fast data collection in tree based wireless sensor networks.bak

Fast Data Collection in Tree-BasedWireless Sensor Networks

Ozlem Durmaz Incel, Amitabha Ghosh, Bhaskar Krishnamachari, and Krishnakant Chintalapudi

Abstract—We investigate the following fundamental question—how fast can information be collected from a wireless sensor network

organized as tree? To address this, we explore and evaluate a number of different techniques using realistic simulation models under

the many-to-one communication paradigm known as convergecast. We first consider time scheduling on a single frequency channel

with the aim of minimizing the number of time slots required (schedule length) to complete a convergecast. Next, we combine

scheduling with transmission power control to mitigate the effects of interference, and show that while power control helps in reducing

the schedule length under a single frequency, scheduling transmissions using multiple frequencies is more efficient. We give lower

bounds on the schedule length when interference is completely eliminated, and propose algorithms that achieve these bounds. We

also evaluate the performance of various channel assignment methods and find empirically that for moderate size networks of about

100 nodes, the use of multifrequency scheduling can suffice to eliminate most of the interference. Then, the data collection rate no

longer remains limited by interference but by the topology of the routing tree. To this end, we construct degree-constrained spanning

trees and capacitated minimal spanning trees, and show significant improvement in scheduling performance over different deployment

densities. Lastly, we evaluate the impact of different interference and channel models on the schedule length.

Index Terms—Convergecast, TDMA scheduling, multiple channels, power control, routing trees.

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1 INTRODUCTION

CONVERGECAST, namely, the collection of data from a set ofsensors toward a common sink over a tree-based

routing topology, is a fundamental operation in wirelesssensor networks (WSNs) [1]. In many applications, it iscrucial to provide a guarantee on the delivery time as well asincrease the rate of such data collection. For instance, insafety and mission-critical applications where sensor nodesare deployed to detect oil/gas leak or structural damage, theactuators and controllers need to receive data from all thesensors within a specific deadline [2], failure of which mightlead to unpredictable and catastrophic events. This fallsunder the category of one-shot data collection. On the otherhand, applications such as permafrost monitoring [3] requireperiodic and fast data delivery over long periods of time,which falls under the category of continuous data collection.

In this paper, we consider such applications and focus onthe following fundamental question: “How fast can data bestreamed from a set of sensors to a sink over a tree-based topology?”We study two types of data collection: 1) aggregatedconvergecast where packets are aggregated at each hop, and

2) raw-data convergecast where packets are individuallyrelayed toward the sink. Aggregated convergecast is applic-able when a strong spatial correlation exists in the data, orthe goal is to collect summarized information such as themaximum sensor reading. Raw-data convergecast, on theother hand, is applicable when every sensor reading isequally important, or the correlation is minimal. We studyaggregated convergecast in the context of continuous datacollection, and raw-data convergecast for one-shot datacollection. These two types correspond to two extreme casesof data collection. In an earlier work [4], the problem ofapplying different aggregation factors, i.e., data compressionfactors, was studied, and the latency of data collection wasshown to be within the performance bounds of the twoextreme cases of no data compression (raw-data converge-cast) and full data compression (aggregated convergecast).

For periodic traffic, it is well known that contention-freemedium access control (MAC) protocols such as TimeDivision Multiple Access (TDMA) are better fit for fast datacollection, since they can eliminate collisions and retrans-missions and provide guarantee on the completion time asopposed to contention-based protocols [1]. However, theproblem of constructing conflict-free (interference-free)TDMA schedules even under the simple graph-basedinterference model has been proved to be NP-complete. Inthis work, we consider a TDMA framework and designpolynomial-time heuristics to minimize the schedule lengthfor both types of convergecast. We also find lower boundson the achievable schedule lengths and compare theperformance of our heuristics with these bounds.

We start by identifying the primary limiting factors offast data collection, which are: 1) interference in the wirelessmedium, 2) half-duplex transceivers on the sensor nodes, and3) topology of the network. Then, we explore a number ofdifferent techniques that provide a hierarchy of successive

86 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 1, JANUARY 2012

. O.D. Incel is with the Department of Computer Engineering, BogaziciUniversity, Istanbul 34342, Turkey.E-mail: [email protected].

. A. Ghosh is with the Department of Electrical Engineering, PrincetonUniversity, F-310, Engineering Quad, Olden Street, Princeton, NJ 08544.E-mail: [email protected].

. B. Krishnamachari is with the Ming Hsieh Department of ElectricalEngineering, University of Southern California, 3740 McClintock Avenue,EEB 300, Los Angeles, CA 90089. E-mail: [email protected].

. K. Chintalapudi is with Microsoft Research, Bengaluru, India.E-mail: [email protected].

Manuscript received 5 Nov. 2009; revised 14 Oct. 2010; accepted 15 Dec.2010; published online 2 Feb. 2011.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2009-11-0482.Digital Object Identifier no. 10.1109/TMC.2011.22.

1536-1233/12/$31.00 � 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

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improvements, the simplest among which is an interfer-ence-aware, minimum-length TDMA scheduling that en-ables spatial reuse. To achieve further improvement, wecombine transmission power control with scheduling, anduse multiple frequency channels to enable more concurrenttransmissions. We show that once multiple frequencies areemployed along with spatial-reuse TDMA, the data collec-tion rate often no longer remains limited by interference butby the topology of the network. Thus, in the final step, weconstruct network topologies with specific properties thathelp in further enhancing the rate. Our primary conclusionis that combining these different techniques can provide anorder of magnitude improvement for aggregated converge-cast, and a factor of 2 improvement for raw-data con-vergecast, compared to single-channel TDMA schedulingon minimum-hop routing trees.

