 to 3 at 298.9889 kbps

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Choose Next SourceChoose DestinationClick on bar to choose flow rateRouting…

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Flow 1 from 32 to 3 at 298.9889 kbpsFlow 2 from 2 to 33 at 298.9889 kbps

Lecture 5, 9/14/04

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Choose Next SourceChoose DestinationClick on bar to choose flow rate

Flow Rejected. Insufficient Resources

Lecture 5, 9/14/04

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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

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Shortest Path Methods ?? 1-3 is widest path from

node 1 to 3 Consider path from 1

to 5 Path 1-3-4-5:

FA+FD+FE<=C, so f<=C/3

Path 1-2-3-4-5: FB+FC<=C, FC+FD<=C, FD+FE<=C, so f<=C/2

2

31

45

A

CB

E

Dinterference

E

CD

B

A

Connectivity Graph Conflict Graph

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

Lecture 5, 9/14/04

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40 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)

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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

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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

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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)

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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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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.