bootstrapping a hop-optimal network in the weak...

37

Bootstrapping a Hop-Optimal Network in the Weak

Sensor Model

MARTIN FARACH-COLTON AND ROHAN J. FERNANDES

Rutgers University

AND

MIGUEL A. MOSTEIRO

Rutgers University and Universidad Rey Juan Carlos

Abstract. Sensor nodes are very weak computers that get distributed at random on a surface. Oncedeployed, they must wake up and form a radio network. Sensor network bootstrapping research thushas three parts: One must model the restrictions on sensor nodes; one must prove that the connectivitygraph of the sensors has a subgraph that would make a good network; and one must give a distributedprotocol for finding such a network subgraph that can be implemented on sensor nodes.

Although many particular restrictions on sensor nodes are implicit or explicit in many papers, thereremain many inconsistencies and ambiguities from paper to paper. The lack of a clear model meansthat solutions to the network bootstrapping problem in both the theory and systems literature all violateconstraints on sensor nodes. For example, random geometric graph results on sensor networks predictthe existence of subgraphs on the connectivity graph with good route-stretch, but these results do notaddress the degree of such a graph, and sensor networks must have constant degree. Furthermore,proposed protocols for actually finding such graphs require that nodes have too much memory, whereasothers assume the existence of a contention-resolution mechanism.

We present a formal Weak Sensor model that summarizes the literature on sensor node restrictions,taking the most restrictive choices when possible. We show that sensor connectivity graphs have low-degree subgraphs with good hop-stretch, as required by the Weak Sensor model. Finally, we give aWeak Sensor model-compatible protocol for finding such graphs. Ours is the first network initializationalgorithm that is implementable on sensor nodes.

Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]:Nonnumerical Algorithms and Problems; C.2.1 [Computer Communication Networks]: Network

A preliminary version of this article appeared in Farach-Colton et al. [2005]. This research wassupported in part by DIMACS, Center for Discrete Mathematics and Theoretical Computer Science,grants NSF CCR-00-87022, NSF EIA-02-05116 and Alfred P. Sloan Foundation 99-10-8.Authors’ addresses: M. Farach-Colton, R. J. Fernandes, Department of Computer Science, RutgersUniversity, Piscataway, NJ 08854; email: {farach,rohanf}@cs.rutgers.edu; M. A. Mosteiro, Depart-ment of Computer Science, Rutgers University, Piscataway, NJ 08854 and GSyC-LADyR, Universi-dad Rey Juan Carlos, Mostoles, 28933 Madrid, Spain; email: [email protected] to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies show this notice on the first page or initial screen of a display along with thefull citation. Copyrights for components of this work owned by others than ACM must be honored.Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistributeto lists, or to use any component of this work in other works requires prior specific permission and/ora fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701,New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or [email protected]© 2009 ACM 1549-6325/2009/10-ART37 $10.00DOI 10.1145/1597036.1597040 http://doi.acm.org/10.1145/1597036.1597040

ACM Transactions on Algorithms, Vol. 5, No. 4, Article 37, Publication date: October 2009.

37:2 M. FARACH-COLTON ET AL.

Architecture and Design—Wireless communication; distributed networks; C.2.4 [Computer

Communication Networks]: Distributed Systems; C.2.2 [Computer Communication Networks]:Network Protocols

General Terms: Algorithms, Design, Theory

Additional Key Words and Phrases: Ad hoc network, contention resolution, maximal independent set,radio network, random geometric graphs, sensor network, weak sensor model

ACM Reference Format:

Farach-Colton, M., Fernandes, R. J., and Mosteiro, M. A. 2009. Bootstrapping a hop-optimal networkin the weak sensor model. ACM Trans. Algor. 5, 4, Article 37 (October 2009), 30 pages.DOI = 10.1145/1597036.1597040 http://doi.acm.org/10.1145/1597036.1597040.

1. Introduction

Advances in technology have made it possible to integrate sensing, processing,and communication in a low-cost device, popularly known as a sensor node. Sen-sor nodes are randomly deployed over an area and must self-organize as a radio-communication network called a sensor network. Even though communicationamong sensor nodes is through radio broadcast, it is useful to set up explicit linksbetween nodes in order to establish routing paths and prevent flooding. The maincontribution of this work is to show that it is possible to establish those links fromscratch, in such a way that the network obtained has “good” geometric properties,even in the most constrained scenario.

A sensor network is capable of achieving large tasks through the coordinatedeffort of sensor nodes, but individual nodes have severe limitations on memorysize, life cycle, range of communication, etc. Any sensor network initializationalgorithm must be fast and distributed, and must resolve channel contention is-sues. The network constructed by such an algorithm must be connected and musthave low degree and diameter. The limitations on individual sensors nodes makethis problem nontrivial, and its adequate resolution is crucial for making sensorsuseful.

There are two main types of issues in sensor network formation: those relatingto geometric properties and those relating to network protocols; and any solutionachieved for either must be compatible with an accurate model of sensor nodes.On the one hand, coverage and connectivity in sensor networks are dependent onthe distribution of nodes in an area and the range of transmission of each node.Additionally, the density of nodes in an area determines the minimum path lengthbetween any two nodes in the induced connectivity graph. The limited range oftransmission makes these properties geometric. On the other hand, protocols forsensor network formation are limited by the fact that sensor nodes share a commonchannel of communication and that they do not typically have access to directionalor positional information. Memory limitations in sensor nodes also impose therestriction that a node can only keep track of O(1) neighbors.

The existing literature on sensor network initialization does not sufficiently han-dle all aspects of the problem. A popular topology model in the sensor networkliterature is the random geometric graph, where n nodes are deployed uniformly atrandom in the unit square and an edge between any two nodes exists if and only ifthey are separated by a Euclidean distance of at most a parameter r . All random ge-ometric graph results related to ad hoc wireless networks require ω(1) degree (see,for example, Muthukrishnan and Pandurangan [2005]). All proposed protocols for


Bootstrapping a Hop-Optimal Network in the Weak Sensor Model 37:3

sensor network formation include some inappropriate hardware assumptions. Forexample, the sensor network formation protocol in Song et al. [2004] builds aconstant-degree network, but relies on positional information hardware. The pro-tocol proposed in Blough et al. [2003] also builds a constant-degree network, butrelies on the preexistence of a scheme for channel-contention resolution. The dif-ferent models implicit in such results are inadequate and poorly reflect the variouslimitations under which sensor nodes operate, and indeed, there seems to be consid-erable confusion in the literature as to what are or are not reasonable assumptionsabout the capabilities of sensor nodes.

Sensor network initialization research has three parts: (i) to specify a compre-hensive model that captures all the restrictions present in sensor nodes; (i i) giventhat under those restrictions is not possible to establish all the links of the con-nectivity graph, to show that there exists a subgraph of the connectivity graph thatwould make a connected network without asymptotically increasing the cost ofdelivering messages; and (i i i) to give a fast distributed protocol that works underthe constraints of the specified model.

In this article, we present a formal Weak Sensor model that summarizes the liter-ature on sensor node restrictions, taking the most restrictive choices when possible.Given the Weak Sensor model, we argue that a good sensor network must have con-stant degree and low hop-stretch, which we define shortly. Our main contribution isto show that any appropriate random geometric graph has such a subgraph. Finally,we complete this work by giving a Weak Sensor model-compatible protocol forfinding such a subgraph. Ours is the first network initialization algorithm that isimplementable on sensor nodes.

1.1. ROADMAP. The remainder of this article is organized as follows. InSection 2 we review previous work in the sensor network area and related prob-lems, in Section 2.4 we survey the existing literature in sensor network initializationand in Section 3 we survey the different models used and finally present the WeakSensor model in Section 4. In Section 5 we analyze geometric properties of goodsensor networks. In Section 6 we present and analyze a distributed algorithm forfinding good sensor networks. We conclude in Section 7.

2. Related Work

The sensor networks area is very active and includes a vast body of theoretical andempirical research work impossible to completely include here. We overview someof this work in this section.

2.1. THRESHOLD PROPERTIES IN Gn,r AND Gn,r,�. Gupta and Kumar [1998], ina seminal paper in the field of random geometric graphs, computed the minimumradius needed to obtain a large connected component with high probability. Thisand other results [Penrose 2003] give us a critical radius such that each node willhave many neighbors. Of course, sometimes, a two-dimensional model may beinadequate when the terrain in which the sensors are positioned is uneven. In thiscase an extension to three-dimensional random geometric graphs may be needed.