Although the techniques of transmission power controland multichannel scheduling have been well studied foreliminating interference in general wireless networks, theirperformances for bounding the completion of data collec-tion in WSNs have not been explored in detail in theprevious studies. The fundamental novelty of our approachlies in the extensive exploration of the efficiency oftransmission power control and multichannel communica-tion on achieving fast convergecast operations in WSNs.Besides, we evaluate the impact of routing trees on fast datacollection and to the best of our knowledge, this has notbeen the topic of previous studies. As we will discuss inSection 2, some of the existing work had the objective ofminimizing the completion time of convergecasts. How-ever, none of the previous work discussed the effect ofmultichannel scheduling together with the comparisonsof different channel assignment techniques and the impactof routing trees and none considered the problems ofaggregated and raw convergecast, which represent twoextreme cases of data collection, together.

As the new concepts in this paper, we introducepolynomial-time heuristics for TDMA scheduling for bothtypes of data collection, i.e., Algorithms 1 and 2, and provethat they do achieve the lower bound of data collection timeonce interference is eliminated. Besides, we elaborate on theperformance of our previous work, a receiver-basedchannel assignment (RBCA) method, and compare itsefficiency with other channel assignment methods andintroduce heuristics for constructing optimal routing treesto further enhance data collection rate. The following listsour key findings and contributions:

. Bounds on convergecast scheduling. We show thatif all interfering links are eliminated, the schedulelength for aggregated convergecast is lowerbounded by the maximum node degree in therouting tree, and for raw-data convergecast bymaxð2nk � 1; NÞ, where nk is the maximum numberof nodes on any branch in the tree, and N is thenumber of source nodes. We then introduce optimaltime slot assignment schemes under this scenariowhich achieve these lower bounds.

. Evaluation of power control under realistic setting.It was shown recently [5] that under the idealizedsetting of unlimited power and continuous range,transmission power control can provide an un-bounded improvement in the asymptotic capacity

of aggregated convergecast. In this work, weevaluate the behavior of an optimal power controlalgorithm [6] under realistic settings considering thelimited discrete power levels available in today’sradios. We find that for moderate size networks of100 nodes, power control can reduce the schedulelength by 15-20 percent.

. Evaluation of channel assignment methods. Usingextensive simulations, we show that schedulingtransmissions on different frequency channels ismore effective in mitigating interference as com-pared to transmission power control. We evaluatethe performance of three different channel assign-ment methods: 1) Joint Frequency Time Slot Scheduling(JFTSS), 2) Receiver-Based Channel Assignment [7], and3) Tree-Based Multichannel Protocol (TMCP) [8]. Thesemethods consider the channel assignment problemat different levels: the link level, node level, orcluster level. We show that for aggregated conver-gecast, TMCP performs better than JFTSS and RBCAon minimum-hop routing trees, while performsworse on degree-constrained trees. For raw-dataconvergecast, RBCA and JFTSS perform better thanTMCP, since the latter suffers from interferenceinside the branches due to concurrent transmissionson the same channel.

. Impact of routing trees. We investigate the effect ofnetwork topology on the schedule length, and showthat for aggregated convergecast, the performancecan be improved by up to 10 times on degree-constrained trees using multiple frequencies ascompared to that on minimum-hop trees using asingle frequency. For raw-data convergecast, multi-channel scheduling on capacitated minimal span-ning trees (CMSTs) can reduce the schedule lengthby 50 percent.

. Impact of channel models and interference. Underthe setting of multiple frequencies, one simplifyingassumption often made is that the frequencies areorthogonal to each other. We evaluate this assump-tion and show that the schedules generated maynot always eliminate interference, thus causingconsiderable packet losses. We also evaluate andcompare the two most commonly used interferencemodels: 1) the graph-based protocol model, and2) the Signal-to-Interference-plus-Noise Ratio(SINR)-based physical model.

The rest of the paper is organized as follows: in Section 2,we discuss related works. In Section 3, we describe theproblem formulation and state our assumptions. In Section4, we analyze the lower bounds on the schedule length foraggregated and raw convergecast, and propose algorithmsthat achieve the corresponding bounds. In Section 5, wefocus on power control and multichannel scheduling asmechanisms to eliminate interference. Section 6 explains theimpact of routing topologies, and Section 7 presentsdetailed evaluation results. Finally, we draw our conclu-sions in Section 8.

2 RELATED WORK

Fast data collection with the goal to minimize the schedulelength for aggregated convergecast has been studied by us

INCEL ET AL.: FAST DATA COLLECTION IN TREE-BASED WIRELESS SENSOR NETWORKS 87


in [7] and [9], and also by others in [5], [10], and [11]. In [7],we experimentally investigated the impact of transmissionpower control and multiple frequency channels on theschedule length, while the theoretical aspects were dis-cussed in [9], where we proposed constant factor andlogarithmic approximation algorithms on geometric net-works (disk graphs). Raw-data convergecast has beenstudied in [1], [12], [13], and [14], where a distributed timeslot assignment scheme is proposed by Gandham et al. [1]to minimize the TDMA schedule length for a single channel.The problem of joint scheduling and transmission powercontrol is studied by Moscibroda [5] for constant anduniform traffic demands. Our present work is differentfrom the above in that we evaluate transmission powercontrol under realistic settings and compute lower boundson the schedule length for tree networks with algorithms toachieve these bounds. We also compare the efficiency ofdifferent channel assignment methods and interferencemodels, and propose schemes for constructing specificrouting tree topologies that enhance the data collection ratefor both aggregated and raw-data convergecast.

