a probabilistic algorithm for efficient and robust data propagation in wireless sensor networks

Ad Hoc Networks 4 (2006) 621–635

www.elsevier.com/locate/adhoc

A probabilistic algorithm for efficient and robustdata propagation in wireless sensor networks q

Ioannis Chatzigiannakis a,*, Tassos Dimitriou b,Sotiris Nikoletseas a, Paul Spirakis a

a Computer Technology Institute (CTI) and Patras University, P.O. Box 1122, 261 10 Patras, Greeceb Athens Information Technology (AIT), Athens, Greece

Received 22 July 2004; received in revised form 23 June 2005; accepted 29 June 2005Available online 8 August 2005

Abstract

We study the problem of data propagation in sensor networks, comprised of a large number of very small and low-cost nodes, capable of sensing, communicating and computing. The distributed co-operation of such nodes may lead tothe accomplishment of large sensing tasks, having useful applications in practice. We present a new protocol for datapropagation towards a control center (‘‘sink’’) that avoids flooding by probabilistically favoring certain (‘‘close to opti-mal’’) data transmissions. Motivated by certain applications (see [I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, E.Cayirci, Wireless sensor networks: a survey, Journal of Computer Networks 38 (2002) 393–422], [C. Intanagonwiwat,R. Govindan, D. Estrin, Directed diffusion: a scalable and robust communication paradigm for sensor networks, in: 6thACM/IEEE Annual International Conference on Mobile Computing (MOBICOM 2000), 2000, pp. 56–67]) and also asa starting point for a rigorous analysis, we study here lattice-shaped sensor networks. We however show that this latticeshape emerges even in randomly deployed sensor networks of sufficient sensor density. Our work is inspired and buildsupon the directed diffusion paradigm of [C. Intanagonwiwat, R. Govindan, D. Estrin, Directed diffusion: a scalable androbust communication paradigm for sensor networks, in: 6th ACM/IEEE Annual International Conference on MobileComputing (MOBICOM 2000), 2000, pp. 56–67].

This protocol is very simple to implement in sensor devices, uses only local information and operates under totalabsence of co-ordination between sensors. We consider a network model of randomly deployed sensors of sufficient den-sity. As shown by a geometry analysis, the protocol is correct, since it always propagates data to the sink, under idealnetwork conditions (no failures). Using stochastic processes, we show that the protocol is very energy efficient. Also,

1570-8705/$ - see front matter � 2005 Published by Elsevier B.V.doi:10.1016/j.adhoc.2005.06.006

q A preliminary version of this work has appeared in the 5th European Wireless Conference on Mobile and Wireless Systems beyond3G (EW, 2004 [7]).

* Corresponding author. Tel.: +306 944552452; fax: +302 610960442.E-mail addresses: [email protected] (I. Chatzigiannakis), [email protected] (T. Dimitriou), [email protected] (S. Nikoletseas), [email protected]

(P. Spirakis).

mailto:[email protected]




622 I. Chatzigiannakis et al. / Ad Hoc Networks 4 (2006) 621–635

when part of the network is inoperative, the protocol manages to propagate data very close to the sink, thus in this senseit is robust. We finally present and discuss large-scale simulation findings validating the analytical results.� 2005 Published by Elsevier B.V.

Keywords: Wireless sensor networks; Data propagation; Protocol design; Probabilistic algorithms; Average case analysis; Simulation

1. Introduction, our results and related work

Recent dramatic developments in micro-electro-mechanical (MEMS) systems, wireless communi-cations and digital electronics have led to thedevelopment of small in size, low-power, low-costsensor devices. Such extremely small devices inte-grate sensing, data processing and communicationcapabilities. Examining each such device individu-ally might appear to have small utility, howeverthe effective distributed co-ordination of large num-bers of such devices may lead to the efficientaccomplishment of large sensing tasks.

Large numbers of sensors can be deployed inareas of interest (such as inaccessible terrains ordisaster places) and use self-organization and col-laborative methods to form a sensor network. Theirwide range of applications is based on the use ofvarious sensor types (i.e. thermal, visual, seismic,acoustic, radar, magnetic, etc.) to monitor a widevariety of conditions (e.g. temperature, objectpresence and movement, humidity, pressure, noiselevels etc.). Thus, sensor networks can be used forcontinuous sensing, event detection, location sens-ing as well as micro-sensing. Hence, sensor net-works have important applications, including (a)military (like forces and equipment monitoring,battlefield surveillance, targeting, nuclear, biologi-cal and chemical attack detection), (b) environ-mental applications (such as fire detection, flooddetection, precision agriculture), (c) health appli-cations (like telemonitoring of human physiologi-cal data) and (d) home applications (e.g. smartenvironments and home automation). For a sur-vey of wireless sensor networks see [1] and also[13,16].

Note however that the efficient and robust real-ization of such large, highly-dynamic, complex,non-conventional networks is a challenging algo-

rithmic and technological task. Features including

the huge number of sensors involved, the severepower, computational and memory limitations,their dense deployment and frequent failures, posenew design and implementation aspects which areessentially different not only to distributed com-puting and systems approaches but also toad-hoc networking techniques.

1.1. Problem description

We focus on an important problem under amodel of sensor networks that we present. Morespecifically, we study the problem of local detection

and propagation, i.e. the local sensing of a crucialevent and the energy and time efficient propaga-tion of data reporting its realization to a controlcenter. This center could be some human authori-ties responsible of taking action upon the realiza-tion of the crucial event. We use the term ‘‘sink’’

for this control center (other terms used in the rel-evant literature [9,8] include the term ‘‘wall’’ whichsuggests the possibility of a moving, maybe along aline, control center). The protocol we present herecan also be used for the more general problem ofdata propagation in sensor networks (see [15]).