In the Gn,r,� model, tight thresholds for connectivity, coverage, and route stretchwere shown by Muthukrishnan and Pandurangan [2005] using an overlapping dis-section technique called bin-covering. More recently, Goel et al. [2004] showed



that in fact all monotone graph properties have a sharp threshold for random geo-metric graphs. Other properties of random geometric graphs such as vertex degreeor k-connectivity were studied in Appel and Russo [1997a, 1997b] and Penrose[1999].

2.2. BLUETOOTH. Bluetooth [Bray and Sturman 2001; Miller and Bisdikian2000],which also limits the local connectivity of nodes, is a local area wirelesstechnology designed to enable voice and data communication between variouselectronic devices. In these networks the nodes have less restrictive constraints(like power supply, range of transmission, memory capacity, etc.) than in sensornetworks. In Bluetooth, a group of devices sharing a common channel is called apiconet. Each piconet has a master unit that selects a frequency hopping sequencefor the piconet and controls the access to the channel. Other participants of the groupknown as slave units are synchronized to the hopping sequence of the piconet master.The maximum number of slaves that can simultaneously be active in a piconet isseven. A slave in one piconet can be a master or slave in another piconet. Piconetscan also be interconnected via bridge nodes to form a bigger ad hoc network knownas a scatternet.

There has been considerable work on schemes for the formation of scatternets.Barriere et al. [2003] proposed a distributed construction technique for Bluetoothscatternets of low degree and fixed diameter. This technique is useful even in thedynamic case where nodes are assumed to come alive and drop dead from timeto time. However, this technique is restricted to networks where all nodes arewithin transmission range of each other and hence is unrealistic for the purposeof sensor network formation. Salonidis et al. [2001] earlier proposed an algorithmfor constructing scatternets, but this technique suffers from the same limitations asexplained before and further is restricted to 32 nodes and static layout. Schemesproposed for scatternet formation in Law and Siu [2001], Salonidis et al. [2001],Wang et al. [2002], Zaruba et al. [2001], and Ferraguto et al. [2004] are designedto work in the more general case where all nodes may not be within transmissionrange of each other. Techniques proposed in these are strictly heuristic or do not fitin the Weak Sensor model.

2.3. CELLULAR SYSTEMS. There are various reasons why medium access con-trol protocols used in cellular systems can not be used in sensor networks. In acellular system, mobile nodes are a single hop away from distinguished nodescalled base stations and the base stations form a wired backbone. The primary goalof a medium access scheme in a cellular system is to guarantee quality of serviceand efficient bandwith use, but power efficiency has a secondary role given thatthe base stations have constant power supply and the users can replenish the bat-teries of the mobile nodes. In sensor networks there is no central control such as abase station and power efficiency dominates the lifecycle of the network, thereforeexistent solutions for cellular systems cannot be applied.

2.4. SENSOR NETWORKS INITIALIZATION AND RELATED PROBLEMS. The so-lutions proposed in the literature do not sufficiently handle all the aspects of thesensor network initialization problem. We overview some of them in this section.

A protocol called Self-Organizing Medium Access Control for Sensor Networkswas presented in Sohrabi et al. [2000]. This protocol builds a flat topology with nolocal or global masters achieving good load balance. The model for this protocol



is as follows. Due to the short transmission range of the sensor nodes, it is as-sumed that a reception consumes the same energy as the transmission. Therefore,nodes cannot have their radios on permanently. There are enough channels as toaccomodate each link among neighbors in a different frequency in order to avoidcollisions. Furthermore, the number of available channels is assumed to be bigenough so that if two nodes choose a channel at random the probability of choosingthe same channel is low. Another key assumption is that nodes have memory ofsize �(�), where � is the maximum degree in the connectivity graph. Finally,nodes are assumed to start running the protocol (wake up) at random times un-der some distribution such that the probability of two nodes being synchronized islow.

The protocol K-Neigh [Blough et al. 2003] builds a network where every nodehas at most k neighboring nodes, where k is tuned to ensure connectivity withhigh probability (w.h.p.). The model under which this protocol works includesthe following assumptions. Nodes are deployed in the plane uniformly at random.Although the transmission power can be adjusted, all nodes are constrained to thesame maximum P . This maximum transmission power is a function of n and it ischosen so that the network is connected with high probability. The protocol alsorelies in some distance estimation mechanism such as measuring the radio signalstrength received or comparing the time of arrival of different kinds of signals.Given that information about all neighboring nodes is collected, the memory sizeis assumed to be in ω(1). Although the synchronization is local, the differencebetween node wake-up times is upper bounded by a constant �.

An energy-efficient topology control scheme called OrdYaoGG was presented inSong et al. [2004]. The assumptions of this model are as usual n nodes distributeduniformly in the R

2 plane, each node with a maximum transmission range normal-ized to 1. Therefore, the connectivity graph is a UDG. All nodes have different IDnumbers and each node knows its position information by means of a GPS or otherspecialized hardware such as a directional antenna and signal strength measurementcapabilities. At a minimum, the assumption is that every node knows in advance orwill collect the position information of all its neighbors. Therefore, the memory sizeis assumed to be in ω(1). Global synchronization is also necessary given that theproposed algorithm works in phases. Finally, an underlying contention resolutionmechanism is assumed in order to collect information of neighboring nodes. Thetopology obtained by this protocol has optimal power stretch, constant degree, andit is planar. The power stretch is defined to be the ratio between the energy cost ofa path connecting two nodes to the optimal, that is, the cost in an optimal path inthe potential connectivity graph. Obtaining a planar topology is a requirement ofmany routing algorithms to guarantee message delivery.

In all these schemes, no contention resolution mechanism is given. Among otherassumptions, the number of channels available, memory size, nonarbitrary start-ing times, distance estimation hardware, etc., make these solutions infeasible forthe most general sensor network setting. More general information about sensornetworks can be obtained from the surveys Rentala et al. [2001], Akyildiz et al.[2002], Karl and Willig [2003], Ponduru and Bharathidasan [2003], Younis et al.[2006], and Li [2004].

A technique frequently used in the sensor networks initialization literature isto include an initial phase that gives structure to the network. This goal is usu-ally achieved creating a hierarchy of nodes, for instance, defining a Connected



Dominating Set (CDS) among them. Unfortunately, most of the literature on con-structing a CDS distributedly include results that do not work in a model with strongmemory limitations. The CDS algorithms in Parthasarathy and Gandhi [2004] donot work under a model with harsh memory size restrictions, given that the algo-rithms presented there require that nodes have information about all neighboringnodes. Similarly, the algorithm presented in Alzoubi [2003] requires the availabilityof a MAC layer and the degree of the spanner obtained is in ω(1).

3. Models

In this section, we review models used in radio networks that are usually found inthe literature. As explained before, in radio networks, it is not always the case thatevery node is connected directly to every other node. Furthermore, in many casesthis connection is not even symmetrical. Therefore, a model for the topology of thenetwork needs to be defined. Also, depending on the application, radio networkshave very different node constraints, for example, in some networks nodes haveternary feedback but in others the feedback is just binary. Therefore, a detailedmodel of the constraints present in the nodes forming the network is also needed.We summarize in this section models of topology and node constraints used in theradio networks area and in Section 4 we focus in sensor networks, describing indetail our harsh Weak Sensor model [Farach-Colton et al. 2005]. More details aboutsensor networks classification and taxonomies can be found in Tilak et al. [2002].

3.1. TOPOLOGY MODELS. Regarding the topology of a network, a well-knownspecification is given by a directed graph. A directed graph is a pair of sets {V, E},where V is a set of points or nodes and E is a set of ordered pairs of distinctpoints taken from V . Any such pair is called an arc or an edge. In our context, thepoints model the nodes of the network and the arcs represent the ability to sendmessages directly (in one hop) from one node to another. If the communication in anetwork is achieved through wires, an edge AB in the graph represents the link thatfacilitates the communication from A to B. If on the other hand the communicationin a network is wireless, an edge AB in the graph implies that B is in the range oftransmission of A. Whenever this relation is symmetric, an undirected graph canbe used as a model. For example, in a wireless network where all nodes have thesame range of transmission, an undirected graph is a suitable model because if anode B is reachable from a node A, A is also reachable from B.

3.1.1. Topology in Radio Networks. The connectivity model widely used inradio networks where all nodes have the same range of transmission is the geometricgraph. The specification of a geometric graph includes a pair of sets {V, E} anda number r ∈ R

+. The set of nodes are points in R2 and an edge AB ∈ E if and

only if A and B are separated by an Euclidean distance of at most r . As mentionedbefore, the graph is undirected because the range of transmission of all nodes isassumed to be the same. If this is not the case, more sophisticated models areneeded.