The use of orthogonal codes to eliminate interference hasbeen studied by Annamalai et al. [10], where nodes areassigned time slots from the bottom of the tree to the top suchthat a parent node does not transmit before it receives all thepackets from its children. This problem and the oneaddressed by Chen et al. [11] are for one-shot raw-dataconvergecast. In this work, since we construct degree-constrained routing topologies to enhance the data collectionrate, it may not always lead to schedules that have lowlatency, because the number of hops in a tree goes up as itsdegree goes down. Therefore, if minimizing latency is also arequirement, then further optimization, such as constructingbounded-degree, bounded-diameter trees, is needed. Astudy along this line with the objective to minimize themaximum latency is presented by Pan and Tseng [15], wherethey assign a beacon period to each node in a Zigbee networkduring which it can receive data from all its children.

For raw-data convergecast, Song et al. [12] presented atime-optimal, energy-efficient packet scheduling algorithmwith periodic traffic from all the nodes to the sink. Onceinterference is eliminated, their algorithm achieves thebound that we present here; however, they briefly mentiona 3-coloring channel assignment scheme, and it is not clearwhether the channels are frequencies, codes, or any othermethod to eliminate interference. Moreover, they assume asimple interference model where each node has a circulartransmission range and cumulative interference fromconcurrent multiple senders is avoided. Different fromtheir work, we consider multiple frequencies and evaluatethe performance of three different channel assignmentmethods together with evaluating the effects of transmis-sion power control using realistic interference and channelmodels, i.e., physical interference model and overlappingchannels and considering the impact of routing topologies.Song et al. [12] extended their work and proposed a TDMA-based MAC protocol for high-data-rate WSNs in [16].TreeMAC considers the differences in load at differentlevels of a routing tree and assigns time slots according tothe depth, i.e., the hop count, of the nodes on the routingtree, such that nodes closer to the sink are assigned moreslots than their children in order to mitigate congestion.

However, TreeMAC operates on a single channel andachieves 1=3 of the maximum throughput similar to thebounds presented by Gandham et al. [1] since the sink canreceive every three time slots.

The problem of minimizing the schedule length for raw-data convergecast on single channel is shown to be NP-complete on general graphs by Choi et al. [13]. Maximizingthe throughput of convergecast by finding a shortest-length,conflict-free schedule is studied by Lai et al. [14], where agreedy graph coloring strategy assigns time slots to thesenders and prevents interference. They also discussed theimpact of routing trees on the schedule length and proposed arouting scheme called disjoint strips to transmit data overdifferent shortest paths. However, since the sink remains asthe bottleneck, sending data over different paths does notreduce the schedule length. As we will show in this paper, theimprovement due to the routing structure comes from usingcapacitated minimal spanning trees for raw-data converge-cast, where the number of nodes in a subtree is no more thanhalf the total number of nodes in the remaining subtrees.

The use of multiple frequencies has been studiedextensively in both cellular and ad hoc networks; however,in the domain of WSN, there exist a few studies that utilizemultiple channels [8], [17], [18]. To this end, we evaluate theefficiency of three particular schemes that treat the channelassignment at different levels.

3 MODELING AND PROBLEM FORMULATION

We model the multihop WSN as a graph G ¼ ðV ;EÞ, whereV is the set of nodes, and E ¼ fði; jÞ j i; j 2 V g is the set ofedges representing the wireless links. A designated nodes 2 V denotes the sink. The euclidean distance between twonodes i and j is denoted by dij. All the nodes except s aresources, which generate packets and transmit them over arouting tree to s. We denote the spanning tree on G rootedat s by T ¼ ðV ;ET Þ, where ET � E represents the treeedges. Each node is assumed to be equipped with a singlehalf-duplex transceiver, which prevents it from sending andreceiving packets simultaneously. We consider a TDMAprotocol where time is divided into slots, and consecutiveslots are grouped into equal-sized nonoverlapping frames.

We use two types of interference models for ourevaluation: the graph-based protocol model and the SINR-based physical model. In the protocol model, we assumethat the interference range of a node is equal to itstransmission range, i.e., two links cannot be scheduledsimultaneously if the receiver of at least one link is withinthe range of the transmitter of the other link. In the physicalmodel, the successful reception of a packet from i to jdepends on the ratio between the received signal strength atj and the cumulative interference caused by all otherconcurrently transmitting nodes and the ambient noiselevel. Thus, a packet is received successfully at j if thesignal-to-interference-plus-noise ratio, SINRij, is greaterthan a certain threshold �, i.e.,

SINRij ¼Pi � gijP

k6¼i Pk � gkj þN; ð1Þ

where Pi is the transmitted signal power at node i, N is theambient noise level, and gij is the propagation attenuation



(link gain) between i and j. We use a simple distancedependent path-loss model to calculate the link gains asgij ¼ d��ij , where the path-loss exponent � is a constantbetween 2 and 6, whose exact value depends on externalconditions of the medium (humidity, obstacles, etc.), as wellas the sender-receiver distance. We assume that the level ofinterference is static and does not change over time. Forsimplicity and ease of illustration, we use the protocolmodel in all the figures.

We study aggregated convergecast in the context ofperiodic data collection where each source node generates apacket at the beginning of every frame, and raw-dataconvegecast for one-shot data collection where each nodehas only one packet to send. We assume that the size of eachpacket is constant. Our goal is to deliver these packets to thesink over the routing tree as fast as possible. Morespecifically, we aim to schedule the edges ET of T using aminimum number of time slots while respecting thefollowing two constraints:

. Adjacency constraint. Two edges ði; jÞ 2 ET andðk; lÞ 2 ET cannot be scheduled in the same timeslot if they are adjacent to each other, i.e., iffi; jg

Tfk; lg 6¼ �. This constraint is due to the half-

duplex transceiver on each node which prevents itfrom simultaneous transmission and reception.

