graph theory in networks lecture 5, 9/14/04 ee 228a, fall 2004 rajarshi gupta university of...
Post on 21-Dec-2015
216 views
TRANSCRIPT
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
2
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
3
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
Rajarshi Gupta
4
Goal
Support quality of service for flows
over ad-hoc networks
Collaborators: John Musacchio Zhanfeng Jia Prof. Jean Walrand
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
5
Ad-Hoc Networks
No base station Multi-hop transmissions Distributed and dynamic operations
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
6
Application Scenarios
Disaster Relief
Convention Center
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
7
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
EE 228A, Fall 2004
Rajarshi Gupta
8
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
12
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
31
45
A
CB
E
Dinterference
Connectivity Graph
2
31
45
A
CB
E
D
Connectivity Graph
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
13
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:
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
14
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
15
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
16
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
EE 228A, Fall 2004
Rajarshi Gupta
17
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
18
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
20
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
EE 228A, Fall 2004
Rajarshi Gupta
21
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
22
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
23
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
24
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
26
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
27
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.
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
28
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
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
30
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
EE 228A, Fall 2004
Rajarshi Gupta
31
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)
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
32
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
Rajarshi Gupta
33
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
EE 228A, Fall 2004
Rajarshi Gupta
34
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
50%
100%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2122
23
24
25
26
27
28
29
30
31
32
33
3435
36
3738
3940
41
42
43
44
45
46
47
48
49
50
X position in km
Y p
ositi
on in
km
0 kbps 1000 kbps500 kbps
Choose SourceChoose DestinationClick on bar to choose flow rateRouting…
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
35
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
50%
100%
0%
1
2
34
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2122
23
24
25
26
27
28
29
30
31
32
33
3435
36
3738
3940
41
42
43
44
45
46
47
48
49
50
X position in km
Y p
ositi
on in
km
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
36
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
50%
100%
0%
1
2
34
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2122
23
24
25
26
27
28
29
30
31
32
33
3435
36
3738
3940
41
42
43
44
45
46
47
48
49
50
X position in km
Y p
ositi
on in
km
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
37
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
50%
100%
0%
1
2
34
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2122
23
24
25
26
27
28
29
30
31
32
33
3435
36
3738
3940
41
42
43
44
45
46
47
48
49
50
X position in km
Y p
ositi
on in
km
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
38
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
39
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
EE 228A, Fall 2004
Rajarshi Gupta
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)
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
41
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
42
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 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
43
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)
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
44
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
Lecture 5, 9/14/04
EE 228A, Fall 2004
Rajarshi Gupta
45
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.