1. Algorithms for Ad Hoc Networks
2005
Algorithms for Ad Hoc Networks
Roger Wattenhofer
MedHocNet 2005

2. Distributed Algorithms vs. Ad Hoc Networking
Small community
O(…), (…), (…)
Everybody knows best paper
New algorithm: Compare it with the best previous
Sometimes study the wrong problem; propose protocols that are way too complicated
Big community
Milliseconds
Everybody knows first* paper
New protocol: Compare it with the first that was proposed
Reinvent the wheel; many papers do not offer any progress
Roger Wattenhofer, ETH MedHocNet 2005

3. Algorithmic Research in Ad Hoc and Sensor Networking
Link Layer
Network Layer
Services
Theory/Models
Clustering (Dominating Sets, etc.)
MAC Layer and Coloring
Topology and Power Control
Interference and Signal-to-Noise-Ratio
Deployment (Unstructured Radio Networks)
New Routing Paradigms (e.g. Link Reversal)
Geo-Routing
Broadcast and Multicast
Data Gathering
Location Services and Positioning
Time Synchronization
Modeling and Mobility
Lower Bounds for Message Passing
Selfish Agents, Economic Aspects, Security
Roger Wattenhofer, ETH MedHocNet 2005

4. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Ad Hoc and Sensor Networks
Routing / Broadcasting
Clustering
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

5. Routing in Ad Hoc Networks
Multi-Hop Routing
Moving information through a network from a source to a destination if source and destination are not within mutual transmission range
Reliability
Nodes in an ad-hoc network are not 100% reliable
Algorithms need to find alternate routes when nodes are failing
Mobile Ad-Hoc Network (MANET)
It is often assumed that the nodes are mobile
Roger Wattenhofer, ETH MedHocNet 2005

6. Simple Classification of Ad hoc Routing Algorithms
Proactive Routing
Small topology changes trigger a lot of updates, even when there is no communication  does not scale
Reactive Routing
Flooding the whole network does not scale
Distance Vector Routing:
as in a fixnet nodes
maintain routing tables
using update messages
Flooding:
when node received
message the first time,
forward it to all neighbors
no mobility
critical mobility
mobility very high
Source Routing (DSR, AODV):
flooding, but re-use old routes
Roger Wattenhofer, ETH MedHocNet 2005

7. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Discussion
Lecture “Mobile Computing”: 10 Tricks  210 routing algorithms
In reality there are almost that many!
Q: How good are these routing algorithms?!? Any hard results?
A: Almost none! Method-of-choice is simulation…
Perkins: “if you simulate three times, you get three different results”
Flooding is key component of (many) proposed algorithms
At least flooding should be efficient
Roger Wattenhofer, ETH MedHocNet 2005

8. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

9. Finding a Destination by Flooding
Roger Wattenhofer, ETH MedHocNet 2005

10. Finding a Destination Efficiently
Roger Wattenhofer, ETH MedHocNet 2005

11. (Connected) Dominating Set
A Dominating Set DS is a subset of nodes such that each node is either in DS or has a neighbor in DS.
A Connected Dominating Set CDS is a connected DS, that is, there is a path between any two nodes in CDS that does not use nodes that are not in CDS.
It might be favorable to have few nodes in the (C)DS. This is known as the Minimum (C)DS problem.
Roger Wattenhofer, ETH MedHocNet 2005

12. Formal Problem Definition: M(C)DS
Input: We are given an (arbitrary) undirected graph.
Output: Find a Minimum (Connected) Dominating Set, that is, a (C)DS with a minimum number of nodes.
Problems
M(C)DS is NP-hard
Find a (C)DS that is “close” to minimum (approximation)
The solution must be local (global solutions are impractical for mobile ad-hoc network) – topology of graph “far away” should not influence decision who belongs to (C)DS
Roger Wattenhofer, ETH MedHocNet 2005

13. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

14. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Algorithm Overview
Input:
Local Graph
Fractional
Dominating Set
Dominating
Set
Connected
Dominating Set
0.2
0.2
0.5
0.3
0.3
0.8
0.5
0.2
0.1
Phase A:
Distributed
linear program
rel. high degree
gives high value
Phase B:
Probabilistic
algorithm
Phase C:
Connect DS
by “tree” of
“bridges”
Roger Wattenhofer, ETH MedHocNet 2005

15. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

16. Phase A is a Distributed Linear Program
Nodes 1, …, n: Each node u has variable xu with xu ¸ 0
Sum of x-values in each neighborhood at least 1 (local)
Minimize sum of all x-values (global)
= 1.5 ¸ 1
Linear Programs can be solved optimally in polynomial time
But not in a distributed fashion! That’s what we do here…
Linear Program
0.2
0.2
0.5
0.3
0.3
0.8
0.5
0.2
0.1
Adjacency matrix
with 1’s in diagonal
Roger Wattenhofer, ETH MedHocNet 2005

17. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Phase A Algorithm
Roger Wattenhofer, ETH MedHocNet 2005

18. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Result after Phase A
Distributed Approximation for Linear Program
Instead of the optimal values xi* at nodes, nodes have xi(), with
The value of  depends on the number of rounds k (the locality)
Roger Wattenhofer, ETH MedHocNet 2005

19. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

20. Dominating Set as Integer Program
What we have after phase A
What we want after phase B
Roger Wattenhofer, ETH MedHocNet 2005

21. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Phase B Algorithm
Each node applies the following algorithm:
Calculate (= maximum degree of neighbors in distance 2)
Become a dominator (i.e. go to the dominating set) with probability
Send status (dominator or not) to all neighbors
If no neighbor is a dominator, become a dominator yourself
From phase A
Highest degree in distance 2
Roger Wattenhofer, ETH MedHocNet 2005

22. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Result after Phase B
Randomized rounding technique
Expected number of nodes joining the dominating set in step 2 is bounded by  log(+1) ¢ |DSOPT|.
Expected number of nodes joining the dominating set in step 4 is bounded by |DSOPT|.
Theorem: E[|DS|] · O( ln  ¢ |DSOPT|)
Roger Wattenhofer, ETH MedHocNet 2005

23. Related Work on (Connected) Dominating Sets
Global algorithms
Johnson (1974), Lovasz (1975), Slavik (1996): Greedy is optimal
Guha, Kuller (1996): An optimal algorithm for CDS
Feige (1998): ln  lower bound unless NP 2 nO(log log n)
Local (distributed) algorithms
“Handbook of Wireless Networks and Mobile Computing”: All algorithms presented have no guarantees
Gao, Guibas, Hershberger, Zhang, Zhu (2001): “Discrete Mobile Centers” O(loglog n) time, but nodes know coordinates
MIS-based algorithms (e.g. Alzoubi, Wan, Frieder, 2002) that only work on unit disk graphs.
Kuhn, Wattenhofer (2003): Tradeoff time vs. approximation
Roger Wattenhofer, ETH MedHocNet 2005

24. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Recent Improvements
Improved algorithms (in submission):
O(log2 / 4) time for a (1+)-approximation of phase A with logarithmic sized messages.
If messages can be of unbounded size there is a constant approximation of phase A in O(log n) time, using the graph decomposition by Linial and Saks.
An improved and generalized distributed randomized rounding technique for phase B.
Works for quite general linear programs.
Is it any good…?
Roger Wattenhofer, ETH MedHocNet 2005

25. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

26. Lower Bound for Dominating Sets: Intuition…
Two graphs (m << n). Optimal dominating sets are marked red.
complete n n n …
n-1 m m m n
|DSOPT| = 2.
|DSOPT| = m+1.
Roger Wattenhofer, ETH MedHocNet 2005

27. Lower Bound for Dominating Sets: Intuition…
In local algorithms, nodes must decide only using local knowledge.
In the example green nodes see exactly the same neighborhood.
So these green nodes must decide the same way! …
n-1 m m n
Roger Wattenhofer, ETH MedHocNet 2005

28. Lower Bound for Dominating Sets: Intuition…
But however they decide, one way will be devastating (with n = m2)!
complete n n n …
n-1 m m m n
|DSOPT| = 2.
|DSOPT without green| ¸ m.
|DSOPT| = m+1.
|DSOPT with green| > n
Roger Wattenhofer, ETH MedHocNet 2005

29. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
The Lower Bound
Lower bounds (Kuhn, Moscibroda, PODC 2004):
Model: In a network/graph G (nodes = processors), each node can exchange a message with all its neighbors for k rounds. After k rounds, node needs to decide.
We construct the graph such that there are nodes that see the same neighborhood up to distance k. We show that node ID’s do not help, and using Yao’s principle also randomization does not.
Results: Many problems (vertex cover, dominating set, matching, etc.) can only be approximated (nc/k2 / k) and/or (1/k / k).
It follows that a polylogarithmic dominating set approximation (or maximal independent set, etc.) needs at least (log  / loglog ) and/or ((log n / loglog n)1/2) time.
Roger Wattenhofer, ETH MedHocNet 2005

30. Graph Used in Dominating Set Lower Bound
The example is for k = 3.
All edges are in fact special bipartite graphs with large enough girth.
Roger Wattenhofer, ETH MedHocNet 2005

31. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
A Theory of “Locality”?
Ad hoc and sensor networks
The largest network in the world?!?
Managing organizations? Society?!?
Matrix multiplication, etc.
Roger Wattenhofer, ETH MedHocNet 2005

32. A better and faster algorithm
Assume that nodes know their position (GPS)
Assume that nodes are in the plane; two nodes are within their transmission radius if and only if their Euclidean distance is at most 1 (UDG, unit disk graph) 1 u v
Roger Wattenhofer, ETH MedHocNet 2005

33. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Then…
half of tx radius
Roger Wattenhofer, ETH MedHocNet 2005

34. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Algorithm
Beacon your position
If, in your virtual grid cell, you are the node closest to the center of the cell, then join the CDS, else do not join.
That’s it.
1 transmission per node, O(1) approximation, even for CDS
If you have mobility, then simply “loop” through algorithm, as fast as your application/mobility wants you to.
Roger Wattenhofer, ETH MedHocNet 2005

35. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Comparison
First algorithm (distributed linear program)
Algorithm computes CDS
k2+O(1) transmissions/node
O(O(1)/k log ) approximation
General graph
No position information
Second algorithm (virtual grid)
Algorithm computes CDS
1 transmission/node
O(1) approximation
Unit disk graph (UDG)
Position information (GPS)
Roger Wattenhofer, ETH MedHocNet 2005

36. Let’s talk about models…
General Graph
Captures obstacles
Captures directional radios
Often too pessimistic
UDG & GPS
UDG is not realistic
GPS not always available
Indoors
2D  3D?
Often too optimistic
too pessimistic
too optimistic
Are there any models in
between these extremes?
Roger Wattenhofer, ETH MedHocNet 2005

37. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Models
UDG
No GPS
UDG
GPS
General
Graph
too pessimistic
too optimistic
Bounded
Growth
Unit Ball
Graph
Quasi
UDG
In a doubling
metric:
Number of
independent
neighbors
is bounded
(UDG: 5) 1 d
Roger Wattenhofer, ETH MedHocNet 2005

38. Another Algorithm 1: MIS
Build maximal independent set (MIS), then connect MIS for CDS
Proposed by many, patented(!) by Alzoubi et al.
A MIS is by definition also a DS
Connecting with independent 1- and 2-hop bridges
Slow! Works well only on UDGs; robust for general graphs
Roger Wattenhofer, ETH MedHocNet 2005

39. Another Algorithm 2: Election
Every node elects a leader; every elected node goes into DS
First analyzed by Jie Gao et al.
1 round of communication for DS only; lots of practical appeal
In the worst case very bad, even for UDGs only a √n approximation 9 2 6 8 5 4 1 7 3
Roger Wattenhofer, ETH MedHocNet 2005

40. Another Algorithm 3: Non-neighboring neighbors
If a node has neighbors who are not neighbors, join CDS
Proposed by Jie Wu et al.
Renders a CDS directly
Almost as bad as choosing all nodes, even for random UDGs
Only DS algorithm reviewed in several books
Lots of improvements, also proposed by Jie Wu et al. ?
Roger Wattenhofer, ETH MedHocNet 2005