. Interfering constraint. The interfering constraint de-pends on the choice of the interference model. In theprotocol model, two edges ði; jÞ 2 ET and ðk; lÞ 2 ETcannot be scheduled simultaneously if they are at 2-hop distance of each other. In the physical model, anedge ði; jÞ 2 ET cannot be scheduled if the SINR atreceiver j is not greater than the threshold �.

Since we consider data collection to be periodic inaggregated convergecast, each of the edges in ET isscheduled only once within each frame, and this scheduleis repeated over multiple frames. Thus, a pipeline isestablished after a certain frame, and then onward the sinkcontinues to receive aggregated packets from all the sourcenodes once per frame. We explain further details about thepipelining in the next section. On the other hand, in one-shotdata collection for raw-data convergecast, the edges in ETmay be scheduled multiple times and no pipelining takesplace. We use the terms link scheduling and node schedulinginterchangeably as they are equivalent in our case. Note thatthe two other scenarios, which we do not consider in thispaper due to space constraints, are one-shot aggregatedconvergecast and periodic raw-data convergecast.

The key difference in terms of scheduling betweenperiodic and one-shot data collection is that a node in theperiodic case does not have to wait for data from its childrenbefore being scheduled. This is because a link is scheduledonly once within each frame and each node generates a packetin the beginning of every frame, so a pipelining is eventuallyestablished. However, in the case of one-shot data collection,a node needs to wait for data from its children before beingscheduled, which we refer to as the causality constraint.

To summarize the steps in our design, we start with treeconstruction and then continue with interference-awarescheduling. If the nodes can control their transmissionpower, scheduling phase is coupled with a transmissionpower control algorithm. If the nodes can change their

operating frequency, channel scheduling can be coupledwith time slot scheduling as it is the case with the JFTSSalgorithm (Section 5.2.1) or first channels are assigned andthen time slot scheduling continues as in the case of RBCAexplained in Section 5.2.3. However, the TMCP algorithm(Section 5.2.2) considers tree construction and channelassignment jointly and then does the scheduling of time slots.

4 TDMA SCHEDULING OF CONVERGECASTS

In this section, we first focus on periodic aggregatedconvergecast and then on one-shot raw-data convergecast.Our objective is to calculate the minimum achievableschedule lengths using an interference-aware TDMA pro-tocol. We first consider the case where the nodes commu-nicate on the same channel using a constant transmissionpower, and then discuss improvements using transmissionpower control and multiple frequencies in the next section.

4.1 Periodic Aggregated Convergecast

In this section, we consider the scheduling problem wherepackets are aggregated. Data aggregation is a commonlyused technique in WSN that can eliminate redundancy andminimize the number of transmissions, thus saving energyand improving network lifetime [19]. Aggregation can beperformed in many ways such as by suppressing duplicatemessages; using data compression and packet mergingtechniques; or taking advantage of the correlation in thesensor readings.

We consider continuous monitoring applications whereperfect aggregation is possible, i.e., each node is capable ofaggregating all the packets received from its children aswell as that generated by itself into a single packet beforetransmitting to its parent. The size of aggregated datatransmitted by each node is constant and does not dependon the size of the raw sensor readings. Typical examples ofsuch aggregation functions are MIN, MAX, MEDIAN,COUNT, AVERAGE, etc.

In Figs. 1a and 1b, we illustrate the notion of pipeliningin aggregated convergecast and that of a schedule length ona network of six source nodes. The solid lines represent treeedges, and the dotted lines represent interfering links. Thenumbers beside the links represent the time slots at whichthe links are scheduled to transmit, and the numbers insidethe circles denote node ids. The entries in the table list thenodes from which packets are received by their correspond-ing receivers in each time slot. We note that at the end offrame 1, the sink does not have packets from nodes 5 and 6;however, as the schedule is repeated, it receives aggregatedpackets from 2, 5, and 6 in slot 2 of the next frame. Similarly,the sink also receives aggregated packets from nodes 1 and4 starting from slot 1 of frame 2. The entries f1; 4g andf2; 5; 6g in the table represent single packets comprisingaggregated data from nodes 1 and 4, and from nodes 2, 5,and 6, respectively. Thus, a pipeline is established fromframe 2, and the sink continues to receive aggregatedpackets from all the nodes once every six time slots. Thus,the minimum schedule length is 6.

4.1.1 Lower Bound on Schedule Length

We first consider aggregated convergecast when all theinterfering links are eliminated by using transmission powercontrol or multiple frequencies. Although the problem of



minimizing the schedule length is NP-complete on generalgraphs, we show in the following that once interference iseliminated, the problem reduces to 1 on a tree, and can besolved in polynomial time. To this end, we first give a lowerbound on the schedule length, and then propose a time slotassignment scheme that achieves the bound.

Lemma 1. If all the interfering links are eliminated, the schedulelength for aggregated convergecast is lower bounded by �ðT Þ,where �ðT Þ is the maximum node degree in the routing tree T .

Proof. If all the interfering links are eliminated, thescheduling problem reduces to 1 on a tree. Now sinceeach of the tree edges needs to be scheduled only oncewithin each frame, it is equivalent to edge coloring on agraph, which needs number of colors at least equal to themaximum node degree. tu

Once all the interfering links are eliminated, concurrencyis still limited by the adjacency constraint due to the half-duplex transceivers, which prevents a parent from trans-mitting when it is already receiving from its children, orwhen its parent is transmitting.

4.1.2 Assignment of Time Slots

Given the lower bound �ðT Þ on the schedule length in theabsence of interfering links, we now present a time slotassignment scheme in Algorithm 1, called BFS-TIMESLO-

TASSIGNMENT, that achieves this bound.