Our protocol can be applied to any networktopology. However, we so far managed to analyzeits performance in exactly lattice shaped networks.In fact, the lattice shape assumption allows as tomake a concrete first step towards a rigorous anal-ysis of the protocol. We believe that is important,since such analyses are missing in the related liter-ature, where protocols are validated (both theirperformance and correctness) by using simula-tions. Moreover, this analysis extends to ‘‘approx-imate’’ lattice-shaped networks of sufficientdensity (i.e. networks satisfying event F, presentedin the start of Section 5).

We furthermore note that even exactly lattice-shaped networks are useful in practice in certain

S

E

Particles thatparticiapate in

forwarding path

Fig. 1. Thin zone of particles.

S

E

p1

p2

1

2

Fig. 2. Angle / and closeness to optimal line.

I. Chatzigiannakis et al. / Ad Hoc Networks 4 (2006) 621–635 623

applications, i.e. in a pre-deployed sensor network,where sensors are put (possibly by a human or arobot) in a way that they form a two-dimensional

lattice. Note indeed that such sensor networks, de-ployed in a structured way, might be useful in pre-cise agriculture applications, where humans orrobots may want to deploy the sensors in a latticestructure to monitor in a rather homogenous anduniform way certain conditions in the spatial areaof interest. Certainly, exact terrain monitoring inmilitary applications may also need some sort ofa grid-like shaped sensor network. Note also thatAkyildiz et al. in a very recent (2002) state of theart survey published in the Journal of ComputerNetworks ([1]) do not exclude the pre-deploymentpossibility. Also Intanagonwiwat, Govindan andEstrin, a group of pioneering researchers in wire-less sensor networks (from the technological pointof view, which by its flavor pays special attentionto modelling assumption) in a recent paper ap-peared in ACM MOBICOM 2002 ([15]) explicitlyrefer to the lattice case.

1.2. Our protocol

For the above problem we propose a new proto-col which tries to minimize energy consumption byprobabilistically favoring certain paths of local data

transmissions towards the sink. Thus we call thisprotocol PFR (Probabilistic Forwarding Protocol).

The basic idea behind this protocol is to avoid

flooding by favoring (in a probabilistic manner)data propagation along sensors which lie ‘‘close’’to the (optimal) transmission line, ES, that con-nects the sensor node detecting the event, E, andthe sink, S (see Fig. 1). This is implemented bylocally calculating the angle / ¼ ðdEPSÞ, whosecorner point P is the sensor currently running thelocal protocol, having received a transmissionfrom a nearby sensor, previously possessing theevent information. If / is equal or greater to a pre-determined threshold, then p will transmit (andthus propagate the event information further).Else, it decides whether to transmit with probabil-ity equal to /

p (see Fig. 2). The way to estimate / isexplained in Section 4.1. Because of the probabilis-tic nature of data propagation decisions and in or-der to prevent the data propagation process from

early failing, we initially use (for a short time per-iod which we evaluate) a flooding mechanism thatleads to a sufficiently large ‘‘front’’ of sensors pos-sessing the data under propagation. When such a‘‘front’’ is created, we perform probabilisticforwarding.

The protocol, as shown by a geometric analysis,always propagates data to the sink, under idealnetwork conditions (no failures), thus it is prov-ably correct. Using properties of stochastic pro-cesses, we show that the protocol is very energyefficient. Also, when part of the network is inoper-ative (which is more realistic, because sensors areprone to faults), the protocol manages to propa-gate data very close to the sink, thus it is robust.The above analytical results are validated by a


large scale simulation we have carried out afterimplementing the protocol in Section 6.

Essentially, our protocol captures the intuitive,deterministic idea ‘‘if my distance from ES issmall, then send, else do not send’’. We have cho-sen to enhance this idea by random decisions(above a threshold) to allow some local floodingto happen with small probability and thus to copewith local sensor failures.

1.3. Related work

Our protocol is inspired by the relevant work of[15], where ‘‘directed diffusion’’, an approach toattribute-based data communication for wirelesssensor networks is proposed. The goal of directeddiffusion is to establish communication betweensources and sinks. Directed diffusion consists ofseveral elements. Data is named using attribute-valued pairs. A sensing task is disseminatedthroughout the sensor network as an interest fornamed data. This dissemination sets up gradientswithin the network designed to ‘‘draw’’ events(i.e. data matching the interest). Events start flow-ing towards the originators of interests along mul-tiple paths. The sensor network reinforces one, or asmall number of these paths. Our protocol is in-spired by the Directed diffusion communicationparadigm in [15] where it is mentioned the possibil-ity of ‘‘multi-path delivery with probabilistic for-warding’’. However, no such protocol is designedin [15]. Furthermore, we here propose and analyzea particular probabilistic multi-path delivery. Aproper modification of our protocol may be usedin the gradient set-up and dissemination phase ofDirected diffusion.