41. Another Algorithm 4: Covering connected neighbors
If higher priority neighbors are connected and cover all other neighbors, then don’t join CDS, else join CDS
This talk, inspired by an improvement of Jie Wu
2 rounds of communication for CDS only; lots of practical appeal
In the worst case very bad, even for UDGs only a √n approximation
However, on random UDGs, this gives a O(1) approximation 9 2 6 8 5 4 1 7 3
Roger Wattenhofer, ETH MedHocNet 2005

42. ? Result Overview UDG = Unit Disk Graph UBG = Unit Ball Graph
GBG = Growth Bounded G.
/GPS = With Position Info
/D = With Distance Info
UDG5
quality √n
UDG67
General Graph2
better
Lower Bound for General Graphs9
log ?
loglog
GBG8
O(1)
UDG/GPS1
UBG/D3
UDG4 1 2
better
O(log*)
O(log)
tx / node
Roger Wattenhofer, ETH MedHocNet 2005

43. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
References
Folk theorem, e.g. Kuhn, Wattenhofer, Zhang, Zollinger, PODC 2003
Kuhn, Wattenhofer, PODC 2003; improvement submitted
CDS improvement by Dubhashi et al, SODA 2003
Kuhn, Moscibroda, Wattenhofer, PODC 2005
Alzoubi, Wan, Frieder, MobiHoc 2002
Wu and Li, DIALM 1999
Gao, Guibas, Hershberger, Zhang, Zhu, SCG 2001
This Talk, improving on Wu and Li
Kuhn, Moscibroda,Nieberg, Wattenhofer, submitted
Kuhn, Moscibroda, Wattenhofer, PODC 2004
Roger Wattenhofer, ETH MedHocNet 2005

44. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
More Models
Random Distribution
for all geometric models
“Infocom vs. PODC”
Related Problems
e.g. (Connected) Domatic Partition  Moscibroda et al., WMAN 2005
Facility Location  Moscibroda et al., PODC 2005
Weighted Graph Models
Signal-to-Interference-and-Noise-Ratio (SINR)
Communication Models
Message Size
Unstructured Radio Network (no established MAC layer)
Roger Wattenhofer, ETH MedHocNet 2005

45. Clustering for Unstructured Radio Networks
“Big Bang” (deployment) of a sensor and/or ad-hoc network:
Nodes wake up asynchronously (very late, maybe)
Neighbors unknown
Hidden terminal problem
No global clock
No established MAC protocol
No reliable collision detection
Limited knowledge of the number of nodes or degree of network.
We have randomized algorithms that compute DS (or MIS) in polylog(n) time even under these harsh circumstances, where n is an upper bound on the number of nodes in the system.
[Kuhn, Moscibroda, MobiCom 2004]
[Moscibroda, PODC 2005]
Roger Wattenhofer, ETH MedHocNet 2005

46. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Overview
Introduction
Clustering
Conclusions
Roger Wattenhofer, ETH MedHocNet 2005

47. Big Research Opportunities
Link Layer
Network Layer
Services
Theory/Models
Clustering (Dominating Sets, etc.)
MAC Layer and Coloring
Topology and Power Control
Interference and Signal-to-Noise-Ratio
Deployment (Unstructured Radio Networks)
New Routing Paradigms (e.g. Link Reversal)
Geo-Routing
Broadcast and Multicast
Data Gathering
Location Services and Positioning
Time Synchronization
Modeling and Mobility
Lower Bounds for Message Passing
Selfish Agents, Economic Aspects, Security
Roger Wattenhofer, ETH MedHocNet 2005

48. Check yourself: www.dcg.ethz.ch  Reading List…
Roger Wattenhofer, ETH MedHocNet 2005

49. Conclusions & Open Problems
You don’t have to do algorithms and proofs…
… but it would be good to be aware of them.
Open Problems and Research Directions
Fast good algorithm (for standard UDG) or new lower bound
Study problems for models in-between UDG and general graph
Mobility and dynamics
Study new models: e.g. SINR
Real implementations
If the networking community wants to make progress…
- The related work section should always give new insights! Try to see things in new light… try to uncover new related work
- Don’t have to do algorithms/analysis… but be aware of them
Roger Wattenhofer, ETH MedHocNet 2005

50. Roger Wattenhofer, ETH Zurich @ MedHocNet 2005
Questions? Comments?
Thank you for your attention
Roger Wattenhofer, ETH MedHocNet 2005