Algorithm 1. BFS-TIMESLOTASSIGNMENT

1. Input: T ¼ ðV ; ET Þ2. while ET 6¼ � do

3. e next edge from ET in BFS order

4. Assign minimum time slot t to edge e respecting

adjacency and interfering constraints

5. ET ET n feg6. end while

In each iteration of BFS-TIMESLOTASSIGNMENT (lines 2-6), an edge e is chosen in the Breadth First Search (BFS)order starting from any node, and is assigned the minimumtime slot that is different from all its adjacent edgesrespecting interfering constraints. Note that, since weevaluate the performance of this algorithm also for the casewhen the interfering links are present, we check for thecorresponding constraint in line 4; however, when inter-ference is eliminated this check is redundant. The algorithmruns in OðjET j2Þ time and minimizes the schedule lengthwhen there are no interfering links, as proved in Theorem 1.

To illustrate, we show the same network of Fig. 1a in Fig. 1c

with all the interfering links removed, and so the network is

scheduled in three time slots.Although BFS-TIMESLOTASSIGNMENT may not be an

approximation to ideal scheduling under the physical

interference model, it is a heuristic that can achieve the

lower bound if all the interfering links are eliminated.

Therefore, together with a method to eliminate interference,

the algorithm can optimally schedule the network.

Theorem 1. If all the interfering links are eliminated, the

schedule length for aggregated convergecast achieved by BFS-

TIMESLOTASSIGNMENT is the minimum, i.e., �ðT Þ.Proof. The proof is by induction on i. Let T i ¼ ðV i; Ei

T Þdenote the subtree of T in the ith iteration constructed in

the BFS order, where EiT comprises all the edges that are

assigned a slot, andV i comprises the set of nodes on which

the edges inEiT are incident. Note that, jEi

T j ¼ i, because at

every iteration, exactly one edge is assigned a slot. For

i ¼ 1, clearly the number of slots used is 1, equal to �ðT 1Þ.Now, assume that the number of slots NðiÞ needed to

schedule the edges in T i is �ðT iÞ. In the ðiþ 1Þstiteration, after assigning a slot to the next edge in BFSorder, the number of slots needed in T iþ1 can eitherremain the same as before, or increase by 1. Thus,

Nðiþ 1Þ ¼ max NðiÞ; NðiÞ þ 1f g: ð2Þ

If it remains the same, Nðiþ 1Þ is still the maximumdegree of T iþ1 at end of ðiþ 1Þst iteration. Otherwise, if itincreases by 1, the new edge must be incident on a nodev�, common to both T i and T iþ1, such that the number ofincident edges on v� that were already assigned a timeslot at the end of ith iteration was �ðT iÞ. This is sobecause in the BFS traversal, all the edges incident on anode are assigned a slot first before moving on to thenext node, and because the slot assigned to the new edgeis the minimum possible that is different from all thatalready assigned to the edges incident on v� until the ithiteration. Thus, at the end of ðiþ 1Þst iteration, thenumber of slots used NðiÞ þ 1 is equal to the number ofassigned edges incident on v� which, in turn, equals�ðT iþ1Þ. This proves the inductive step. Therefore, itholds at every iteration of the algorithm until the endwhen i ¼ jV j � 2, yielding a schedule length equal to themaximum degree �ðT Þ ¼ �ðT jV j�1Þ. Now, since assign-ing different time slots to the adjacent edges of T isequivalent to edge coloring T , which requires at least�ðT Þ colors, the schedule length is minimum. tu


Fig. 1. Aggregated convergecast and pipelining: (a) Schedule length of 6 in the presence of interfering links. (b) Node ids from which (aggregated)packets are received by their corresponding parents in each time slot over different frames. (c) Schedule length of 3 using BFS-TIMESLOTASSIGNMENT when all the interfering links are eliminated.


4.2 One-Shot Raw-Data Convergecast

In this section, we consider one-shot data collection whereevery sensor reading is equally important, and so aggrega-tion may not be desirable or even possible. Thus, each of thepackets has to be individually scheduled at each hop en routeto the sink. As before, we focus on minimizing the schedulelength. Unlike in the case of periodic aggregated converge-cast where a pipelining takes place and each of the tree edgesis scheduled only once within each frame, here the edgescould be scheduled multiple times and there is no pipelining.

The problem of minimizing the scheduling length forraw-data convergecast is proved to be NP-complete evenunder the protocol interference model by a reduction fromthe well known Partition Problem [13]. Before getting into thedetails, we first define the following terms: a branch isdefined as a subtree containing the sink as an endpoint; atop-subtree is defined as a subtree that has a child of the sinkas its root. For instance, in Fig. 3, the branches are fs; 1; 4g,fs; 2; 5; 6g, and fs; 3; 7g, while the top-subtrees are f1; 4g,f2; 5; 6g, and f3; 7g.

4.2.1 Lower Bound on Schedule Length

As mentioned in Section 4.1.1, if all the interfering links areeliminated using multiple frequencies, the only limitingfactor in minimizing the schedule length is the half-duplextransceivers. In the following, we give a lower bound on theschedule length under this scenario.

Lemma 2. If all the interfering links are eliminated, the schedulelength for one-shot raw-data convergecast is lower bounded bymaxð2nk � 1; NÞ, where nk is the maximum number of nodesin any top-subtree of the routing tree, and N is the number ofsources in the network.