A different approach for propagating informa-tion to the sink is to use existing routing techniquesfor mobile ad-hoc routing protocols ([22]) in sensornetworks. However, although protocols for mobilead-hoc networks take into consideration energyconservation issues, most of them are not reallysuitable for sensor networks. In [17] a routing pro-tocol is presented that is suitable for sensor net-works that makes greedy forwarding decisionsusing only information about a node�s immediateneighbors in the network topology. This approachachieves high scalability as the density of the net-

work increases. In [14] a clustering-based protocolis given that utilizes randomized rotation of localcluster heads to evenly distribute the energy loadamong the sensors in the network. In [19] a new en-ergy efficient routing protocol is introduced thatdoes not provide periodic data monitoring (as in[14]), but instead nodes transmit data only whensudden and drastic changes are sensed by thenodes. As such, this protocol is well suited for timecritical applications and compared to [14] achievesless energy consumption and response time.

Furthermore, this work is related to previousresearch of [9,8], where new local detection andpropagation protocols are proposed, that are veryenergy and time efficient, as shown by a rigorousaverage case analysis performed in these works un-der certain simplifying assumptions. In particular,the Local Target Protocol LTP protocol uses localoptimization to select the next-hop neighbor, whilethe Sleep–Awake Protocol SWP protocol intro-duces alternation between sleep–awake modes tosave energy. Also [2] uses variable transmissionrange to better perform in cases of low density ofparticles, obstacle presence and to avoid overusingcritical (close to the sink) sensors. In [21] we intro-duce sleep–awake mechanisms in the PFR protocoland compare it to hierarchical clustering ap-proaches. In [10], random geometric graphs areused to analyze performance and fault-toleranceproperties of sensor networks.

For a detailed discussion on data propagationprotocols in wireless sensor networks see also [4].For a comparative study of the multi-path deliveryPFR protocol to single path delivery (the LocalTarget Protocol, LTP), see [5,6,9]. For otherimportant aspects of energy efficiency (such as en-ergy balance) see [12,3]. For a discussion of chal-lenges in wireless sensor networks research see [24].

2. The model

Sensor networks are comprised of a vast num-ber of ultra-small homogenous sensors, which wecall ‘‘grain’’ particles. Each grain particle is afully-autonomous computing and communicationdevice, characterized mainly by its available powersupply (battery) and the energy cost of computa-


tion and transmission of data. Such particles (inour model here) cannot move.

Each particle is equipped with a set of monitors(sensors) for light, pressure, humidity, temperatureetc. and has a broadcast (digital radio) beacon mode.

As a starting point and also motivated by cer-tain applications (precise agriculture, militaryapplications, see also Section 1), we adopt here atwo-dimensional (plane) lattice deployment ofgrain particles (see Fig. 5).

We assume an n · n lattice deployment of n2

grain particles, each one of maximum transmissionrange R, put in (horizontal and vertical) distancesd apart. Note that d depends on R and also d char-acterizes the density of the network. In particularwe choose R ¼ d

ffiffiffi2p

. In the sequel we use d = 1for simplicity. Thus each particle (excluding theparticles on the boundaries of the lattice) is sur-rounded by eight immediate neighbors that liewithin its transmission range.

There is a single point in the network area,which we call the sink S, and represents a controlcenter where data should be propagated to. Notethat, although in the basic case we assume the sinkto be static, in a variation it may be allowed tomove around its initial base position, to possiblyget data that failed to reach it but made it closeenough to it.

We assume that each grain particle has thefollowing abilities:

1. It can estimate the direction of a received trans-mission (e.g. via the technology of direction-sensing antennae).

2. It can estimate the distance from a nearbyparticle that did the transmission (e.g. via esti-mation of the attenuation of the receivedsignal).

3. It knows the direction towards the sink S (i.e.not the exact co-ordinates). This can be imple-mented during a set-up phase, where the (verypowerful in energy) sink broadcasts the infor-mation about itself to all particles.

4. All particles have a common co-ordinatessystem.

The above assumptions suggest a strong model,that however does not trivialize the problem; we

believe that even assuming such a model, the de-sign of our protocol is still a challenging task. Fur-thermore some of our assumptions are realistic(e.g. it is indeed possible to apply smart antennasin tiny sensors, e.g. see [11]) or may become realis-tic in the near future. Furthermore, some model-ling assumptions can be relaxed by alternatives,like introducing a preprocessing phase where sen-sors use localization techniques by executing anunderlaying protocol f to decide on fictitious

virtual co-ordinates [23] (i.e. relax assumption 4)and thus be able to estimate directions (i.e. relaxassumption 1) and distances of transmissions (i.e.relax assumption 2). In any case, our assumptionsabove are simplifying ones, allowing us to simplifythe analysis of the protocol properties.

Notice that GPS information is not needed forour protocol. Also, there is no need to know theglobal structure of the network (i.e. the size ofthe network, the positions of other particles etc.).

3. The problem

Assume that a single particle, P, senses the real-ization of a local crucial event E. Then the propaga-

tion problem P is the following:

‘‘How can particle P, via co-operation with therest of the grain particles, efficiently propagateinformation infoðEÞ, reporting realization of eventE, to the sink S?’’

To minimize the energy consumption in the sen-sor network we wish to avoid flooding and to min-imize the number of hops (directed transmissions)

performed in the data propagation process, whilestill managing to reach the sink.

Definition 1. Let E be the position of P whichsenses event E. Let S be the position of the sink S.

Any protocol P solving the propagation prob-lem P must satisfy:

• Correctness. P must guarantee that data arrivesto the position S, given that the whole networkexists and is operational.

• Robustness. P must guarantee that data arrivesat enough points in a small interval around S, in


cases where part of the network has becomeinoperative, without however isolating the sink.