There are also some variations of a geometric graph in the literature. When thedistance r , modeling the range of transmission is normalized to 1, the graph iscalled Unit Disk Graph (UDG). For cases in which the connectivity beyond somedistance r ∈ (0, 1] is uncertain, there is a generalization of a geometric graph called



in the literature Quasi-Unit Disk Graph (QUDG). The QUDG includes all edgesshorter than a parameter 0 ≤ d ≤ 1 and no edge longer than 1, but the existenceof edges of length in (d, 1] is not specified. The latter model can be extended witha distribution on the probability of being connected when the separation distanceis bigger than the uncertainty threshold. Also, the uncertainty threshold can bedefined as a function of the angle with respect to some direction of reference forcases where directional antennas are used.

Of course, any of these models can be also extended with node sets in higherdimensional spaces and with threshold distances under different metrics. The par-ticular extension depends on the setting we are modeling. A usual simple extensionis to consider the points in R3 to model the deployment of the network in the realworld. Another possible extension is to consider a distribution on the probability oftwo nodes being connected. Such an extension would imply a combination of theclassical random graph model [Erdos and Renyi 1959] with a geometric graph. Amore appropriate application of randomness to the geometric graph model in thespecific area of sensor networks is explained in the next section.

3.1.2. Topology in Sensor Networks. In addition to a comprehensive model forthe various constraints present in sensor networks, a formal model of the potentialconnectivity of the network needs to be defined. In the past, computer networkshave been modeled by means of classical random graphs. Starting in 1959 witha paper by Erdos and Renyi [1959], the field of random graphs has been widelyexplored. The classical Bernoulli random graph model is denoted as Gn,p, where nis the number of nodes and p is the probability of existence of each edge. Randomgraph models have been used, for instance, to model the Web-graph [Aiello et al.2002; Kleinberg et al. 1999] where the structure of the random graph gives in-sight into the behavior of the Web-graph. However, the classical random graphmodel is not adequate for the sensor network setting because the probability ofhaving an edge AB is either 0 or 1, depending on the Euclidean distance betweenA and B.

Regarding the deployment of nodes in a sensor network, deterministic deploy-ment, that is, the placement of nodes at specific locations, is only possible for smallnetworks in a friendly environment. However, this scenario is not reallistic for mostof the intended applications of sensor networks where a large area is expected tobe covered and the environment is expected to be either hostile or remote. Hence,random deployment is used. Nevertheless, such deployment of nodes is not theresult of some uncontrolled experiment where any arbitrary distribution of nodeshas some positive probability of occurring. Thus, a specific distribution is also afrequent assumption. Two models of random deployment of nodes are used. In onemodel, n nodes are assumed to be distributed uniformly at random so that each nodeis equally likely to fall in any location of the area of interest, independently of theother nodes. The other model is a stationary Poisson point process with intensity nwhere the number of nodes in disjoint regions is Poisson distributed and mutuallyindependent. In this article, we use the first model, although the same bounds canbe proved for the latter. As shown in Farach-Colton and Mosteiro [2007], theseresults can be extended to arbitrary upper bounds on the density of nodes in anydisc of radius r .

Thus, sensor networks are best modeled by Random Geometric Graphs (RGGs)in R

2 [Penrose 2003]. In the Random Geometric Graph model Gn,r,�, n nodes are



FIG. 1. A random geometric graph.

distributed uniformly at random in [0, �]2, and nodes are connected by an edge ifand only if they are at an Euclidean distance of at most r ≤ �, the connectivityradius (Figure 1). The node density depends on the relative values of n, r and �.A specific instance of Gn,r,� is a Random Geometric Graph (RGG), also referred toas G(n, r, �). A popular instance of this model is Gn,r,1 or simply Gn,r . Of course,sometimes, a two-dimensional model may be inadequate when the terrain in whichthe sensors are positioned is uneven. In this case an extension to three-dimensionalrandom geometric graphs may be needed.

Some properties commonly studied for random geometric graphs within thecontext of sensor networks are as follows.

Physical coverage. For the region in question, what fraction of the region iscovered by balls of radius r , centered on the points thrown randomly into theregion with uniform distribution? More specifically we are interested in the numberof nodes we must throw such that the fraction of the region covered is 1 − o(1).

Graph connectivity. What is the relation among n, r and � when a graph G(n, r, �)becomes connected? In keeping with the random nature of the model we say thatG(n, r, �) is connected when it is connected with high probability.

Route stretch. Given two nodes u, v in a graph G(n, r, �), stretch(u, v) is definedas the ratio of the shortest distance between u and v in the graph to the normed dis-tance between the two points in the plane. The stretch of G(n, r, �) is the maximumof the stretch over all pairs of points (u, v) in G(n, r, �).

The theory of random geometric graphs is a key tool to study some of the un-derlying properties in sensor networks such as connectivity or coverage. However,the results obtained in this field cannot be directly applied to sensor networks dueto the additional constraints present in them.

Throughout the article, we study the sensor network initialization problem as-suming that no obstacles are present in the area of interest for clarity. Nevertheless,the results obtained can be straightforwardly extended to a model with obstaclesdue to the following argument. Instead of the straight-line Euclidean distance, inpresence of obstacles, the minimum distance between any pair of nodes is still



the length of some shortest curve connecting them that does not cross the obsta-cle. Using techniques similar to those included in Section 5.2, the existence of ahop-optimal path with respect to this curve can be proved.

3.2. NODE CONSTRAINTS MODELS. Radio networks is a vast area and there isa myriad of applications of such a technology, such as cellular phones, wirelesscomputer networks, ad hoc networks, etc. Depending on the specific applicationthe nodes forming the network have very different constraints on their processingand communication capabilities, that is, range of transmission, life cycle, storagesize, etc. In addition to formal models of the topology or the potential connectivityamong nodes, an appropriate model of the constraints of the nodes in the networkhas to be defined in order to properly design and analyze protocols. We summarizehere some of the models used in radio networks and sensor networks.

In a seminal paper [Bar-Yehuda et al. 1992], Bar-Yehuda et al. presented aformal model of a radio network that specifies many of the important restrictionson sensor nodes, including, for example, limits on contention resolution, but theymake no mention of computational limits, such as small memory. More precisely,the model consists of an arbitrary multihop undirected network. The nodes areassumed to be locally synchronous, that is, they all have the same clock frequencybut perhaps different starting times. Each node either receives or transmits withineach time slot, but not both. A node receives a message successfully in a time slotif exactly one of its neighboring nodes transmits in that time slot. If more thanone neighboring node transmits in the same time slot, the messages are garbledand the node receives noise. It is not possible to detect collisions, hence, a nodecannot distinguish the case in which no neighboring node transmits from the casein which there is more than one transmit in the same time slot. The topology of thenetwork is not known a priori. The main difficulty in this model, as well as in mostof the models in radio networks, is the possibility of message collision, therefore,any protocol for this model has to include contention resolution in order to beuseful.

After this model was introduced, some papers [Nakano and Olariu 2000; Kumaret al. 2004] have added more restrictions, although often such restrictions are im-plicit in the text or algorithms rather than fully specified. In the following sectionwe elucidate a complete and comprehensive model for sensor networks.

4. The Weak Sensor Model

As explained before, nodes in sensor networks are designed with the goal of ob-taining a device as small as possible and at a very low cost. Therefore, sensor nodeshave very harsh constraints in each of its main capabilities, processing, communi-cation, and sensing. These strong constraints are the main reason why problems insensor networks are challenging, because the typical solutions utilized in computernetworks are not suitable in such a harsh scenario. Therefore, in order to approachany problem in sensor networks, and in addition of formal models of the connec-tivity of the network, a formal model of the various sensor node constraints has tobe defined.

Given the various limitations of sensor nodes and the absence of a reliable com-munication structure after deployment, any sensor network protocol must work un-der difficult conditions. In this section, we specify the formal Weak Sensor model



that summarizes the literature on sensor node restrictions, taking the most restrictivechoices when possible.

Memory size. Sensor nodes may have limited memory size. In fact, asymptoticallyspeaking, if we assume that the memory size is any function in ω(1) we would beassuming that nodes can have a memory of infinite size. Therefore, in the WeakSensor model sensor nodes may store only a constant number of O(log n)-bitwords.

Short transmission range. Due to costs and size restrictions, sensor nodes may nothave a large range of transmission. Consequently, not all nodes are reachablefrom a given node, leading to the well known hidden-terminal problem. Thislimitation has an impact on the density of sensor node deployment.

Discrete transmission range. Some of the extant literature [Song et al. 2004] as-sumes that nodes can vary their power of transmission. However, assuming thatany number of levels can be reached is unrealistic; in particular to analyze theasymptotic behavior of the algorithm. In this model, sensor nodes can adjusttheir power of transmission to only a O(1) number of levels. As we argue later,specific conditions on the relation among the transmission range, the number ofnodes, and the area where those nodes are deployed will be necessary in orderto achieve connectivity.

