1

An O(log n) Dominating Set An O(log n) Dominating Set Protocol for Wireless Ad-Protocol for Wireless Ad-Hoc Networks under the Hoc Networks under the

Physical Interference ModelPhysical Interference ModelAndrea W. Richa

Arizona State University

Joint work with Christian Scheideler and Paolo Santi

2

Wireless Ad-hoc Networks Mobile stations communicating over wireless medium

Challenges: ● design appropriate models● design and analyze algorithms under these models

3

Wireless Ad-hoc Networks

● Wireless communication very difficult to model accurately– Signal propagation – Interference – Mobility– Physical Carrier Sensing

● Algorithms are very difficult to analyze under a very accurate model

Find balance between accuracy and provability.

4

UDG: What is the problem?

Unit-Disk Graph (UDG)● Given a transmission radius R,

nodes u, v are connected iff d(u,v) ≤ R

Problems:

● Transmission range could be of highly nonuniform shape

● Does not consider interference

uR

v

5

● Can handle arbitrary transmission shapes● Nodes u, v can communicate directly iff they are

connected.● Interference Model:

– (interference range) = (transmission range)

● Problem: linear slowdown if interference range is larger than transmission range

u

v

w

v'

Packet Radio Network: What is the problem?

6

●While in the PRN model, s can send a message to t in 2 steps, no uniform protocol can successfully send a message in expected o(n) number of steps: linear slowdown

PRN: What is the problem?

v

n-2 nodes

st ≤

rt

≤ rt

≤ ri

≥ rt

7

Transmission and Interference Ranges:● Separate values.● Interference range constant times bigger

than transmission range.

● Problem? …

Bounded Interference Models

u rtri

does not cause interference at u (even if all nodes outside transmit at the same time)

may cause interference at u

8

Physical Interference

u

Reality looks more like this: transmission range

interference

9

Bounded x Physical Interference: Bad News

Bad news:● Blough, Canali, Resta, and Santi’08:

combined interference from far-away nodes cannot be neglected– bounded interference model: neglected interference

can be two orders of magnitude greater than noise floor

– simulations: 210% loss in throughput when interference from far away nodes taken into account

(We will see some good news later…)

10

Dominating Set Problem

Classical dominating set problem:

Given a graph G=(V,E) , find a subset U V of minimum size so that for every node v in V, either v is in U or v has a neighbor in U.

11

Dominating Set Problem

1

Wireless setting:

First formally analyzed for unit disk graph model.

12

Is dominating set problem still relevant in general setting?

● Studies fundamental problem of selecting local leaders of constant density that cover entire network area.

● Building block for many other problems in wireless networks.

● constant density: at most a constant number of nodes in any constant size area.

Our goal:Construct node set U of constant density via simple, local-control algorithm under the physical interference model so that all nodes v in V\U can receive messages from a node in U (i.e., U is coordinator set).

13

Bounded x Physical Interference: Good news

● Blough, Canali, Resta and Santi ’08:If nodes have constant density, then physical (SINR) interference model reduces to bounded interference model.

u

14

Overview of Talk

• Our model– Signal propagation– Interference model– Physical carrier sensing

• The dominating set problem• Our contribution• TWIN protocol

– Algorithm– Analysis

• Future Work

15

Signal Propagation

Log-normal shadowing model:

• d0: reference distance

• : path loss coefficient

• Signal loss at distance d in dB:

-10 log(d/d0) + X

for some Gaussian RV X

16

Signal Propagation

Log-normal shadowing model without X :

• P: signal strength at d0=1

• signal strength at distance d>1: P/d

17

Signal Propagation

Our model:• Non-symmetric function

c(v,w) [(1+)-1 d(v,w), (1+) d(v,w)]• accounts for nonuniform variations of

communication environment

• Received power (or signal strength) from v at w: Pw(v)=P/c(v,w)