• Efficiency. If P activates k particles during itsoperation then P should have a small ratio ofthe number of activated over the total numberof particles r ¼ k

N. Thus r is an energy efficiencymeasure of P.

4. The probabilistic forwarding protocol (PFR)

As already mentioned in Section 1, the basic ideaof the protocol lies in probabilistically favoringtransmissions towards the sink within a thin zone

of particles around the line connecting the particlesensing the event E and the sink (see Fig. 1). Notethat transmission along this line is energy optimal.However it is not always possible to achieve thisoptimality, basically because certain sensors on thisdirect line might be inactive, either permanently(because their energy has been exhausted) ortemporarily (because these sensors might enter asleeping mode to save energy). Further reasonsinclude (a) physical damage of sensors, (b) deliber-

ate removal of some of them (possibly by an adver-sary in military applications), (c) changes in the

position of the sensors due to a variety of reasons(weather conditions, human interaction etc.) and(d) physical obstacles blocking communication.

The protocol evolves in two phases:

Phase 1: The ‘‘front’’ creation phase. Initially webuild (by using a limited, in terms of rounds,flooding) a sufficiently large ‘‘front’’ of parti-cles, in order to guarantee the survivability ofthe data propagation process. During thisphase, each particle having received the datato be propagated, deterministically forwardsthem towards the sink. In particular, and for asufficiently large number of steps s ¼ 180

ffiffiffi2p

,each particle broadcasts the information to allits neighbors, towards the sink. The particularselection of s is explained further in Section5.1 where we analyze the correctness of the pro-tocol. Remark that to implement this phase,and in particular to count the number of steps,we use a counter in each message. This counterneeds at most ½log 180

ffiffiffi2p� bits. Before a particle

performing a broadcast, it reduces the counterof the message by one. As long as the counterin a message is non-zero particles execute Phase1; when this counter becomes zero, particlesreceiving the message switch to Phase 2.Phase 2: The probabilistic forwarding phase.

During this phase, each particle P possessingthe information under propagation, calculatesan angle / by calling the sub-protocol ‘‘/-calcu-lation’’ (see description below) and broadcastsinfoðEÞ to all its neighbors with probabilityPfwd (or it does not propagate any data withprobability 1� Pfwd) defined as follows:

Pfwd ¼1 if / P /threshold;

/p

otherwise;

8<:

where / is the angle defined by the line EP andthe line PS and /threshold = 134� (the selectionreasons of this /threshold will become more evi-dent in Section 5.1).

In both phases, if a particle has already broad-cast infoðEÞ and receives it again, it ignores it. Alsothe PFR protocol is presented for a single eventtracing. Thus no multiple paths arise and packetsizes do not increase with time. However, PFRcan be applied to multiple events as well by assum-ing an appropriate resolution layer (or MAC) anda coding scheme that uniquely identifies events.Also in the case where a single event is sensed bymore than one sensors, one may apply data aggre-gation techniques during Phase 1 to avoid unnec-essary repetitions of messages in Phase 1.

Remark that when / = p then P lies on the lineES and vice-versa (and always transmits).

If the density of particles is appropriately large(see Sections 5, 6), then for a line ES there is (withhigh probability) a sequence of points ‘‘closely sur-rounding ES’’ whose angles / are larger than/threshold and so that successive points are withintransmission range. All such points broadcast andthus essentially they follow the line ES (see Fig. 2).

4.1. The /-calculation sub-protocol

Let Pprev the particle that transmitted infoðEÞ toP (see Fig. 3).

SE

Pprev P

Fig. 3. Angle / calculation example.

S

EParticles

LatticeDissection

Fig. 4. A lattice dissection G.


1. When Pprev broadcasts infoðEÞ, it also attachesthe info jEPprevj and the direction P prevE

��!.

2. P estimates the direction and length of line seg-ment PprevP, as described in the model.

3. P now computes angle ð dEP prevP Þ, and computes

jEPj and the direction of PE�!

(this will be usedin further transmission from P).

4. P also computes angle ð dP prevPEÞ and by sub-

tracting it from ð dP prevPSÞ it finds /.

Notice the following:

i. The direction and distance from activatedsensors to E is inductively propagated (i.e.P becomes Pprev in the next phase).

ii. Our protocol needs only messages of lengthbounded by logA, where A is some measureof the size of the network area, since (becauseof i. above) there is no cumulative effect onmessage lengths.

Note that the number of steps in the forwardingphase of the protocol depends on the /threshold ofthe protocol as it can be seen from the analysisin Section 5. For /threshold = 134� the number offlooding steps must be at least 180

ffiffiffi2p

for correct-ness reasons. We can increase the /threshold; thiswill increase also the number of flooding steps.This also implies a tradeoff between energy effi-ciency and robustness.

5. Properties of PFR

Consider a partition of the network area intosmall squares of a fictitious grid G (see Fig. 4).Let the length of the side of each square be l. Let

the number of squares be q. The area covered isbounded by ql2. Assuming that we randomlythrow in the area at least aq logq = N particles(where a > 0 a suitable constant), then the proba-bility that a particular square is avoided is

1� 1

q

� �aq log q

6 e�a log q ¼ q�a.

So the probability that all squares get particles is atleast

1� q � q�a ¼ 1� q�ða�1Þ

¼ 1� HN

log N

� �� ða�1Þ

.

We condition all the analysis on this event, call itF, of at least one particle in each square.

