1. Graph Theory in Networks
Lecture 4, 9/9/04
EE 228A, Fall 2004
Rajarshi Gupta
University of California, Berkeley

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

3. Support quality of service
Goal
Support quality of service
for flows
over ad-hoc networks
Collaborators:
John Musacchio
Zhanfeng Jia
Prof. Jean Walrand
Lecture 4, 9/9/04

4. Ad-Hoc Networks No base station Multi-hop transmissions
Distributed and dynamic operations
Lecture 4, 9/9/04

5. Application Scenarios
Disaster Relief
Convention Center
Lecture 4, 9/9/04

6. 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

7. 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

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

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)
Lecture 4, 9/9/04

10. Conflict Graph Three Links: F1 + F2 <= C and F2 + F3 <= C
Two Links:
F1 + F2 <= C
Single Link:
F1 <= C L1
Interference
Radius L2 L3
Lecture 4, 9/9/04

11. Independent Set Solution
Construct Conflict Graph L1 L2 L3 L4 L5
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.
Lecture 4, 9/9/04

12. 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

13. 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

14. Conflict Graph Three Links: F1 + F2 <= C and F2 + F3 <= C
Two Links:
F1 + F2 <= C
Single Link:
F1 <= C L1
Interference
Radius L2
Alternatively:
F1 + F2 + F3 <= C L3
Lecture 4, 9/9/04

15. Row Constraints
Each row in the Conflict Graph incidence matrix yields a constraint
At Node 2: F2 + F1 <= C
At Node 1:
F1 + F2 + F3 + F4 + F5 <= 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
Lecture 4, 9/9/04

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

17. 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

18. Cliques Definitions Observe 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
Lecture 4, 9/9/04

19. Clique Constraints Identify All Maximal Cliques Write Constraints
{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]
Lecture 4, 9/9/04

20. 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’
Lecture 4, 9/9/04

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
Lecture 4, 9/9/04

22. 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
Lecture 4, 9/9/04

23. Extensions to Realistic Networks
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
Lecture 4, 9/9/04

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
Lecture 4, 9/9/04

25. 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

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
Lecture 4, 9/9/04

27. 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 4, 9/9/04

28. 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

29. 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

30. Choose Source Choose Destination Routing…
0.5 1
1.5 2
2.5
Choose Source
Choose Destination
Routing…
Click on bar to choose flow rate
0 kbps
1000 kbps
500 kbps
50%
100% 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Y position in km
X position in km
Lecture 4, 9/9/04

31. Click on bar to choose flow rate
Choose Destination
Choose Next Source
Click on bar to choose flow rate
Routing…
0 kbps
1000 kbps
500 kbps
Lecture 4, 9/9/04

32. Lecture 4, 9/9/04

33. Click on bar to choose flow rate
Choose Destination
Choose Next Source
Click on bar to choose flow rate
0 kbps
1000 kbps
500 kbps
Flow Rejected. Insufficient Resources
Lecture 4, 9/9/04

34. Overview Introduction and Motivation QoS in Ad-Hoc Networks
Model and Related Work
Row Constraints
Clique Constraints
Computing Cliques
Implementation of Algorithms
Simulations of b
Interference-based QoS Routing
Lecture 4, 9/9/04

35. Shortest Path Methods ?? 1-3 is widest path from node 1 to 3
Consider path from 1 to 5
Path : FA+FD+FE<=C, so f<=C/3
Path : FB+FC<=C, FC+FD<=C, FD+FE<=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
Lecture 4, 9/9/04

36. 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 4, 9/9/04

37. 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

38. SWP Tradeoffs (contd) Short Paths Wide 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

39. 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

40. 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 1 2 3 4 5 6 7 8
F23+F34+F45 <= avlbw
F45+F56+F67 <= avlbw
Lecture 4, 9/9/04

41. Row Constraints Keep everything same1 2 3 4 5 6 7 8
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
Lecture 4, 9/9/04

42. 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
Lecture 4, 9/9/04

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 4, 9/9/04