One channel of communication. Although it is assumed in some papers that ω(1)channels are available in order to avoid collisions, this assumption is unrealistic,specially in order to analyze the asymptotic behavior of protocols. We constrainthe number of channels of communication to exactly one.

Locality. Sensor nodes are distributed over a large area and may not be reachableby a central controller. Hence, each sensor node must be capable of configuringitself automatically.

Low-information channel contention.Shared channel of communication. Given that this is a radio network and that

there is only one channel available, the communication with neighboring nodesis through broadcast in a shared channel.

Contention resolution mechanism. If more than one message is placed on amultiple-access communication channel at the same time, a collision occursand no message is delivered. Hence, sensor nodes have to implement a con-tention resolution mechanism to access the channel.

No initial infrastructure. Right after deployment, the nodes of a sensornetwork have no communication infrastructure available (MAC layer).Therefore, before any exchange of information can be carried out, nodeshave to self-organize a medium access scheme bringing structure to thenetwork.

No collision detection. Although in many radio networks it is possible to detecta collision, it has been also argued that a collision cannot be detected in thepresence of noisy channels [Bar-Yehuda et al. 1992]. In this model, only twochannel states are feasible, namely single transmission and silence/collision.This scenario is popularly known as binary channel or it is said that nodeshave binary feedback.

Nonsimultaneous reception and transmission. A sensor node may not be able toreceive while transmitting because, in its vicinity, its own signal overwhelms



any signal transmitted by other nodes. Therefore, transmitters also cannotdetect collisions.

Asynchronicity. No global clock or other synchronizing mechanism is assumed, butall sensor nodes have the same clock frequency. We assume that time is dividedinto slots. The use of a slotted scenario instead of a more realistic unslotted onewas justified in Roberts [1975], where it was shown that they differ only by afactor of 2 because a packet can interfere in no more than 2 time slots. This typeof synchronicity is usually called local synchronism.

Limited lifecycle. Sensor nodes may be powered by sources such as solar energy.These sensors may go down from time to time to recharge. This necessitatessimpler and fast computations and energy-efficient protocols.

No position information. Due to cost and size restrictions, sensor nodes may nothave position information obtained using a global or local positioning system,directional antenna, or other specialized hardware.

Adversarial node wake-up schedule. Given that the sensor nodes are deployed overlarge areas and given the lack of a centralized controller, we can not expect allsensor nodes to start the execution of protocols in the same time slot. Therefore,in order to analyze these protocols in a worst-case scenario, we assume theexistance of an adversary that determines the wake-up schedule.

Unreliability. In addition to the lack of guarantees of a constant power supply, dueto low cost sensor nodes are unreliable. Hence, sensor network protocols haveto be designed to be robust in the case of failures of one or more sensors.

Throughout this article, our node constraints model is the Weak Sensor model andthe potential connectivity of the nodes is modeled by a random geometric graph. Asexplained in Section 3.1.2, the deployment of nodes in a random geometric graphcan also be interpreted as a Poisson process in the plane where the number of pointsin [0, �]2 is given by the Poisson distribution with mean n. In our proofs, we assumethe uniform deployment, that is, each of the sensors is equally likely to fall at anylocation in [0, �2] independently of the other sensors, although the results hold forthe Poisson model as well with almost no change in the proof techniques.

5. Geometric Analysis of Sensor Networks

Recall that sensor nodes may only set up links with a constant number of neighbors,a consequence of the memory size limitation in the Weak Sensor model, and sincesensor nodes are distributed uniformly at random, the potential connectivity rela-tion defines a Random Geometric Graph (RGG). Hence, any protocol for networkformation must set up links defining a constant-degree spanning subgraph of theRGG. However, ignoring potential links may result in an increase in path lengths inthe subgraph. This increase in path length can be measured in two ways: in termsof increase in the number of hops or increase in route stretch.

In applications where the propagation delay is significant, route stretch is anappropriate measure of optimality. However, sensor networks have small internodedistances, and propagation delay is low. One of our primary concerns in the WeakSensor model is that we should minimize energy consumption at each node so asto maximize the lifecycle. Thus, a sensor network is optimal when it minimizes thenumber of transmissions, which is to say, minimizes the number of hops in each



path, rather than the weighted path length. Note that schemes have been proposedthat attempt to minimize energy consumption [Song et al. 2004], and these favormany short hops over a few long ones. However, any such scheme requires anω(1) number of transmission power levels and, furthermore, ignores the contentionresolution overhead of the extra hops. A formal definition of stretch in terms ofhops follows.

Let the length of a path connecting two nodes in a given graph be the numberof edges of such a path. Let dmin(u, v) be the shortest path between two nodes uand v in the RGG G(n, r, �). Let D(u, v) be the Euclidean distance between u andv in the plane. Note that in G(n, r, �), �D(u, v)/r� is a lower bound on dmin(u, v).Call this lower bound, dopt (u, v). The hop-stretch of (u, v) is defined as the ratiodmin(u, v)/dopt (u, v). The hop-stretch of G(n, r, �) is the maximum of the hop-stretch of (u, v) over all pairs of points (u, v) in G(n, r, �). In the rest of this sectionwe will outline a scheme to obtain a constant degree hop-optimal subgraph from asufficiently dense random geometric graph.

5.1. DISK COVERING SCHEME FOR SENSOR NETWORK FORMATION. The diskcovering scheme presented in this section shows the existence of a bounded degree,bounded stretch subgraph of a RGG. Roughly, the main idea of the scheme is todefine a cluster graph of constant degree and to connect nodes within each cluster sothe resulting subgraph has constant degree and low stretch. Similar approaches havebeen followed in Das and Narasimhan [1997] and Gudmundsson et al. [2002] fora different model of computation yielding superlinear complexity. The descriptionand analysis of a distributed algorithm is presented later in Sections 6.1, 6.2, and6.3. Before describing the scheme, we introduce some necessary terminology.

Definition 5.1. A Random Geometric Graph or G(n, r, �) is an instance ofGn,r,� where r is the connectivity radius.

Given a sufficiently dense G(n, r, �) = 〈V, E〉, the goal of the disk coveringscheme is to produce as output a spanning subgraph 〈V ′, E ′〉 such that V ′ = V ,E ′ ⊂ E , the maximum degree is bounded by a constant and the path length isasymptotically optimal. The precise nature of the path length optimality is given inthe proof of Theorem 5.11.

Definition 5.2. The graph obtained as a result of the disk covering scheme iscalled the Constant-degree Hop-optimal Spanning Graph (CHSG).

The following definitions will be relevant here and their meaning will be clearafter the disk covering scheme is fully described.

Definition 5.3. All nodes covered by the same disk at the end of the diskcovering scheme are connected to each other in the RGG and will be referred to asa disk-clique.

Definition 5.4. Some (possibly all) of the nodes covered by the same disk atthe end of the disk covering scheme are connected to each other by a spanner in theCHSG and will be referred to as a disk-spanner.

Definition 5.5. A bridge is a node, lying at the center of a disk, that is designatedto communicate between two or more disk-cliques.



(a) the connectivity graph (b) step 1

(c) step 2 (d) step 3

(e) step 4 (f) step 5 (partially)

FIG. 2. Illustration of the disk covering scheme.

The pseudocode in Algorithm 1 summarizes the disk covering scheme. Note thata and b are tunable parameters that affect the maximum degree and hop-stretch ofthe CHSG. Figure 2 illustrates this protocol.

Algorithm 1. Disc Covering Scheme.

1 Add all nodes from the RGG to the CHSG.2 Lay down small disks of radius ar/2, 0 < a < 1 centered on nodes, such that no central node is

covered by more than one small disk and no node is left uncovered. We call each central node abridge. Note that the bridges form a Maximal Independent Set (MIS) of the spanning subgraphG(n, ar/2, �) ⊆ G(n, r, �).

3 Add to the CHSG all edges from the RGG that connect bridges.4 Expand the small disks into big disks of radius br/2, a < b ≤ 1.5 Add to the CHSG the necessary edges to form a disk-spanner of constant degree among nodes

covered by the same big disk.

5.2. ANALYSIS OF THE DISK COVERING SCHEME. In this section the disk cov-ering scheme described in Section 5.1 is proved to produce a CHSG with asymptot-ically optimal path length. In Section 5.2.1 we establish a bound on the maximumdegree of a node in the CHSG. In Section 5.2.2 two useful results for a connectedG(n, r, �) are established: a bound on hop-stretch and bounds on the node den-sity. Finally, in Section 5.2.3 we prove a theorem on the hop-optimality of theCHSG.