Proof. Let ni denote the number of nodes in top-subtree i.Order the top-subtrees in nonincreasing order of theirsizes: nk � nk�1 � � � � � n1. Consider the routing treeshown in Fig. 2. Since the nodes cannot receive multiplepackets simultaneously, N is a trivial lower bound toreceive all the packets. Next, consider the largest top-subtree k, the root of which has to transmit nk packets tothe sink, and the children of this root have to forwardnk � 1 packets in total. Because of the half-duplextransceivers, time slots assigned to the root of this top-subtree must be distinct from all those assigned to itschildren. Thus, in total, we need at least nk þ ðnk � 1Þ ¼2nk � 1 distinct time slots. tu

We note that this bound of maxð2nk � 1; NÞ, whichapplies only when all the interfering links are removed, issmaller than the lower bound of 3N for general networksand that of maxð3nk � 3; NÞ for tree networks, as computedby Gandham et al. [1] for the 2-hop interference model. Theyproposed a time slot assignment scheme for tree networks,

which requires each node to maintain a buffer that stores atmost two packets and minimizes the schedule length. In thefollowing, we describe a time slot assignment scheme thatcomputes a schedule of length exactly equal to the lowerbound when interference is eliminated and does not requireto store more than one packet in buffers at any time.

4.2.2 Assignment of Time Slots

We now describe a time slot assignment scheme inAlgorithm 2, called LOCAL-TIMESLOTASSIGNMENT, whichis run locally by each node at every time slot. The key idea isto: 1) schedule transmissions in parallel along multiplebranches of the tree, and 2) keep the sink busy in receivingpackets for as many time slots as possible. Because the sinkcan receive from the root of at most one top-subtree in anytime slot, we need to decide which top-subtree should bemade active. We assume that the sink is aware of thenumber of nodes in each top-subtree. Each source nodemaintains a buffer and its associated state, which canbe either full or empty depending on whether it contains apacket or not. Our algorithm does not require any of thenodes to store more than one packet in their buffer at anytime. We initialize all the buffers as full, and assume thatthe sink’s buffer is always full for the ease of explanation.

Algorithm 2. LOCAL-TIMESLOTASSIGNMENT

1. node.buffer ¼ full2. if {node is sink} then

3. Among the eligible top-subtrees, choose the one with

the largest number of total (remaining) packets, say

top-subtree i

4. Schedule link ðrootðiÞ; sÞ respecting interfering

constraint5. else

6. if {node.buffer == empty} then

7. Choose a random child c of node whose buffer

is full

8. Schedule link ðc; nodeÞ respecting interfering

constraint

9. c.buffer = empty

10. node.buffer = full

11. end if

12. end if

The first block of the algorithm in lines 2-4 gives thescheduling rules between the sink and the roots of the top-subtrees. We define a top-subtree to be eligible if its root has atleast one packet to transmit. For a given time slot, weschedule the root of an eligible top-subtree which has thelargest number of total (remaining) packets. If none of the top-subtrees are eligible, the sink does not receive any packetduring that time slot.

Inside each top-subtree, nodes are scheduled accordingto the rules in lines 5-12. We define a subtree to be active ifthere are still packets left in the subtree (excluding its root)to be relayed. If a node’s buffer is empty and the subtreerooted at this node is active, we schedule one of its childrenat random whose buffer is not empty. Our algorithmguarantees (as proved in Lemma 3) that in an active subtree,there will always be at least one child whose buffer is notempty, and so whenever a node empties its buffer, it will


Fig. 2. Raw-data convergecast: largest top-subtree with nk nodes.


receive a packet in the next time slot, thus emptying buffersfrom the bottom of the subtree to the top.

We run through an example shown in Fig. 3a to explainthe algorithm. In the first time slot, since the eligible top-subtree containing the largest number of remaining packetsis f2; 5; 6g, we schedule the link ð2; sÞ, and the sink receives apacket from node 2 in slot 1. In the second time slot, theeligible top-subtrees are f1; 4g and f3; 7g, both of which havetwo remaining packets. We choose one of them at random,say f1; 4g, and schedule the link ð1; sÞ. Also, in the same timeslot since node 2’s buffer is empty, it chooses one of itschildren at random, say node 5, and schedule the link ð5; 2Þ.In the third time slot, the eligible top-subtrees are f2; 5; 6gand f3; 7g, both of which have two remaining packets. Wechoose the first one at random and schedule the link ð2; sÞ,and so the sink receives node 5’s packet (relayed by node 2).We also schedule the link ð4; 1Þ in the third time slot becausenode 1’s buffer is empty at this point. This process continuesuntil all the packets are delivered to the sink, yielding anassignment that requires seven time slots. Note that, in thisexample, 2nk � 1 ¼ 5, and so maxð2nk � 1; NÞ ¼ 7. In Fig. 3b,we show an assignment when all the interfering links arepresent, yielding a schedule length of 10.

In the following, we prove that the algorithm requiresexactly maxð2nk � 1; N) slots when all the interfering linksare eliminated. Before giving the details of the proof, wefirst highlight the two key insights of the algorithm: 1) thesink is kept busy in receiving packets for as many time slotsas possible, and 2) a node’s buffer is not empty for two ormore consecutive time slots so long as the subtree rooted atthis node is active. The first one is evident from thescheduling rule between the sink and the top-subtrees. Weprove the second insight in the following lemma:

Lemma 3. In an active subtree, a node with an empty bufferalways has a child and a parent whose buffers are full.

Proof. We prove it by induction on time slot t. The parentand grandparent of node i are denoted by pðiÞ and gpðiÞ;similarly, a child and a grandchild of i are denoted bycðiÞ and gcðiÞ, respectively. Slightly abusing notation, wealso use these symbols to denote the state of the bufferson the respective nodes.

