graph theory in networks
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Graph Theory in Networks. Lecture 4, 9/9/04 EE 228A, Fall 2004 Rajarshi Gupta University of California, Berkeley. 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 - PowerPoint PPT PresentationTRANSCRIPT
Graph Theory in Networks
Lecture 4, 9/9/04EE 228A, Fall 2004
Rajarshi Gupta
University of California, Berkeley
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
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 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
3
Goal
Support quality of service for flows
over ad-hoc networks
Collaborators: John Musacchio Zhanfeng Jia Prof. Jean Walrand
Lecture 4, 9/9/04
EE 228A, Fall 2004
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Ad-Hoc Networks
No base station Multi-hop transmissions Distributed and dynamic operations
Lecture 4, 9/9/04
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Application Scenarios
Disaster Relief
Convention Center
Lecture 4, 9/9/04
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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
Interference-based QoS Routing
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
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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 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
8 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 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
9 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 4, 9/9/04
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Rajarshi Gupta
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Single Link:F1 <= C
Two Links:F1 + F2 <= C
Three Links:F1 + F2 <= C
andF2 + F3 <= C
Conflict Graph
L3
L2
L1
Interference Radius
L1
L2
L3
ConflictGraph:
Lecture 4, 9/9/04
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Rajarshi Gupta
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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.
Lecture 4, 9/9/04
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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
Interference-based QoS Routing
Lecture 4, 9/9/04
EE 228A, Fall 2004
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Single Link:F1 <= C
Two Links:F1 + F2 <= C
Three Links:F1 + F2 <= C
andF2 + F3 <= C
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 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
16 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 4, 9/9/04
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 4, 9/9/04
EE 228A, Fall 2004
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18
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
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
21 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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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 unit disk graph
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
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
24 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
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Overview
Introduction and Motivation QoS in Ad-Hoc Networks
Model and Related Work Row Constraints Clique Constraints Computing Cliques [time ?] Implementation of Algorithms
Interference-based QoS Routing
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
26 Representing a Link by its Center
Approximate the interference of a link by a circle centred at mid-point
Since Ix > Tx, the extra area is small
S D
Interferencerange of S
Interferencerange of D
Interferencerange of link
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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 4, 9/9/04
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
Lecture 4, 9/9/04
EE 228A, Fall 2004
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Choose SourceChoose DestinationClick on bar to choose flow rateRouting…
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Choose Source, click on clear area to quit
Flow 1 from 32 to 3 at 298.9889 kbps
0 kbps 1000 kbps500 kbps
Choose Next SourceChoose DestinationClick on bar to choose flow rateRouting…
Lecture 4, 9/9/04
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ositi
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Choose Source, click on clear area to quit
Flow 1 from 32 to 3 at 298.9889 kbpsFlow 2 from 2 to 33 at 298.9889 kbps
Lecture 4, 9/9/04
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ositi
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Choose Source, click on clear area to quit
Flow 1 from 32 to 3 at 298.9889 kbpsFlow 2 from 2 to 33 at 298.9889 kbps
0 kbps 1000 kbps500 kbps
Choose Next SourceChoose DestinationClick on bar to choose flow rate
Flow Rejected. Insufficient Resources
Lecture 4, 9/9/04
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 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 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
37 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
Lecture 4, 9/9/04
EE 228A, Fall 2004
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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
Lecture 4, 9/9/04
EE 228A, Fall 2004
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Source Routing Heuristic
Link state exchange allows src to know Topology Available capacity on all links i
New flow (src, dest, bw) arrives Choose several candidate paths by source
routing Shortest Path (SP) SP compliment Shortest Feasible Path Shortest Widest Path (SWP) Weighted Path Cost (OSPF)
Send probe packets along each path Final path chosen and confirmed by destination
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
41 Distributed Path Evaluation
Paths compared via suitable (monotone) metrics
Probe packets Evaluate clique constraints
along path Check for violated constraints Accumulate path metric
Destination chooses amongst multiple viable paths
Once path confirmed, avlbw updated in network
F23+F34+F45 <= avlbwF45+F56+F67 <= avlbw
1
2
3
4
5
6
7
8
Lecture 4, 9/9/04
EE 228A, Fall 2004
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Row Constraints Keep everything same Except evaluate row
constraints along path
Guaranteed to find distributed schedule
Could be employed only for high priority flows F23 + F34 + F45 +
F56 + F67 <= avlbw
1
2
3
4
5
6
7
8
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EE 228A, Fall 2004
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Measurement-based Link state protocol used to compute cliques as before But measurement-based avlbw instead of clique-
based avlbw = Idle / (Transmitting + Listening + Noisy + Idle) Accounts for
distributed scheduling invisible interference
Cliques still used by probe packets to estimate effect of new flow on avlbw
e.g. new flow uses 3 links on my worst clique, so need 3 x flowbw
Once flow admitted, true effect recallibrated by avlbw measurements
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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)