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Graph Theory in Networks

- Lecture 5, 9/14/04
- EE 228A, Fall 2004
- Rajarshi Gupta
- University of California, Berkeley

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.

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 ?)

Goal

- Support quality of service
- for flows
- over ad-hoc networks

- Collaborators
- John Musacchio
- Zhanfeng Jia
- Prof. Jean Walrand

Ad-Hoc Networks

- No base station
- Multi-hop transmissions
- Distributed and dynamic operations

Application Scenarios

Disaster Relief

Convention Center

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

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

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

GuptaKumar (2000), GrossglauserTse (2002), El

Gamal et. al. (2003)

Whats 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) gt Transmission Range (Tx)

Representing a Link by its Center

- Approximate the interference of a link by a

circle centered at mid-point

- Since Ix gt Tx, the extra area is small

Conflict Graph (CG)

- Every link in G is represented by a node in CG
- Edge in CG if the two links interfere

Constraints on Conflict Graph

Three Links F1 F2 lt C and F2 F3 lt C

Single Link F1 lt C

Two Links F1 F2 lt C

Independent Set Solution

- Construct Conflict Graph

- Identify All Maximal Independent Sets
- L1, L3

, 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.

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 10s of minutes for simple graph (lt100

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.

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

Constraints on Conflict Graph

Three Links F1 F2 lt C and F2 F3 lt C

Single Link F1 lt C

Two Links F1 F2 lt C

Alternatively F1 F2 F3 lt C

Row Constraints

- Each row in the Conflict Graph incidence matrix

yields a constraint

- At Node 2 F2 F1 lt C
- At Node 1
- F1 F2 F3 F4 F5 lt C

- 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

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) lt T

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

Cliques

- Definitions
- Clique Complete Subgraph
- Maximal Clique Clique not a subset of any other

- 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

Clique Constraints

Clique

- 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 lt C, F1 F5 lt C, ...
- Necessary conditions for a feasible schedule MSR

2003

Insufficiency of Clique Constraints

- But, clique constraints are not sufficient
- F1F2F3F4F5 C/2 satisfy clique constraints
- But, we see that only 2 of 5 nodes may be on at

once - F1F2F3F4F5 2C/5 is the max possible

allocation - Sufficient only for Perfect Graphs

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

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

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

Imperfection Ratio

- 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,

Graph Imperfection I, S. Gerke and C.

McDiarmid, Journal of Combinatorial Theory,

Series B, vol. 83 (2001), pp. 58-78.

Extensions to Realistic Networks

- 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

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

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

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

HararyRoss (1957), Bierstone (1960s), Augustson

et. al. (1970), BronKerbosch (1973)

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

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

Choose Destination

Click on bar to choose flow rate

Routing

Choose Source

Y position in km

X position in km

Choose Next Source

Choose Destination

Click on bar to choose flow rate

Routing

(No Transcript)

Choose Next Source

Choose Destination

Click on bar to choose flow rate

Flow Rejected. Insufficient Resources

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

Shortest Path Methods ??

- 1-3 is widest path from node 1 to 3
- Consider path from 1 to 5
- Path 1-3-4-5 FAFDFEltC, so fltC/3
- Path 1-2-3-4-5 FBFCltC, FCFDltC, FDFEltC,

so fltC/2

- Violates Bellmans 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

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)

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

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

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)

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

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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