FIG. 3. Illustration for Lemma 5.6.

5.2.1. Degree Bound.

LEMMA 5.6. At the end of the disk covering scheme, each edge of length atmost (b − a)r/c has both endpoints within a single big disk with high probability,for any constant c > 1.

PROOF. For the sake of contradiction, assume there exists such an edge e oflength l ≤ (b − a)r/c not covered completely by one big disk. All nodes arecovered by small disks. Each endpoint of e has to be covered by a different bigdisk, otherwise e is already covered. Call C the center of e. Call D the center ofany big disk partially covering e. Since e has at least one point outside the big disk,the distance d(D, C) > br/2 − l/2 as shown in Figure 3.

Therefore, all centers of big disks that partially cover e are located outside a circleof radius (r − l)/2 centered on C . Then, the corresponding small disks leave anuncovered area bigger than the area of a circle of radius r ′ > br/2 − l/2 − ar/2 ≥(b − a)(c − 1)r/2c. Since there is no small disk in this area, there is no node inthis area, otherwise it would be a disk center. But, as proved in Lemma 5.9, inany circle of radius �(r ) there are �(log l) nodes with high probability. This is acontradiction.

LEMMA 5.7. The degree of any node in the CHSG is in O(1).

PROOF. All bridges are separated by a distance of at least ar/2. Connectedbridges are at a distance of at most r . In Figure 4(a) consider the smallest regularhexagon whose side is a multiple of ar/2 and covers completely a circle of radius r .Consider a tiling of such hexagon with equilateral triangles of side ar/2. As provedby Fejes-Toth [1940], the hexagonal lattice is indeed the densest of all possibleplane packings. Therefore, the number of vertices in such a tiling is an upper boundon the number of bridges that connect to a bridge located in the center of such ahexagon. That number is

3

⌈4

a√

3

⌉ (⌈4

a√

3

⌉+ 1

). (1)

There is an extra edge that is needed to connect a bridge with its disk-spanner. Sincea is any constant such that 0 < a < 1, the degree of any bridge is in O(1).



(a) bridge nodes

(b) nonbridge nodes

FIG. 4. Illustration of the upper bound on the degree.

Using a simple geometric packing argument, it can be proved that a nonbridgenode is covered by at most π/ arcsin(a/2b) big disks. By construction, a nonbridgenode is connected to a constant number of neighbors within the same big disk (seeFigure 4(b)). Therefore, the degree of any node is in O(1).

5.2.2. Hop-Stretch and Density in G(n, r, �). Theorem 5.8 demonstrates theexistence of a path with an asymptotically optimal hop-stretch. The proof of thetheorem uses an overlapping dissection technique, called bin-covering, presented inMuthukrishnan and Pandurangan [2005]. A crucial requirement in a sensor networkis that the whole network must have a unique connected component. In other words,every pair of nodes must be able to communicate, either directly or relying onretransmissions of other nodes. The conditions on the relation among r , n and �,required in the following theorem, were shown to be necessary conditions to achieveconnectivity in the same paper.

THEOREM 5.8. Given a G(n, r, �) where the following conditions are satisfied:r2n = k�2 ln �, r = θ (�ε f (�)), f (�) ∈ o(�γ ), γ > 0, 0 ≤ ε < 1, and 0 < α ≤ 1 isa fixed constant. For any constant k > 5 4+α2

α, the hop-stretch is 1 + √

α2 + 4 withhigh probability.

PROOF. It is enough to show that for any pair of nodes (u, v), there is a pathP defined by a sequence of nodes 〈u = x0, x1, . . . , xm = v〉 such that the ratiobetween the number of hops m of such a path and �D(u, v)/r� is bounded upwardsby 1 + √

α2 + 4 with high probability.



u

v

s

r /2

r/2

αr/2

slices

FIG. 5. Strip between nodes u and v showing bin covering and slices.

For a given pair of nodes (u, v), the bin-covering technique is applied as follows.Let r ′ be the shortest horizontal projection of a segment of length r contained inthe strip, namely r ′ = r/

√1 + (α/2)2. The line connecting u and v is covered with

overlapping bins of dimension r ′/2 × αr ′/2 with a spacing parameter s, as shownin Figure 5. This bin layout will be referred to as a strip.

The coordinate system is rotated such that the line segment u, v is parallel tothe x axis. In what follows all distances are specified within this rotated frameof reference. Let Dh(x, y) and Dv (x, y) be the horizontal and vertical distances,respectively, between the nodes x and y.

Given a node x j in the path P the node x j+1 is selected using the followingcriteria.

—The node x j+1 lies within the strip.—Dh(x j , x j+1) ≤ r ′.—The horizontal distance Dh(x j+1, v) is minimized.

A hole is a rectangle of dimension r ′/2 × αr ′/2, within a strip, that is devoid ofnodes and adjoins a node on the side closest to u.

Consider any 3 consecutive nodes along the path xi−1, xi , xi+1 where 0 < i < m,and assume that along any strip there is no hole, then Dh(xi−1, xi ) ≥ r ′/2. To seethat this claim is true, assume for the sake of contradiction that Dh(xi−1, xi ) < r ′/2.The distance Dh(xi−1, xi+1) > r ′, otherwise xi+1 would have been selected as thesuccessor of xi−1. Thus, the distance Dh(xi , xi+1) > r ′/2. Since there cannot be anyhole in the strip, there exists a node y such that Dh(xi , y) < r ′/2. This implies thatDh(xi−1, y) < r ′. Note that Dh(y, v) < Dh(xi , v), therefore y should have beenchosen as the successor of xi−1 by the construction criteria, which is a contradiction.The initial assumption of Dh(xi−1, xi ) < r ′/2 is thus proven false which proves thetruth of the claim.

Since Dh(xi−1, xi ) ≥ r ′/2 for 0 < i < m − 1, the number of hops in the path Pis

m ≤⌈

D(u, v)

r ′/2

⌉=

⌈√α2 + 4

D(u, v)

r

⌉.



If D(u, v) ≤ r the path is simply the edge connecting u and v and the hop-stretchis trivially 1. Otherwise, D(u, v) > r and so, the hop-stretch is 1 + √

α2 + 4.It remains to show that there is no hole with high probability.To bound the probability that there is a hole in any strip, consider the sequence

of small rectangles (call them slices) defined by the spacing parameter, of sizes × αr ′/2. The slices are numbered in ascending order from u to v .

For any node xi that is contained in some slice j , let Ei be the event that the nodexi+1 is contained in the slice j − 1 + �r ′/2s� at a horizontal distance greater thanr ′ from xi . Then,

Pr [Ei ] ≤(

n − 1

1

)αr ′s2�2

(1 − αr ′2

4�2

)n−2

.

If xi+1 is contained in a slice closer to xi then there is no hole. If xi+1 is containedin a slice farther than j − 1 + �r ′/2s� then there is at least one empty bin in thestrip. The probability that some bin is empty is bounded by

Pr [EmptyBin] ≤ max(u,v) D(u, v)

s

(1 − αr ′2

4�2

)n

.

Therefore, the probability that there is a hole within any strip is

Pr [Hole] ≤(

n2

)(n(n − 1)

αr ′s2�2

(1 − αr ′2

4�2

)n−2

+ max(u,v) D(u, v)

s

(1 − αr ′2

4�2

)n)

≤ n2 1

enαr ′2/4�2

(n2αr ′s

2�2eαr ′2/2�2 +

√2�

s

).

This expression is minimized when

s =(

2√

2�3

n2αr ′eαr ′2/2�2

)1/2

.

Then,

Pr [Hole] ≤ 2k3�6 ln3 �

r6�1+(kα/(4+α2))

(αr ′eαr ′2/2�2

2√

2�

)1/2

∈ O(�−γ ) for k > 54 + α2

α.

A simpler proof of Theorem 5.8 is also possible and follows, though the constantobtained is worse.

PROOF. Consider a strip Sj , the probability that a node xi is contained in Sj isat most αr ′/

√2�. The probability that there is a hole within Sj adjoining xi is at



most (1 − αr ′2/4�2)n−1. Then, the probability that there is a hole in any strip is

Pr [ Hole] ≤(

n2

)n

αr ′√

2�

(1 − αr ′2

4�2

)n−1

∈ O(�−γ ) for k > 6(4 + α2)

α.

LEMMA 5.9. In a G(n, r, �) satisfying the parameter conditions ofTheorem 5.8, the number of nodes contained in a circle of radius �(r ) is �(log �)with high probability.