At t ¼ 1, the lemma is trivially true because all thebuffers are full. Suppose the lemma holds for t ¼ k, i.e.,every node whose buffer is empty has a child and aparent whose buffers are full. At t ¼ kþ 1, each nodewith an empty buffer schedules one of its children whosebuffer is full. The following two situations can occur:

. Node i is full, while pðiÞ and cðiÞ are both empty.

. Nodes i and pðiÞ are both full, while cðiÞ is empty.

For the first case, we need to show that both pðiÞ andcðiÞ (since now they are empty) have a child and a parentwhose buffers are full. Clearly, pðiÞ has a child with a fullbuffer because i is now full. Similarly, pðiÞ also has aparent with a full buffer because a transmission tookplace from pðiÞ to its parent at t ¼ kþ 1. For the latter,cðiÞ has a parent with a full buffer because transmissiontook place from cðiÞ to i at t ¼ kþ 1. If the child of cðiÞ,i.e., gcðiÞ, was empty at t ¼ k, then gcðiÞ also had a childwith a full buffer because the lemma was true at t ¼ k.Therefore, at t ¼ kþ 1, the child of gcðiÞ transmits andfills up its parent’s buffer. Otherwise, if gcðiÞ was full att ¼ k, then it also remains full at t ¼ kþ 1 because itcannot transmit to its parent cðiÞ, which was full at t.

For the second case, cðiÞ transmitted and pðiÞ did not.For this to happen, gpðiÞ was full at t ¼ k and eitherempties or remains full at t ¼ kþ 1. If it empties, gpðiÞhas a parent with a full buffer because it transmitted att ¼ kþ 1, and also has a child with a full buffer becausepðiÞ did not transmit. If it remains full, at t ¼ kþ 1 nodesi, pðiÞ, and gpðiÞ are full, cðiÞ is empty, and gcðiÞ is full aswe showed in the first case. So, the lemma holds fort ¼ kþ 1, and the proof follows. tu

Theorem 2. If all the interfering links are eliminated, theschedule length for raw-data convergecast achieved byalgorithm LOCAL-TIMESLOTASSIGNMENT is the minimum,i.e., maxð2nk � 1; NÞ.

Proof. Let ni be the number of nodes in top-subtree i.Order the top-subtrees in nonincreasing order of theirsizes: nk � nk�1 � � � � � n1. Suppose nk >

Pk�1i¼1 ni; then

maxð2nk � 1; NÞ ¼ 2nk � 1. From Lemma 1, we knowthat it takes at least 2nk � 1 slots to schedule all thepackets originated in top-subtree k. Out of these, the sinkcan use at most nk � 1 slots to receive packets from theother top-subtrees, which have a total of at most nk � 1packets. Also, when nk >

Pk�1i¼1 ni, the root of the largest

top-subtree k gets scheduled once in every two timeslots. Therefore, the schedule length is at most 2nk � 1.

Now suppose nk �Pk�1

i¼1 ni; then maxð2nk � 1; NÞ ¼N . We need to show that there always exists an eligibletop-subtree to complement for the largest one when it isnot eligible. In this case, the sink will receive packets inevery slot, because otherwise it remains idle during sometime slots and the first condition nk >

Pk�1i¼1 ni will be met.

Thus, we will prove that the algorithm keeps theinequality nk �

Pk�1i¼1 ni as an invariant.

In any given time slot t, the algorithm schedules aneligible top-subtree that has the largest number ofremaining packets. At slot tþ 1, therefore, we havenk ¼ nk � 1, and the following three cases might arise:

. Top-subtree k still has the largest number ofremaining packets with nk � nk�1 � � � � � n1.Then, the root of k is again chosen to transmitat tþ 1, and the inequality still holds as nk � 1 �Pk�1

i¼1 ni.

. Top-subtree k and at least another one, say j, havean equal number of remaining packets. Then, the


Fig. 3. Raw-data convergecast using algorithm LOCAL-TIMESLOTAS-

SIGNMENT: (a) Schedule length of 7 when all the interfering links areremoved. (b) Schedule length of 10 when the interfering links arepresent.


root of j is chosen, and the inequality still holdsbecause nj � 1 �

Pk�1i¼1 ni � 1 (since nj ¼ nk � 1).

. Top-subtree k does not have the largest number ofremaining packets, implying that there were othertop-subtrees with an equal number of packets leftas k in slot t. Then, the root of a new largest top-subtree j is chosen, and the inequality holds sincenj � 1 �

Pk�1i¼1 ni � 1 (since nj ¼ nk).

Thus, the algorithm keeps the inequality as aninvariant, and there always exists a top-subtree that canbe alternately scheduled with the largest top-subtree.When nk ¼ 1,

Pk�1i¼1 ni � 1 ¼ 1, which means that there are

two packets left at two different top-subtrees that can bescheduled in alternate slots. Since this inequality holds forall the N steps, the sink always finds a top-subtree toreceive packets from, and therefore it takes N slots.Moreover, Lemma 1 implies that a top-subtree becomeseligible after a transmission because its root is filled up inthe next slot. Therefore, the theorem follows. tu

5 IMPACT OF INTERFERENCE

So far, we have focused on computing spatial-reuse TDMAschedules where transmissions take place on the samefrequency at a constant transmission power. In this section,we focus on different methods to mitigate the effects ofinterference on the schedule length. First, we discuss thebenefits of using transmission power control and explainthe basics of a possible algorithm. Then, we discuss theadvantages of using multiple channels by considering threedifferent channel assignment schemes.

