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alg. hoc. net. 2005. Algorithms for Ad Hoc Networks. Roger Wattenhofer MedHocNet 2005. Small community O(…),  (…),  (…) Everybody knows best paper New algorithm: Compare it with the best previous

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algorithms for ad hoc networks

alg

hoc

net

2005

Algorithms for Ad Hoc Networks

Roger Wattenhofer

MedHocNet 2005

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 waytoo 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

Distributed Algorithms vs. Ad Hoc Networking

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

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

Algorithmic Research in Ad Hoc and Sensor Networking

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

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

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

routing in ad hoc networks
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 Zurich @ MedHocNet 2005

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

Simple Classification of Ad hoc Routing Algorithms

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

critical mobility

mobility very high

Source Routing (DSR, AODV):

flooding, but re-use old routes

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

discussion
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 Zurich @ MedHocNet 2005

overview1
Overview
  • Introduction
  • Clustering
    • Flooding vs. Dominating Sets
    • Algorithm Overview
    • Phase A
    • Phase B
    • Lower Bounds
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

finding a destination by flooding
Finding a Destination by Flooding

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

finding a destination efficiently
Finding a Destination Efficiently

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

connected dominating set
(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.

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

formal problem definition m c ds
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 Zurich @ MedHocNet 2005

overview2
Overview
  • Introduction
  • Clustering
    • Flooding vs. Dominating Sets
    • Algorithm Overview
    • Phase A
    • Phase B
    • Lower Bounds
  • Topology Control
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

algorithm overview
Algorithm Overview

Input:

Local Graph

Fractional

Dominating Set

Dominating

Set

Connected

Dominating Set

0.2

0.2

0

0.5

0.3

0

0.8

0.3

0.5

0.1

0.2

Phase B:

Probabilistic

algorithm

Phase C:

Connect DS

by “tree” of

“bridges”

Phase A:

Distributed

linear program

rel. high degree

gives high value

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

overview3
Overview
  • Introduction
  • Clustering
    • Flooding vs. Dominating Sets
    • Algorithm Overview
    • Phase A
    • Phase B
    • Lower Bounds
  • Topology Control
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

phase a is a distributed linear program
Phase A is a Distributed Linear Program
  • Nodes 1, …, n: Each node u has variable xuwith xu¸ 0
  • Sum of x-values in each neighborhood at least 1 (local)
  • Minimize sum of all x-values (global)

0.5+0.3+0.3+0.2+0.2+0 = 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

0.5

0.3

0

0.8

0.3

0.5

0.1

0.2

Adjacency matrix

with 1’s in diagonal

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

phase a algorithm
Phase A Algorithm

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

result after phase a
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 Zurich @ MedHocNet 2005

overview4
Overview
  • Introduction
  • Clustering
    • Flooding vs. Dominating Sets
    • Algorithm Overview
    • Phase A
    • Phase B
    • Lower Bounds
  • Topology Control
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

dominating set as integer program
Dominating Set as Integer Program
  • What we have after phase A
  • What we want after phase B

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

phase b algorithm
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

Highest degree in distance 2

From phase A

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

result after phase b
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 Zurich @ MedHocNet 2005

related work on connected dominating sets
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 Zurich @ MedHocNet 2005

recent improvements
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 Zurich @ MedHocNet 2005

overview5
Overview
  • Introduction
  • Clustering
    • Flooding vs. Dominating Sets
    • Algorithm Overview
    • Phase A
    • Phase B
    • Lower Bounds
  • Topology Control
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

lower bound for dominating sets intuition
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 Zurich @ MedHocNet 2005

lower bound for dominating sets intuition1
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 Zurich @ MedHocNet 2005

lower bound for dominating sets intuition2
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 Zurich @ MedHocNet 2005

the lower bound
The Lower Bound
  • Lower bounds (Kuhn, Moscibroda, Wattenhofer @ 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 Zurich @ MedHocNet 2005

graph used in dominating set lower bound
Graph Used in Dominating Set Lower Bound
  • The example is for k = 3.
  • All edges are in fact special bipartite graphswith large enough girth.

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

a theory of locality
A Theory of “Locality”?
  • Ad hoc and sensor networks
  • The largest network in the world?!?
  • Managing organizations? Society?!?
  • Matrix multiplication, etc.

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

a better and faster algorithm

v

1

u

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)

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

slide33
Then…

half of tx radius

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

algorithm
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 Zurich @ MedHocNet 2005

comparison
First algorithm (distributed linear program)

Algorithm computes CDS

k2+O(1) transmissions/node

O(O(1)/klog ) approximation

General graph

No position information

Second algorithm (virtual grid)

Algorithm computes CDS

1transmission/node

O(1) approximation

Unit disk graph (UDG)

Position information (GPS)

Comparison

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

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

Let’s talk about models…

too pessimistic

too optimistic

Are there any models in

between these extremes?

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

models
Models

UDG

No GPS

UDG

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

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

another algorithm 1 mis
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 Zurich @ MedHocNet 2005

another algorithm 2 election
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 Zurich @ MedHocNet 2005

another algorithm 3 non neighboring neighbors
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 Zurich @ MedHocNet 2005

another algorithm 4 covering connected neighbors
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 Zurich @ MedHocNet 2005

result overview
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

√n

General Graph2

better

Lower Bound for General Graphs9

log

?

loglog

GBG8

O(1)

UDG4

UBG/D3

UDG/GPS1

1

2

O(log*)

O(log)

tx / node

better

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

references
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 Zurich @ MedHocNet 2005

more models
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 Zurich @ MedHocNet 2005

clustering for unstructured radio networks
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 @ MobiCom 2004]
  • [Moscibroda, Wattenhofer @ PODC 2005]

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

overview6
Overview
  • Introduction
  • Clustering
  • Conclusions

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

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

Big Research Opportunities

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

check yourself www dcg ethz ch reading list
Check yourself: www.dcg.ethz.ch  Reading List

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

conclusions open problems
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

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005

questions comments

Questions?Comments?

Thank you for your attention

Roger Wattenhofer, ETH Zurich @ MedHocNet 2005