PROOF. To prove this lemma it is enough to show that the probability that thenumber of nodes, within any circle of radius βr for some constant β, deviates fromlog � by more than a constant factor, is polynomially small. Consider the randomprocess of dropping nodes in a square of side length �. Define the random variableX as the number of nodes contained in that circle. For a given node, the probabilityof falling in the circle is πβ2r2/�2. Using Chernoff bounds

Pr(

X ≥ (1 + ε)πβ2r2

�2n)

≤ e− ε2

3 n πβ2r2

�2

Pr(

X ≤ (1 − ε)πβ2r2

�2n)

≤ e− ε2

2 n πβ2r2

�2 .

Using the parameter conditions

Pr (X ≥ (1 + ε)πβ2k ln �) ≤ �− ε2πβ2k3

Pr (X ≤ (1 − ε)πβ2k ln �) ≤ �− ε2πβ2k2 .

5.2.3. Hop Optimality of the CHSG. The following lemma shows the hop-optimality of the CHSG. The proof is based in a geometric argument. Similaroverall ideas have been used before [Das and Narasimhan 1997; Gudmundssonet al. 2002] but the details differ.

LEMMA 5.10. Consider the RGG G(n, r, l), where n satisfies the parameterconditions of Theorem 5.8 for a reduced connectivity radius of r ′ = (b−a)r/c. Forany pair of nodes (u, v) in the RGG at Euclidean distance D(u, v), there exists a pathbetween them in the CHSG of at most �c

√α2 + 4D(u, v)/(b −a)r�−1+ O(log �)

edges with high probability.

PROOF. Theorem 5.8 states that in the RGG that satisfies the parameter con-ditions of Theorem 5.8, there exists a path of �√α2 + 4D(u, v)/r� edges withhigh probability. We can thus imply that if the RGG satisfies the same parameterconditions for a reduced connectivity radius of r ′ = (b − a)r/c, there exists apath between u and v using �c

√α2 + 4D(u, v)/(b − a)r� edges of length at most

(b − a)r/c. Let p be such a path and e1, e2, . . . , em be its sequence of edges.In the description of the disk covering scheme, two kinds of disks were defined for

clarity: big disks and small disks. In order to prove hop-optimality of the CHSG, we



u v


only refer to big disks and simply call them disks. The rest of the proof is illustratedin Figure 6.

Lemma 5.6 states that every edge in the path p is completely covered by one disk.Therefore, there exists a sequence d1, d2, . . . , dm ′ of overlapping disks, where anyedge ei in p is covered by some disk d j in this sequence. A disk may completelycover more than one edge, hence m ′ ≤ m. Let Di be the bridge (center) of disk di .

Define a path p′ using only edges of the CHSG as follows. Connect u andthe bridge D1 with a path p1 of disk-spanner edges defined by the disk d1. Foreach edge i , 1 ≤ i ≤ m, replace the edge ei in p with the node Di . Connect allconsecutive bridges Di and Di+1 within the path of overlapping disks with edgeDi Di+1. Consecutive bridges are adjacent to each other in the RGG, because theirdisks overlap and the radius of each disk is br/2 with b ≤ 1. Finally, connect thebridge Dm and v with a path pm of disk-spanner edges defined by the disk dm . Thelength of p′ is given by: length(p′) ≤ length(p1) + (m − 1) + length(pm). Usingthe stretch bound, length(p′) ≤ �c

√α2 + 4D(u, v)/(b − a)r� − 1 + length(p1) +

length(pm) with high probability. Only disk-spanner edges are used in p1 and pm . Itis shown in Lemma 5.9 that the number of nodes within a disk is O(log �) with highprobability. Therefore, length(p1) + length(pm) = O(log �) with high probabilitycompleting the proof.

The following theorem shows the main result.

THEOREM 5.11. For every pair of nodes in an RGG, there is a path in theCHSG, whose length is asymptotically optimal with high probability.

PROOF. The optimal path between any pair of nodes (u, v) separated by adistance D(u, v) has at least �D(u, v)/r� edges. If log � is also an asymptotic lowerbound on the length of such a path with high probability, then (D(u, v)/r + log �)/2is also an asymptotic lower bound, and the result proved in Lemma 5.10 is a constantfactor approximation. It remains to show that log � is an asymptotic lower boundon the length of an optimal path in a constant-degree random geometric graph withhigh probability.

In a δ-regular graph, the expected distance between any pair of nodes randomlychosen is at least logδ−1 n. A �(1) degree random geometric graph is a subgraph ofsome regular graph. Hence, in a �(1) degree random geometric graph, the expecteddistance between any pair of nodes randomly chosen is in �(log n). The previous



result is true with high probability because for some constant β

Pr (D(u, v) < β log n) ≤ 1

n − 1

β log n−2∑i=0

δ(δ − 1)i

∈ O(n−γ ).Using the union bound, under the parameter conditions of Lemma 5.10, D(u, v) ∈

�(log �) for all pairs of nodes (u, v) with high probability.

6. Distributed Algorithm

In this section we describe how to distributedly implement the steps of the diskcovering scheme for network formation. Step 2 of the disk covering scheme can beachieved distributedly by means of a Maximal Independent Set (MIS) computationwith nodes transmitting in a range of ar/2. An algorithm to compute an MIS in aweak model is presented in Moscibroda and Wattenhofer [2005]. This algorithmcan be tailored to our setting and can be shown to have a running time of O(log2 �).The details are presented in Section 6.1.

Steps 3 and 4 of the disk covering scheme require uncolliding transmissions ofeach bridge in a radius of r and br/2, respectively. All nodes assigned to the samebridge will participate in a common spanner construction. Additionally, bridgenodes must set up links with all bridge nodes at a distance of at most r . Thedetails are presented in Sections 6.2 and 6.3. Finally, the constant-degree spannerconstruction is described in Section 6.3.

6.1. MIS COMPUTATION (STEP 2).

6.1.1. Algorithm. Step 2 of the disk covering scheme can be achieved distribut-edly by means of an MIS computation with nodes transmitting in a range of ar/2.An algorithm to compute an MIS in a weak model for arbitrary graphs was pre-sented in Moscibroda and Wattenhofer [2005]. This algorithm can be tailored to oursetting and can be shown to have a running time of O(log2 �). Algorithm 2 givesthe details of such MIS computation and we give our analysis in the followingsection.

Algorithm 2. MIS Computation, δ1, δ2, δ3 and δ4 are constants.

1 Set local counter to −�δ2 log ��.2 Transmit the local counter with probability 1/δ1 log �.3 if not transmitting in the current time slot then4 if a neighbor’s counter is received and the difference between the local and

neighbor’s counter is ≤ �δ2 log �� then5 Set local counter to −�δ2 log ��.6 end7 else if a neighbor’s ID is received then8 Set the local state to covered and stop.9 end

10 end11 Increase counter by one if transmitted at least once.12 if the counter is �δ3 log2 �� then13 Set the local state to MIS member.14 Transmit ID forever with probability q = 1/δ4.15 end16 Goto step 2 at end of time slot.



6.1.2. Analysis. The analysis of the MIS algorithm turns out to be difficultbecause nodes running different phases interfere with each other. Hence, necessaryassumptions regarding bounds on the total probability of transmission of nodes inother phases cannot be made, leading to a circular argument. In order to break thecircularity we prove the following lemmas by induction on the time slots in whicha given node joins the MIS.

Before the analysis, we recall the following basic fact [Motwani and Raghavan1995].

Fact 6.1. For all n ≥ 1 and |x | ≤ n

ex(

1 − x2

n

)≤

(1 + x

n

)n≤ ex .

LEMMA 6.2. Given any node that joins the MIS in a given time slot, the counterof all neighboring nodes is at most �δ3 log2 �� − �δ2 log �� in the same time slotwith high probability.

LEMMA 6.3. Every MIS node transmits its MIS status message successfully inthe �δ2 log �� time slots after it joins the MIS with high probability.

PROOF. We prove both preceding lemmas simultaneously by employing induc-tion on the order in which the nodes join the MIS, with ties broken arbitrarily.

Base case. Consider the first node within the whole network, call it μ1, that joinsthe MIS at time t1.

For the sake of contradiction, assume that there is a node x contained in μ1’sneighborhood whose counter is greater than L = �δ3 log2 �� − �δ2 log �� at t1. Bythe definition of the algorithm, μ1 has first transmitted at time t1 −�δ3 log2 �� and xhas first transmitted within the next �δ2 log �� time slots. Afterwards, neither μ1 norx have sent without collision otherwise one of their counters would have been reset.Let E(k) denote the event that neither μ1 nor x have sent without collision withink time slots. Using the fact that there are at most δ6 log � nodes within the 2-hopneighborhood of μ1 with high probability, for some constant δ6 > 0, as shown inLemma 5.9:

Pr [E(L)] ≤[

1 − 21

δ1 log �

(1 − 1

δ1 log �

)δ6 log �]�δ3 log2 ��−�δ2 log ��

∈ O(�−γ1 ) (Using fact 6.1, for some δ3, δ1 >√

δ6/ log � ).