5.1. The correctness of PFR

Without loss of generality, we assume eachsquare of the fictitious lattice G to have side length1.

Lemma 1. PFR succeeds with probability 1 in

sending the information from E to S given the event

F.

Proof. In the (trivial) case where jESj 6 180ffiffiffi2p

,the protocol is clearly correct due to front creationphase.

Let R a unit square of G intersecting ES in someway (see Fig. 6). Since a particle always existssomewhere in R, we will only need to examine the

A

B’B

A’

E

S

D

y

y’

1

1

1

2

Fig. 6. The square R (A 0AB 0B).

Sink

SensorParticles

E

Fig. 5. A lattice sensor network.


worst case of it being in one of the corners of R.Consider vertex A. The line EA is always to the leftof AB (since E is at end of ES). The same is truefor AS (S is to the right of B 0).

Let AD the segment from A perpendicular toES (and D its intersection point) and let yy 0 be theline from A parallel to ES. Then,

/ ¼ ðdEASÞ ¼ 180� � ðdyAEÞ � ðdy 0ASÞ

¼ 90� � ðdyAEÞ þ 90� � ðdy 0ASÞ

¼ ðdyADÞ � ðdyAEÞ þ ðdDAy 0 Þ � ðdy0ASÞ.

Let cx1 ¼ ðdyADÞ � ðdyAEÞ and cx2 ¼ ðdDAy 0 Þ�ðdy 0ASÞ and, without loss of generality, letED < DS. Then always cx1 > 45�, since it includeshalf of the 90�-angle of A in the unit square. Also

sinðdy0ASÞ ¼ sinðdASDÞ ¼ ADAS

<ADDS

but AD 6ffiffiffi2pð¼ ABÞ and DS P

ES2

P 90ffiffiffi2p

) sinðdy0ASÞ 6 1

90() ðdy0ASÞ < 1�.

(Note that here we use the elementary fact thatwhen x < 90� then sin x < 1

90() x < 1�). Thencx2 > 89� since ðdDAy 0 Þ ¼ 90� by construction.

Thus

/ ¼ cx1 þ cx2 > 45� þ 89� ¼ 134�.

There are two other ways to place an intersectingto ES unit square R, whose analysis is similar.

But (a) the initial square (from E) alwaysbroadcasts due to the first protocol phase and (b)any intermediate intersecting square will be noti-fied (by induction) and thus will broadcast. Hence,S is always notified, when the whole grid isoperational. h

5.2. The energy efficiency of PFR

Consider the fictitious lattice G of the networkarea and let the event F hold. We have (at least)one particle inside each square. Now join all‘‘nearby’’ particles (see Fig. 4) of each particle toit, thus by forming a new graph G 0 which is ‘‘lat-tice-shaped’’ but its elementary ‘‘boxes’’ may notbe orthogonal and may have varied length. WhenG 0s squares become smaller and smaller, then G 0

will look like G. Thus, for reasons of analytic trac-tability, we will assume in the sequel that our par-ticles form a lattice (see Fig. 5). We also assumelength l = 1 in each square, for normalization pur-poses. Notice however that when l! 0 then‘‘G 0 ! G’’ and thus all our results in this Sectionhold for any random deployment ‘‘in the limit’’.

Theorem 1. The energy efficiency of the PFR

protocol is H n0

n

� �2�

where n0 = jESj and n ¼ffiffiffiffiNp

.

For n0 = jESj = o(n), this is o(1).

Proof. The analysis of the energy efficiency con-siders particles that are active but are as far as pos-sible from ES. Thus the approximation we do aresuitable for remote particles. h

Here we estimate an upper bound on thenumber of particles in an n · n (i.e. N = n · n)lattice. If k is this number then r ¼ k

n2 (0 < r 6 1) isthe ‘‘energy efficiency ratio’’ of PFR.

We want r to be less than 1 and as small aspossible (clearly, r = 1 leads to flooding). Since, by

E S

n0

Q

LQ

n

Fig. 7. The LQ area.

E SD

Q

x 12

Fig. 9. The QES triangle.


Lemma 1, ES is always ‘‘surrounded’’ by activeparticles, we will now assume without loss ofgenerality that ES is part of a horizontal grid line(see Fig. 7) somewhere in the middle of the lattice.

Recall that jESj = n0 particles of all the active,via PFR, particles that continue to transmit, andlet Q be a set of points whose shortest distancefrom particles in ES is maximum. Then LQ is thelocus (curve) of such points Q. The number ofparticles included in LQ (i.e. the area inside LQ) isthe number k.

Now, if the distance of such points Q of LQ

from ES is x then k 6 2(n0 + 2x)x (see Fig. 8) andthus r 6 2xðn0þ2xÞ

n2 . Notice however that x is arandom variable hence we will estimate itsexpected value EðxÞ and the moment Eðx2Þ to get

EðrÞ 6 4Eðx2Þn2

þ 2n0

n2EðxÞ. ð1Þ

Now, look at points Q : ðdEQSÞ ¼ / < 30�, i.e./ < p

6. We want to use the approximation

jEDj ’ x, where x is a random variable dependingon the particle distribution.

Let jEDjx ¼ 1þ �. Then a bound on the approx-

imation factor � determines a bound on the ‘‘cut-

off’’ angle /0 since jEDj cos /0

2

� ¼ �x.

SE

n0

Fig. 8. The particles inside the LQ area.