18

Signal Propagation

• random function c: approximates well (a truncated form of) the log-normal shadowing model

19

Transmission Range• forward error correction: transition between

being able to correctly receive a message (w.h.p.) and not being able to correctly receive a message (w.h.p.) is less than 1dB

sharp boundarysharp boundary

uv

w

20

Physical Interference (SINR)

• u receives msg from vif and only if

Pu(v)

N+w Pu(w)

N: background noise

• Received power from v at w: Pw(v)=P/c(v,w)

>

u

v

21

Physical Carrier Sensing

• Provided by Clear Channel Assessment (CCA) Circuit

• Monitors the medium as a function of Received Signal Strength Indicator (RSSI)

• Energy Detection (ED) bit set to 1 if RSSI exceeds a certain threshold

• Has a register to set the threshold T

So v can check if N+w Pv(w) > T

22

Overview of Talk

• Our model– Signal propagation– Interference model– Physical carrier sensing

• Prior work and our Contribution• TWIN protocol

– Algorithm– Analysis

• Future Work

23

Prior WorkModelling:• Log-normal shadowing model and physical interference model

common in physical layer community• Gupta and Kumar ’00, Grossglauser and Tse ‘01: capacity of

wireless networks• Brar, Blough, Santi ’06 and Moscibroda, Wattenhofer, Zollinger ‘06:

transmission scheduling• Goussevskaia, Moscibroda, Wattenhofer ’08: broadcasting

Dominating sets:• Luby ’85, Alzoubi et al ’02, Dubhashi et al ’03, Kuhn et al ’03, Huang

et al ’04,…: UDG• Kuhn et al ’04, Partasarathy and Gandhi ’04 : protocols for bounded

interference model (runtime O(log2 n) )• Kothapalli et al ‘05: protocol for more general bounded interference

model with physical carrier sensing (runtime O(log4 n) )

24

Dominating Set Problem

• V: set of n nodes of arbitrary distr. in IR2

• c: non-symmetric cost function

• Find subset U of V of constant density so

that for every v in V:– either v in U

– or there is a w in U with Pv(w) > N.

v can receive msg from wv can receive msg from w

25

Our Contribution

• More general model for theoretical analysis (hopefully closer to reality)

Theorem. TWIN protocol establishes a constant density dominating set in O(log n) time w.h.p.

Main ideas:• Extensive use of physical carrier sensing• Leaders emerge in twins (if possible)

26

Why Physical Carrier Sensing?

• Using physical carrier sensing, we can extract information from the network without relying on successful message transmissions– quite often it is enough just to know if at least one

node is sending a message, rather than receiving the message

– linear speedup

• It comes for “free” v

27

Overview of Talk

• Our model– Signal propagation– Interference model– Physical carrier sensing

• The dominating set problem• Our contribution• TWIN protocol

– Algorithm– Analysis

• Future Work

28

TWIN Protocol• Nodes do not need any prior knowledge• All messages of constant size (signals)• All nodes transmit with same power P• Nodes may be

– inactive: not in dominating set– twin: in dominating set; twins come up in pairs– active single: “isolated” nodes which cannot

form a twin pair but are still needed for coverage

• acc(v) : counter (acc(v)>0 iff v active)

29

TWIN Protocol

• Nodes operate in synchronized rounds that are continuously executed

• Stage 1: announcing active twins• Stage 2: guessing the right density• Stage 3: forming new twins

stage 1 stage 2 stage 3

round

Diff frequency for each time slot: no syncDiff frequency for each time slot: no sync

30

TWIN Protocol

For every node v:Initially, v is inactive and acc(v)=0. Access probability pv may have any value in

(0, pmax], where pmax<<1.

D: maximum density of twin nodes

Stage 1: announcing active twins• Active twin: send ACTIVE signal with prob 1/D • Inactive or active single: if v receives ACTIVE

signal, it terminates and becomes inactive

31

TWIN Protocol0<<1: constant inc/dec step for access probability

Stage 2: guessing the right density• Inactive or active single: v chooses one of two time slots

at random, say s (other slot s’).