Now we must additionally prove that within �δ2 log �� time slots of μ1 joining theMIS, all nodes within range of it receive a message declaring its MIS status. Forat least �δ2 log �� time slots after the node μ1 joins the MIS, no other nodes in itsneighborhood join the MIS with high probability as shown. If in this time its MISstatus message is received by all its neighbors, then they will all stop counting andtransition into the covered state. We will now show that this message is received byall its neighbors with high probability. Let E(k) denote the event that μ1 does nottransmit without collision in k consecutive time slots. The probability of failure in



�δ2 log �� consecutive time slots is

Pr [E(�δ2 log ��)] =[

1 − 1

δ4

(1 − 1

δ1 log �

)δ6 log �]�δ2 log ��

∈ O(�−γ2 ) (using Fact 6.1 and for some δ2).

This shows that μ1 sends its MIS status message without collision successfully in�δ2 log �� time slots with high probability.

Inductive Step. Consider the i th node μi , i > 1, that joins the MIS at time ti .

Inductive hypothesis. For all nodes μ j such that j < i , joining the MIS at timet j , the counters of all nodes in the neighborhood of μ j are at most �δ3 log2 �� −�δ2 log �� at time t j with high probability. Additionally all nodes μ j transmit theirMIS status message successfully within the interval t j . . . t j + �δ2 log �� with highprobability.

Therefore by time t j + �δ2 log �� all nodes in the range of all MIS nodesμ1 . . . μi−1 will be in the covered state. From the previous statements of the in-ductive hypothesis we can conclude that none of the MIS nodes μ j (where j < i)are neighbors of each other with high probability.

We want to show that the counters of all nodes in the neighborhood of μi are atmost �δ3 log2 �� − �δ2 log �� at time ti with high probability and that all neighborsof μi are in the covered state by time ti + �δ2 log �� with high probability.

If μi is out of the two-hop neighborhood of all the previous MIS members, theclaim can be easily proved using the same argument as in the base case. Otherwise,μi is within a two-hop neighborhood of some MIS members. Since all nodes thatpreviously joined the MIS are not in range of each other, μi is within the two-hopneighborhood of at most 12 other MIS members. This is true because a regularpolygon with side of length at least r and distance from the center to the verticesat most 2r has at most 12 sides.

For the sake of contradiction, assume that there is a node y contained in μi ’sneighborhood whose counter is greater than L = �δ3 log2 �� − �δ2 log �� at ti . Bythe definition of the algorithm, μi has first transmitted at time ti −�δ3 log2 �� and yhas first transmitted within the next �δ2 log �� time slots. Afterwards, neither μi nory have sent without collision otherwise one of their counters would have been reset.Let E(k) be the event that neither μi or y send without collision for k consecutivetime slots.

Pr [E(L)] ≤[

1 − 21

δ1 log �

(1 − 1

δ1 log �

)δ6 log � (1 − 1

δ4

)12]�δ3 log2 ��−�δ2 log ��

∈ O(�−γ3 ) (using Fact 6.1, for some δ3, δ1 >√

δ6/ log �).

Now we will show that all neighbors of MIS node μi will be in the covered stateby time slot ti + �δ2 log ��. Any neighbor of an MIS node has a counter that lagsthe MIS node’s counter by at least �δ2 log ��. Additionally no MIS node can bewithin range of any other. Hence every MIS node can be subjected to interferenceby at most 18 other MIS nodes (by a simple geometric packing argument). Let E(k)denote the event that a neighbor of an MIS node does not receive its MIS status



A

C

D

3r/2

r/2


message for k consecutive time slots. Thus the probability that a MIS node doesnot transmit its MIS status message without collision is given by

Pr [E(�δ2 log ��)] ≤[

1 −(

1

δ4

) (1 − 1

δ4

)18 (1 − 1

δ1 log �

)δ6 log �]�δ2 log ��

∈ O(�−γ4 ) (using Fact 6.1 and for some δ2 ).

LEMMA 6.4. No two nodes belonging to the MIS are within transmission rangeof each other with high probability.

PROOF. This is a direct conclusion of Lemmas 6.2 and 6.3.

LEMMA 6.5. For any node running the MIS algorithm with radius r , there is atleast one node, in its immediate r/2 neighborhood, that transmits without collisionwithin �δ5 log2 �� steps with high probability, for some constant δ5 > 0.

PROOF. Consider a node A running the MIS algorithm (refer to Figure 7). SinceA is awake, there is at least one node awake in C at time t . From Lemma 6.4 it canbe seen that no MIS nodes can be within range of each other, therefore there canbe at most 9 MIS nodes within D (If there were more then one of them would bein range of A). Let E(k) denote the event that no node in A’s r/2 neighborhood(including A) transmits without collision in k consecutive time slots. Lemma 5.9shows that there are at most δ6 log � nodes in D with high probability, for someconstant δ6 > 0.

Pr [E(�δ5 log2 ��)] ≤[

1 −(

1

δ1 log �

) (1 − 1

δ1 log �

)δ6 log � (1 − 1

δ4

)9]�δ5 log2 ��

∈ O(�−γ5 ) (using Fact 6.1, δ1 >√

δ6/ log �, for some δ5 ).

THEOREM 6.6. For a given node running the MIS algorithm, at least one nodewithin its transmission range joins the MIS in O(log2 �) time slots and no two MISnodes are within range of each other with high probability.



x1

x2

x3

≤ 3r/2≤ 3r/2

> r

FIG. 8. Illustration for Theorem 6.6.

PROOF. The proof is illustrated in Figure 8. In Lemma 6.5, it was shown thatwithin a circle of radius r/2 centered on any node x1, there will be a node x2,transmitting without collision, in less than �δ5 log2 �� steps with high probability.After this single transmission, there is at least one node, namely x2, within theneigborhood of x1 increasing its counter. If x2 joins the MIS after its counter reachesthe value �δ3 log2 ��, then the statement of the theorem is proved. Otherwise, someother node, call it x3, within range of x2, reaches this value and joins the MIS before.If x3 is within range of x1, then the statement of the theorem is proved. Otherwise,x3 covers at least one node within the r/2 neighborhood of x1, namely x2, withinthe next �δ2 log �� time slots with high probability (as shown in Lemma 6.3).

Note that the distance between x1 and x3 satisfies the following relation.

r < D(x1, x3) ≤ 3r/2 (2)

All uncovered active nodes within the r/2 neighborhood of x1 are still counting.Hence, the same argument can be repeatedly applied with the restriction that thenext MIS node is at least at a distance of r from x3 (by Lemma 6.4). There can beat most 9 MIS nodes around x1 before x1 or one of its neighbors joins the MIS, asexplained in Lemma 6.5. Thus, this process terminates in at most 10(�δ3 log2 �� +�δ5 log2 �� + �δ2 log ��) time slots.

6.2. BROADCAST (STEPS 3 AND 4). After a node is covered by some neighboringMIS node, it needs to be assigned to that MIS node. All nodes assigned to the sameMIS node will participate in a common spanner construction. Additionally, MISnodes must set up links with all MIS nodes at a distance of at most r . Any of thesesteps only require each MIS node to achieve an uncolliding transmission. In thissection, an algorithm for achieving this is detailed and a time bound is proved.

6.2.1. Algorithm. The algorithm is simple to describe.With probability 1/β1, each MIS node transmits its ID, within range β2r .Here β1 and β2 are constants whose values depend on which of the aforemen-

tioned steps is implemented. For informing the non-MIS nodes about assignment,the transmission is made with β2 = b/2. For setting up connections with neighbor-ing MIS nodes, the transmission is made with β2 = 1.



6.2.2. Analysis.

LEMMA 6.7. Any MIS node running the broadcast algorithm achieves a trans-mission without collision within O(log �) steps with high probability.

PROOF. Let � denote the maximum number of interfering MIS neighbors(which depends on β2). Let Pr [fail] denote the probability that any node failsto transmit without collision after β3 log � steps for some constant β3. For appro-priate values of β2 and β3, using the parameter conditions of Theorem 5.8 and theunion bound,

Pr [fail] = n

(1 − 1

β1

(1 − 1

β1

)�)β3 log �

∈ O(�−γ ) For some γ > 0.