ð1þ �Þx cos/0

2

� �¼ x) cos

/0

2

� �¼ 1

1þ �/0 above is constant that can be appropriatelychosen by the protocol implementor. We chose/0 = 30� so � = 0.035. Also, we remark here thatthe energy spent increases with x0 ¼ n0 1� n

2

� �in

the area below x0, where n = tan75�, but decreaseswith x for x > x0. This is a trade-off and one cancarefully estimate the angle /0 (i.e. �) to minimizethe energy spent.

Let /1 ¼ ðdEQDÞ;/2 ¼ ðdDQSÞ where QD?ESand D in ES. In Fig. 9, / = /2 � /1.

We approximate / by sin /2 � sin /1 (note:/ ’ sin / P � sin /1 þ sin /2).

Note sin /1 ’jEDjjQEj ’

jEDjx

and sin /2 ’jDSj

x

) sin / ’ sin /2 � sin /1 ’jESj

x¼ n0

x.

Thus /p ’

n0

px for such points Q. We note that theanalysis is very similar when ðdQESÞ < p

2and thus

/ = /1 + /2.Let M be the stochastic process that represents

the vertical (to ES) distance of active points fromES, and W be the random walk on the vertical lineyy 0 (see Fig. 10) such that, when the walk is atdistance x P x0 from ES then it (a) goes to x + 1with probability n0

px or (b) goes to x � 1 withprobability 1� n0

px and never goes below x0, wherex0 is the ‘‘30�-distance’’ i.e. x0 ¼ n0n

2 .Clearly W dominates M i.e. PM x P x1f g 6

1� PW x P x1f g; 8x1 > x0.Furthermore, W is dominated by the continu-

ous time ‘‘discouraged arrivals’’ birth-death pro-cess W0 (for xPx0) where the rate of going from x

E S

y

y’

x

x

xo

xo

0

Fig. 10. The random walk W.

E D

Q’

x

S

y y’

1 2

Fig. 11. The Q 0ES triangle.


to x + 1 in W0 is ax ¼

n0=px and the rate of returning

to x�1 is 1� n0

px0¼ 1� 2

pn ¼ b.

We know from [18] that for Dx = x � x0

(x > x0)

EW0 ðDxÞ ¼ ab¼ n0

p 1� 2pn

� .

Thus, EW0 ðxÞ ¼ x0 þ ab, hence by domination

EðxÞ 6 EMðxÞ 6 x0 þab

. ð2Þ

Also from [18] the process W0 is a Poisson one and

P Dx ¼ kf g ¼ ða=bÞk

k!e�ða=bÞ. From this, the variance

of Dx is r2 = a/b (again) i.e. for x = x0 + Dx

EW0 ðx2Þ ¼ EW0 ðx0 þ DxÞ2�

¼ x20 þ 2x0EW0 ðDxÞ þ EW0 ðDx2Þ

¼ x20 þ 2

ab

x0 þ r2 þ E2ðDxÞ� �

¼ x20 þ 2

ab

x0 þabþ a

b

� �2

So

EW0 ðx2Þ ¼ 3n20

4þ 2

ab

n0n2þ a

bþ a

b

� �2

where

ab¼ n0

p 1� 2pn

� ¼ n0

s

and

s ¼ p 1� 2

pn

� �¼ p� 2

n

thus

EW0 ðx2Þ ¼ n20

3

4þ n

sþ 1

s2

� �þ n0

sð3Þ

and by domination Eðx2Þ 6 EMðx2Þ 6 EW0 ðx2Þ.So, finally

EðrÞ 64 n2

034þ n

s þ 1s2

� �þ n0

s

�n2

þ n20

n2nþ 2

s

� �ð4Þ

which proves the theorem. h

5.3. The Robustness of PFR

We now consider the robustness of our proto-col, in the sense that PFR must guarantee that dataarrives at enough points in a small interval aroundS, in cases where part of the network has becomeinoperative, without however isolating the sink.

Lemma 2. PFR manages to propagate the crucial

data across lines parallel to ES, and of constant

distance, with fixed nonzero probability (not depend-

ing on n, jESj).

Proof. Here we consider particles very near theline ES. Thus the approximations that we do aresuitable for nearby particles. Let Q 0 an active par-ticle at vertical distance x from ES and D 2 ES

such that Q 0D?ES.Let yy0kES, drawn from Q 0 (see Fig. 11) and

/1 ¼ ðdyQ0EÞ, /2 ¼ ðdSQ0y0 Þ. Then / = 180� �(/1 + /2), i.e. /

p ¼ 1� /1þ/2

p .

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000 8000

||ES||

r

Simulation

Theory

Fig. 12. Energy efficiency measure r for different kESk whenn = 8000.


Since /1, /2 are small (for small x) we use theapproximation /1 ’ sin /1 and /2 ’ sin /2

/1 þ /2 ’ sin /1 þ sin /2 ¼x

EQ0þ x

Q0S.

Since / > 90�, ES is the biggest edge of triangleðdEQ0SÞ, thus EQ 0 6 n0 and Q 0S 6 n0. Hence,

sin /1 þ sin /2 62xn0

and

1� sin /1 þ sin /2

pP 1� 2x

pn0

ðfor small xÞ.

Without loss of generality, assume ES is part of ahorizontal grid line. Here we study the case inwhich some of the particles on ES or very nearES (i.e. at angles >134�) are not operating.

Consider now a horizontal line ‘ in the grid, atdistance x from ES. Clearly, some particles of ‘near E will be activated during the initial broad-casting phase (see Phase 1).