• Slot s: v sends PING signal with prob pv.If not, v senses channel with threshold T

• Slot s’: v senses channel with threshold T

• v does not sense anything: pv:=min{(1+)pv, pmax}• v senses busy channel: pv:=(1+)-1pv

• If pv=pmax then acc(v):=acc(v)+4, else acc(v):=max{acc(v)-1,0} (0: inactive)

v is an active singlev is an active single

32

TWIN Protocol

Stage 3: forming new twins• Inactive or active single: If v sent PING in slot s

and received PING at slot s’in stage 2, then it sends ACK in slot s of this stage. If it receives an ACK signal in slot s’ of this stage, v becomes an active twin.

PING

PING

ACK

ACK

v w

active twin active twin, since w must have received PINGfrom v only (otherwise no ACK from w)

33

TWIN Protocol

• (Stage 3.) If v just became active twin, v sends NEW signal in last slot. If v is inactive or active single and senses a busy channel with threshold T, then v becomes inactive and terminates the protocol

NEWv z

active twin inactive or active singleinactive

sensing range of v

34

Overview of Talk

• Our model– Signal propagation– Interference model– Physical carrier sensing

• The dominating set problem• Our contribution• TWIN protocol

– Algorithm– Analysis

• Towards self-stabilization• Future Work

35

Analysis Overview

• probabilities pv quickly converge to constant in every transmission area

• low runtime: constant chance of twins emerging• constant twin density: twins must receive ACKs,

and NEW signals deactivate local neighborhood• active singles: nodes not covered and not

having node to pair up with eventually become active single; if density of active singles beyond certain constant, active twin will emerge

36

Getting Down to Constant Density

• Sensing area Rs(v):– whenever a node in Rs(v) transmits, v will be

able to sense transmission with threshold T– Rs(v) R(v), where R(v) is the transmission

area of node v

Lemma: After logarithmic many steps, w in RS(v) pw = O(1)for a constant fraction of the rounds.

constant densityconstant density

37

Bounding Far-away Interference

• A round is called good iff w in R(v) pw = O(1) and the interference caused by nodes not in R(v) is less than T-N– “far-away“ noise will not trigger busy channel

• Lemma. For any constant ε>0, at least (1- ε) fraction of time steps are good for v w.h.p., if T sufficiently large.

bounded interferencebounded interference

38

Quick constant density coverage

• Lemma. After a logarithmic number of steps, for a pmax <<1, every node v will

– (coverage:) either be an active single or have an active twin within its transmission range, whp. Moreover,

– (constant density:) have at most a constant number of active singles and twins within its transmission range whp

39

Conclusion

• O(log n)-time protocol for dominating set under more realistic model

• should be implementable in most simple devices• possible building block for many other

applications on top of it

Open questions:• self-stabilization• How does protocol perform in practice???• More robust form (jamming-resistant)

40

Is the model sufficiently realistic?

• Our interference model conservative:– signal cancellation– different signal strengths– bit recovery

• fading and other nondeterministic characteristics of the wireless signal

41

Towards Self-Stabilization

• Initial pv’s can be arbitrary• Initial acc(v)’s can be arbitrary

Problems: 1. Termination of protocol not allowed.

Instead, node should just “fall asleep” for O(log n) many rounds.

2. Initial density of active twins might be too high.

Possible solution for 2.: add another time slot in which active twins check their cumulative signal strength (random decision to send or sense)

Problem: time of stabilization cannot be bounded well, just eventual recovery

42

Questions?

43

Self-Stabilization

• wireless communication too complex: no model will be able to accurately take into account all that can happen

Problem: What happens if things deviate from proposed model?

Solution: Protocols need to be self-stabilizing, i.e., they need to go back to a valid configuration for the model

44

45