6.3. SPANNER CONSTRUCTION (STEP 5). After nodes are covered by one ormore bridges (MIS members), they have to connect locally to neighboring nodescovered by the same bridge, that is, within the same disk. Nodes can be coveredby more than one bridge. Hence, interference of transmissions not only from thelocal disk but also from neighboring disks must be taken into account to ana-lyze the performance of any spanner construction algorithm. However, any nodeis covered by at most a constant number of disks as explained in Lemma 5.7,then the number of interfering transmissions with respect to the local disk is in-creased only by a constant factor that we fold into the constants involved in thisanalysis.

6.3.1. Algorithm. Our goal here is to construct a constant-degree spanner graphon the set of nodes assigned to a given bridge node. Since the diameter is not con-strained, we adopt the simplest topology, namely, a linked list. In order to minimizethe running time, we avoid handshaking among nodes and all the construction isdone by broadcasting. We start with every node choosing an integer index uniformlyat random from the interval [1, �]. Since there are O(log �) nodes within the samerange with high probability as shown before, no two nodes choose the same indexwith high probability.

Algorithm 3. Spanner Construction. β4 is a constant.

1 for each non-bridge node in parallel do

2 predecessor.ID ← bridge.ID;3 successor.ID ← bridge.ID;4 choose an integer index uniformly at random from the interval [1, �];5 while true do

6 transmit ←{

true with probability p = 1/β4 log �

false with probability 1 − p7 if transmit then broadcast 〈index,ID〉;8 else if an index is received then

9 update predecessor.ID or successor.ID accordingly;10 end

11 end

12 end



6.3.2. Analysis.

LEMMA 6.8. Any node running the spanner algorithm joins the spanner withinO(log2 �) steps with high probability.

PROOF. In order to prove this lemma it is enough to show that every node cov-ered by the same bridge that is running the spanner algorithm achieves at least onesingle (i.e., uncolliding) transmission within O(log2 �) steps with high probability.It was shown in Lemma 5.9 that there are �(log �) nodes within any disk of radiusO(r ). Hence, it is enough to show that within any disk with at most β4 log � nodesthere are β4 log � different single transmissions within β5 log2 � steps with highprobability, where β4 and β5 are constants.

To show this, we use the following balls-and-bins analysis. Let each node berepresented by a bin and each transmission step be represented by a ball. A nodeachieving a single transmission at a given step is modeled with the ball representingthat step falling in the bin representing that node. If at a given transmission stepthere is no single transmission, we say that the ball falls outside the bins. Now, toprove this lemma it is enough to show that after dropping β5 log2 � balls in β4 log �bins, no bin is empty with high probability.

For a given ball, the probability of falling in a given bin is the probability ofachieving a single transmission, namely

Pr = 1

β4 log �

(1 − 1

β4 log �

)β4 log �−1

.

Hence, the probability of some empty bin is

Pr (fail) ≤β4 log �∑

i=1

(β4 log �

i

) (1 − i

1

β4 log �

(1 − 1

β4 log �

)β4 log �−1)β5 log2 �

≤(

1 − 1

β4 log �

(1 − 1

β4 log �

)β4 log �−1)β5 log2 � β4 log �∑

i=1

(β4 log �

i

).

Using the binomial theorem,

Pr (fail) ≤(

1 − 1

β4 log �

(1 − 1

β4 log �

)β4 log �−1)β5 log2 �

2β4 log �

∈ O(�−γ ), γ > 0 (using Fact 6.1, for a large enough β5 > eβ4 ).

6.3.3. A Small-Diameter Spanner. In the previous construction, the distancebetween any two nodes is at most the number of nodes within the disk, namelyO(log �). Although a diameter of �(log �) for the disk spanner is optimal(Theorem 5.11) for a constant-degree random geometric graph, a constant-degreespanner with diameter o(log log �) is also possible as shown in this section.

The structure we utilize is popularly known as a butterfly network. Butterflynetworks are used in many parallel computers to provide paths of length log m



00 01 10 11

000

001

010

011

100

101

110

111

FIG. 9. A butterfly network with 32 nodes.

connecting m inputs to m outputs. A labeled instance of a butterfly network withm = 8 is shown in Figure 9. The inputs of the network are on the left and the outputsare on the right. In our case, all nodes have the same role and a message betweenany pair of nodes can be sent in O(log m) hops. Then, given that there are �(log �)nodes in any disk, the diameter obtained is o(log log �). Notice that once uniqueconsecutive labels are assigned to all nodes, each node can easily compute to whichnodes it is connected. Then, our goal is to assign unique consecutive indexes to allnodes within the disk.

The distributed algorithm for nonbridge nodes to construct such a network withinone disk consists of three phases, as follows. First, every node chooses an indexuniformly at random from the interval [1, �]. As explained before, no two nodes willchoose the same index with high probability. Then, every node broadcasts its indexand ID as in Algorithm 3, but in this case they keep track of the ID of its predecessoronly and the process runs for just O(log2 �) steps. As shown in Lemma 6.8, at thispoint all nodes have achieved at least one transmission without collision so, allnodes know who is their predecessor.

To obtain consecutive indexes, the nodes now have to pack the indexes one byone as follows. Upon receiving the new index i of its predecessor, a node redefinesits index as i + 1 and broadcasts its new index and ID with constant probability forO(log �) steps. As shown in Lemma 6.2.2, there will be at least one transmissionwithout collision with high probability. Obviously, the first node in this orderingwill not have any predecessor and will start this phase of the algorithm redefining itsindex as 1. At this point, all nodes have consecutive indexes and have to connect asa butterfly accordingly, but they do not know yet the IDs of their butterfly neighborswith smaller index, so a final round broadcasting the new index and ID is necessary.The details can be seen in Algorithm 4.



Algorithm 4. A Small-Diameter Spanner Construction. β4 is a constant.

1 for each non-bridge node in parallel do2 predecessor.ID ← NULL;3 choose an integer index uniformly at random from the interval [1, �];4 for β6 log2 � steps do

5 transmit ←{


false with probability 1 − p6 if transmit then broadcast 〈index, ID〉;7 else if an index is received then8 update predecessor.ID accordingly;9 end

10 end11 index ← 0;12 if predecessor.ID �= NULL then13 wait until an index from predecessor.ID is received;14 end15 index ← index + 1;16 for β8 log � steps do17 broadcast 〈index, ID〉 with probability 1/β9;18 end19 for β6 log2 � steps do

20 transmit ←{


false with probability 1 − p21 if transmit then broadcast 〈index,ID〉;22 else if an index is received then23 store ID’s of butterfly neighbors according with the index;24 end25 end26 end

The first and third phase take O(log2 �) time by definition of the algorithm. Inthe second phase, each of �(log �) nodes in turn transmit for O(log �) steps. Hence,the overall running time of this algorithm is O(log2 �).

7. Conclusions

The bootstrapping protocol described in this article builds a hop-optimal constant-degree sensor network under the constraints of the Weak Sensor model in O(log2 �)time with high probability. The time bounds are for the MIS algorithm O(log2 �), forthe broadcast algorithm O(log �), and for the spanner algorithm O(log2 �). Hence,the total running time is upper bounded by O(log2 �).

There is a trade-off among the maximum degree, the length of the optimal path,and the density given by the following.

There is a path of ≤⌈

D(u,v)r

c√

4+α2

b−a

⌉− 1 + O(log �) hops with high probability.

The degree of any bridge is ≤ 3� 4a√

3�(� 4

a√

3� + 1

)+ 1 with high probability.

The density of nodes is n�2 > 5 4+α2

α

( cb−a

)2 ln �r2 .

Here, 0 < a < 1, a < b ≤ 1, c > 1, and 0 < α ≤ 1.The longer the edges covered, the lower density and smaller number of hops in

the optimal path, but the degree is bigger.



Notice that in our construction, only three ranges of transmission are used, namelyar/2, br/2, and r . Hence, the specific values of a and b are hardware dependent.

Notice also that for any of the various parts of the bootstrapping algorithm nosynchronicity assumption is needed. Furthermore, neighboring disks do not need tobe running the same phase of the algorithm. Regarding failures, the MIS algorithmand its final broadcast algorithm as well as the linked list spanner constructionalgorithm are also maintenance algorithms, since both bridge and nonbridge nodeskeep broadcasting forever. If a bridge node fails, after some time nonbridge nodeswill detect the absence of their bridge broadcast and will restart the MIS algorithmto obtain a new bridge. On the other hand, if a nonbridge node fails, its successor andpredecessor will interconnect within the next round of the spanner construction. Ifthe butterfly network spanner is used instead and a link is lost, the butterfly networkcan be simply rebuilt locally from scratch.

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