Let A be the event that all the particles of ‘(starting from p) will be activated, until a verticalline from S is reached. Then

PðAÞP 1� 2xpn0

� �n0

’ e�2xp . �

6. Simulation results

We validate the theoretical results and investi-gate the asymptotic behavior of the PFR protocolby conducting a set of large scale simulations.Our implementation follows closely the protocoldescription of Section 4 and is based on C++ andthe Library of Efficient Data types and Algorithms(LEDA) [20].

We start by examining the energy efficiency ofPFR by measuring the ratio r of the number ofactivated particles over the total number of parti-cles r ¼ k

n2. We used lattice shaped sensor fieldscomprised of extremely large number of sensors(n 2 [1000, 8000]) that are put in (horizontal andvertical) distances d = 5 m apart and each parti-cle�s broadcast range R ¼ d

ffiffiffi2p

. In each case, werepeated the simulations for at least 100 times to

get good average results (as indicated by thesmoothness of curves).

In the first set of simulations, we drop a fixednumber of particles (n = 8000) and then positionE and S in a way such that kESk 2 ½1000R;4000R�. Fig. 12 depicts the ratio r for variouskESk.

In the second set of simulations we work in adifferent way: we drop sensor fields comprised ofdifferent number of total particles (n 2 1000,4000) and then position E and S so that theirdistance is always fixed to kESk ¼ 1000R. Fig. 13depicts the ratio r for different n.

In both figures the dashed line depicts the theo-retical result of Theorem 1, i.e. an upper boundr0 ¼ n0

n

� �2on the ratio of activated particles, where

kESk = n0.The above figures indeed validate our analytical

results, since in each case the efficiency ratio r

exhibits a quadratic behavior with respect to kESk(in the first case) and n (in the second). Further-more, notice that our analysis is tight enough,since the curve of the simulation measurements isvery close to the curve of the analytic result (i.ethe upper bound is pretty close to the measuredratio).

We then investigate the robustness of ourprotocol, in the case where some particles

r

Simulation

Theory

1000 1500 2000 2500 3000 3500 4000

n

0

0.2

0.4

0.6

0.8

1

Fig. 13. Energy efficiency measure r for different n whenjjESjj ¼ 1000R.

0

500

1000

1500

2000

2500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

fr

D

fc=0.4

fc=0.45

fc=0.5

Fig. 14. Robustness for jjESjj ¼ 1000R and n = 3000.


(permanently) fail. In particular, the failure proba-bility is taken ‘‘large’’ for particles close to ES line(i.e. fc = {0.4, 0.45,0.5}). The particles that are lo-cated further away from ES line, fail with smallerfailure probability (i.e. fr 2 [0.01,0.7]). The differ-ent failure probabilities for ‘‘close’’ and ‘‘remote’’(to ES line) particles captures the fact that the par-ticles tend to consume high (or low) energy in eachcase respectively, due to the fact that ‘‘close’’ to ES

particles transmit more frequently than ‘‘remote’’ones. To evaluate the robustness of the protocolwe measure in the case of failure to reach the sink,the distance D from it.

Fig. 14 shows that our protocol is very robust.In particular, even for large failure probabilitiesas high as 0.5, the protocol successfully propagatesdata to the sink. As the failure probability in-creases (i.e. fr > 0.4) the protocol manages to getclose enough to the sink (i.e. in the worst casethe final position of the propagation is 1400 m,while kESk = 7000 m). We note that proximity offinal propagation position to the sink also in-creases with failure probability of ‘‘close’’ particles(fc).

Note that the proximity of the final position tothe sink seems to exhibit a certain threshold behav-ior (in the case of values studied, this threshold is

around fr = 0.5). This threshold behavior is prob-ably due to the stochastic process evolution andwe intend to also evaluate it analytically.

A possible intuitive explanation of this behavioris that when particles lying further away from theoptimal transmissions line are more or less opera-tional (i.e. fr < 0.5) then, regardless of the failure ofparticles on the optimal line, information messagesrearch close to the sink (i.e. the distance D issmall); when however the failure probability ofparticles (fr) becomes larger, then it becomesimpossible to get the messages to the sink sinceno sufficient number of particles to propagate dataare operational (and thus distance D becomes verylarge).

7. Conclusions and future work

We presented here a new protocol for informa-tion propagation in sensor networks. This proto-col can be used either directly to solve a localcrucial event detection and propagation problemor as part of a more general information dissemi-nation paradigm (such as in [15]). PFR is basedonly on local information (and thus it is necessarya probabilistic one) and succeeds to efficiently


propagate information to the sink without flood-ing the network (although each transmitting parti-cle broadcasts around it), by probabilisticallyfavoring certain close to optimal ‘‘paths’’ towardsthe sink and also by avoiding any control mes-sages. Our analysis shows that the protocol is veryefficient, in terms of energy consumption, whileachieving high success rates even in the case wherepart of the network fails. We have implementedour protocol and conducted extensive simulationson networks of large size to validate the analysisand also investigate other important performancemeasures, such as proximity of final position tothe sink when a significant part of the networkbecomes inoperative.

We plan to extend our simulation results bystudying the performance of our protocol in moregeneral topologies and for additional networkparameters. We believe that the protocol indeedwould work in random topologies with appropri-ate modifications, such as a different thresholdangle in the probabilistic choice taking intoaccount the network density. Also, to compara-tively evaluate PFR against other protocols in therecent literature.

