Algorithms for Ad Hoc Networks

1
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

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

4
Overview

• Introduction
• Ad Hoc and Sensor Networks
• Routing / Broadcasting
• Clustering
• Conclusions

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

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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

Flooding when node received message the first
time, forward it to all neighbors
Distance Vector Routing as in a fixnet
nodes maintain routing tables using update
messages
no mobility
mobility very high
critical mobility
Source Routing (DSR, AODV) flooding, but re-use
old routes
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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

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Overview

• Introduction
• Clustering
• Flooding vs. Dominating Sets
• Algorithm Overview
• Phase A
• Phase B
• Lower Bounds
• Conclusions

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Finding a Destination by Flooding
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Finding a Destination Efficiently
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(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 tohave few nodes in the
  (C)DS. This is known as theMinimum (C)DS
  problem.

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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

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Overview

• Introduction
• Clustering
• Flooding vs. Dominating Sets
• Algorithm Overview
• Phase A
• Phase B
• Lower Bounds
• Topology Control
• Conclusions

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Algorithm Overview
Input Local Graph
Fractional Dominating Set
Dominating Set
Connected Dominating Set
0.2
0.2
0.5
0
0.3
0
0.8
0.3
0.5
0.1
0.2
Phase C Connect DS by tree of bridges
Phase B Probabilistic algorithm
Phase A Distributed linear program rel. high
degree gives high value
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Overview

• Introduction
• Clustering
• Flooding vs. Dominating Sets
• Algorithm Overview
• Phase A
• Phase B
• Lower Bounds
• Topology Control
• Conclusions

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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)
• 0.50.30.30.20.20 1.5 1
• Linear Programs can be solved optimally in
  polynomial time
• But not in a distributed fashion! Thats what we
  do here

Linear Program
0.2
0.2
0.5
0
0.3
0
0.8
0.3
0.5
0.1
0.2
Adjacency matrix with 1s in diagonal
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Phase A Algorithm
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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)

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Overview

• Introduction
• Clustering
• Flooding vs. Dominating Sets
• Algorithm Overview
• Phase A
• Phase B
• Lower Bounds
• Topology Control
• Conclusions

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Dominating Set as Integer Program

• What we have after phase A
• What we want after phase B

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Phase B Algorithm

• Each node applies the following algorithm

1. Calculate ( maximum degree of neighbors
   in distance 2)
2. Become a dominator (i.e. go to the dominating
   set) with probability
3. Send status (dominator or not) to all neighbors
4. If no neighbor is a dominator, become a dominator
   yourself

From phase A
Highest degree in distance 2
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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 EDS O(? ln ? DSOPT)
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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

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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?

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Overview

• Introduction
• Clustering
• Flooding vs. Dominating Sets
• Algorithm Overview
• Phase A
• Phase B
• Lower Bounds
• Topology Control
• Conclusions

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Lower Bound for Dominating Sets Intuition

• Two graphs (m ltlt n). Optimal dominating sets are
  marked red.

complete
n
n
n

n-1
m
m
m
n
DSOPT 2.
DSOPT m1.
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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
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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 m1. DSOPT with green gt n
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The Lower Bound

• Lower bounds (Kuhn, Moscibroda, Wattenhofer _at_
  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 IDs do not help, and using
  Yaos 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.

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Graph Used in Dominating Set Lower Bound

• The example is for k 3.
• All edges are in fact special bipartite
  graphswith large enough girth.

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A Theory of Locality?

• Ad hoc and sensor networks
• The largest network in the world?!?
• Managing organizations? Society?!?
• Matrix multiplication, etc.

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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)

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Then
half of tx radius
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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.
• Thats 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.

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Comparison

• First algorithm (distributed linear program)
• Algorithm computes CDS
• k2O(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)

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Lets 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?
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Models
UDG GPS
UDG No GPS
General Graph
too pessimistic
too optimistic
Unit Ball Graph
Quasi UDG
Bounded Growth
In a doubling metric
Number of independent neighbors is bounded (UDG
5)
1
d
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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

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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
  vn approximation

9
2
6
8
5
4
1
7
3
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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.

?
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Another Algorithm 4 Covering connected neighbors

• If higher priority neighbors are connected and
  cover all other neighbors, then dont 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
  vn approximation
• However, on random UDGs, this gives a O(1)
  approximation

9
2
6
8
5
4
1
7
3
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Result Overview
UDG Unit Disk Graph UBG Unit Ball Graph GBG
Growth Bounded G. /GPS With Position Info /D
With Distance Info
UDG5
quality
UDG67
vn
General Graph2
better
Lower Bound for General Graphs9
log
?
loglog
GBG8
O(1)
UDG4
UDG/GPS1
UBG/D3
tx / node
1
2
O(log)
O(log)
better
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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

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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)

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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, Wattenhofer _at_ MobiCom 2004
• Moscibroda, Wattenhofer _at_ PODC 2005

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Overview

• Introduction
• Clustering
• Conclusions

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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

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Check yourself www.dcg.ethz.ch ? Reading List

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Conclusions Open Problems

• You dont 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

50
Questions?Comments?
Thank you for your attention