Acknowledgement

This work has been partially supported by theIST Programme of the European Union undercontract numbers IST-2001-33116 (FLAGS) andIST-2004-001907 (DELIS).

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Ioannis Chatzigiannakis is currently aResearcher of Research Unit 1(‘‘Foundations of Computer Science,Relevant Technologies and Applica-tions’’) at the Computer TechnologyInstitute (CTI), Patras, Greece and alsoa Post-doc PHD Researcher at theComputer Engineering and InformaticsDepartment of Patras University,Greece. His research interests includeFundamental Issues in Mobile Com-

puting, Data Propagation in Wireless Sensor Networks, Coop-
erative Mobile Robotic Systems and Algorithmic Engineering.
He received his BEng degree from the Computer Science andEngineering Department of the University of Kent at Canter-

bury, UK in 1997 and his Ph.D degree from the ComputerScience and Engineering Department of the University ofPatras, Greece in 2003. His Ph.D advisor was PaulSpirakis.

He has published scientific articles in international confer-ences and journals. He has participated in EU funded R&Dprojects and project funded by the Private Section.

Tassos Dimitriou is an assistant pro-fessor at Athens Information Tech-nology interested in various aspects ofTheoretical Computer Science likeCombinatorial Optimization, Analysisof Heuristics for difficult to solveproblems, Use of Randomness in algo-rithms and Derandomization tech-niques, Generation of hard instancesfor SATisfiability problems and ana-lysis of SAT heuristics, Algorithms for

Selfish Agents and Smart Dust systems, Cryptography.
He received his B.Sc. degree from the Computer Science and
Engineering Department of the University of Patras, Greeceback in 1990 and his M.Sc. and Ph.D degrees from the Uni-versity of California, San Diego in 1993 and 1996, respectively.His Ph.D advisor was Russell Impagliazzo.

Sotiris Nikoletseas is currently a SeniorResearcher and Managing Director ofResearch Unit 1 (‘‘Foundations ofComputer Science, Relevant Technol-ogies and Applications’’) at the Com-puter Technology Institute (CTI),Patras, Greece and also a Lecturer atthe Computer Engineering and Infor-matics Department of Patras Univer-sity, Greece. His research interestsinclude Probabilistic Techniques and

Random Graphs, Average Case Analysis of Graph Algorithms
and Randomized Algorithms, Fundamental Issues in Paralleland Distributed Computing, Approximate Solutions toComputationally Hard Problems. He has published scientificarticles in major international conferences and journals and hasco-authored (with Paul Spirakis) a book on ProbabilisticTechniques.
He has been invited speaker in important internationalscientific events and Universities. He has been a referee forthe Theoretical Computer Science (TCS) Journal andimportant international conferences (ESA, ICALP). He hasparticipated in many EU funded R&D projects (ESPRIT/ALCOM-IT, ESPRIT/GEPPCOM). He currently partici-pates in 6 Fifth Framework projects: ALCOM-FT, ASPIS,UNIVERSAL, EICSTES (IST), ARACNE, AMORE(IMPROVING).


Paul Spirakis (google: Paul Spirakis)born in 1955, obtained his Ph.Dfrom Harvard University, USA, in1982. Has served as a postdoctoralresearcher at Harvard University andas an Assistant Professor at New YorkUniversity, (the Courant Institute). Hewas appointed as an Associate Profes-sor in the Department of ComputerScience and Engineering of PatrasUniversity (Greece) in 1987 and pro-

moted to Full Professor in the same department in 1990.
He was honored several times with international prizes and
grants (e.g. NSF), also the top prize of the Greek MathematicsSociety. He was appointed as a Distinguished Visiting Scientistof Max Planck Informatik in 2001. His research interestsinclude probabilistic methods in algorithms, combinatorialoptimization, average case analysis of algorithms, parallelalgorithms, algorithms and protocols for distributed systems,algorithms and complexity of graph theoretic problems, parallelcomplexity, approximations to hard problems. Recently,research on Foundational issues of algorithmic game theory.He has done work also in design and applications of networkprotocols, security in computer networks, telematics problemsand services, telematics in Education, Performance analysis ofcomputer systems, performance and algorithms for databases,distributed systems design, Queueing Theory.

He has extensively published in most of the importantComputer Science Journals and most of the significant refereedconferences including the ACM STOC, the ACM/IEEE FOCS,the ACM SPAA, the ACM PODC, most of the other ACMconferences and the major European Conferences like ICALP,STACS, e.g. He has edited various conference proceedings andis currently an Editor of the Mathematical Systems TheoryJournal, the Elsevier Computational Geometry J., the ParallelProcessing Letters Journal, the Journal of Theoretical Com-puter Science and the Journal of Parallel and DistributedComputing.

He had published two books through Cambridge UniversityPress, four books in Greek with Gutenberg publications, twobooks with Patras University and two books with Greek Letterspublications. He was the Greek National Representative in theInformation Society Research Programme (IST) from January1999 till June 2002. He was elected unanimously as one of thetwo vice-President of the Council of the European Associationfor Theoretical Computer Science (EATCS), in July 2002 in theIcalp Conference that took place in Malaga, Spain. He ismember of ISTAG (Information Society Technologies Advi-sory Group) a prestigious body of about 40 individuals advisingEU for research policy, from January 2003 on. He consults forthe Greek State, the European Union and several major GreekComputing Industries.

a probabilistic algorithm for efficient and robust data propagation in wireless sensor networks

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