5.1 Transmission Power Control

In wireless networks, excessive interference can be elimi-nated by using transmission power control [6], [20], i.e., bytransmitting signals with just enough power instead ofmaximum power. To this end, we evaluate the impact oftransmission power control on fast data collection usingdiscrete power levels, as opposed to a continuous rangewhere an unbounded improvement in the asymptoticcapacity can be achieved by using a nonlinear powerassignment [5]. We first explain the basics of one particularalgorithm that we use in our evaluations in Section 7.

The algorithm proposed by ElBatt and Ephremides [6] isa cross layer method for joint scheduling and power controland it is an optimal distributed algorithm to improve thethroughput capacity of wireless networks. The goal is tofind a TDMA schedule that can support as many transmis-sions as possible in every time slot. It has two phases: 1)scheduling and 2) power control that are executed at everytime slot. First, the scheduling phase searches for a validtransmission schedule, i.e., largest subset of nodes, where nonode is to transmit and receive simultaneously, or toreceive from multiple nodes simultaneously. Then, in thegiven valid schedule, the power control phase iterativelysearches for an admissible schedule with power levels chosento satisfy all the interfering constraints. In each iteration, thescheduler adjusts the power levels depending on thecurrent RSSI at the receiver and the SINR thresholdaccording to the iterative rule: Pnew ¼ �

SINR � Pcurrent. Accord-ing to this rule, if a node transmits with a power level

higher than what is required by the threshold value, itshould decrease its power and if it is below the threshold, itshould increase its transmission power, within the availablerange of power levels on the radio. If all the nodes meetthe interfering constraint, the algorithm proceeds with theschedule calculation for the next time slot. On the otherhand, if the maximum number of iterations is reached andthere are nodes which cannot meet the interfering con-straint, the algorithm excludes the link with minimumSINR from the schedule and restarts the iterations with thenew subset of nodes. The power control phase is repeateduntil an admissible transmission scenario is found.

5.2 Multichannel Scheduling

Multichannel communication is an efficient method toeliminate interference by enabling concurrent transmissionsover different frequencies [21]. Although typical WSN radiosoperate on a limited bandwidth, their operating frequenciescan be adjusted, thus allowing more concurrent transmis-sions and faster data delivery. Here, we consider fixed-bandwidth channels, which are typical of WSN radios, asopposed to the possibility of improving link bandwidth byconsolidating frequencies. In this section, we explain threechannel assignment methods that consider the problem atdifferent levels allowing us to study their pros and cons forboth types of convergecast. These methods consider thechannel assignment problem at different levels: the link level(JFTSS), node level (RBCA), or cluster level (TMCP).

5.2.1 Joint Frequency Time Slot Scheduling

JFTSS offers a greedy joint solution for constructing amaximal schedule, such that a schedule is said to bemaximal if it meets the adjacency and interfering constraints,and no more links can be scheduled for concurrenttransmissions on any time slot and channel withoutviolating the constraints. Approximation bounds on JFTSSfor single-channel systems and its comparison with multi-channel systems are discussed in [22] and [23], respectively.

JFTSS schedules a network starting from the link that hasthe highest number of packets (load) to be transmitted.When the link loads are equal, such as in aggregatedconvergecast, the most constrained link is considered first,i.e., the link for which the number of other links violatingthe interfering and adjacency constraints when scheduledsimultaneously is the maximum. The algorithm starts withan empty schedule and first sorts the links according to theloads or constraints. The most loaded or constrained link inthe first available slot-channel pair is scheduled first andadded to the schedule. All the links that have an adjacencyconstraint with the scheduled link are excluded from the listof the links to be scheduled at a given slot. The links that donot have an interfering constraint with the scheduled linkcan be scheduled in the same slot and channel whereas thelinks that have an interfering constraint should be sched-uled on different channels, if possible. The algorithmcontinues to schedule the links according to the mostloaded (or most constrained) metric. When no more linkscan be scheduled for a given slot, the scheduler continueswith scheduling in the next slot. Fig. 4a shows the same treegiven in Fig. 1a which is scheduled according to JFTSSwhere aggregated data are collected. JFTSS starts with link



(2; sink) on frequency 1 and then schedules link (4,1) nexton the first slot on frequency 2. Then, links (5,2) onfrequency 1 and (1; sink) on frequency 2 are scheduled onthe second slot and links (6,2) on frequency 1 and (3; sink)on frequency 2 are scheduled on the last slot.

An advantage of JFTSS is that it is easy to incorporate thephysical interference model; however, it is hard to have adistributed solution since the interference relationshipbetween all the links must be known.

5.2.2 Tree-Based Multichannel Protocol

TMCP is a greedy, tree-based multichannel protocol for datacollection applications [8]. It partitions the network intomultiple subtrees and minimizes the intratree interference byassigning different channels to the nodes residing ondifferent branches starting from the top to the bottom ofthe tree. Fig. 4b shows the same tree given in Fig. 1a which isscheduled according to TMCP for aggregated data collec-tion. Here, the nodes on the leftmost branch are assignedfrequency F1, second branch is assigned frequency F2, andthe last branch is assigned frequency F3 and after thechannel assignments, time slots are assigned to the nodeswith the BFS-TimeSlotAssignment algorithm. The advantageof TMCP is that it is designed to support convergecast trafficand does not require channel switching. However, conten-tion inside the branches is not resolved since all the nodeson the same branch communicate on the same channel.

5.2.3 Receiver-Based Channel Assignment

In our previous work [7], we proposed a channel assign-ment method called RBCA where we statically assigned thechannels to the receivers (parents) so as to remove as manyinterfering links as possible. In RBCA, the children of acommon parent transmit on the same channel. Every nodein the tree, therefore, operates on at most two channels, thusavoiding pairwise, per-packet channel negotiation over-heads. The algorithm initially assigns the same channel toall the receivers. Then, for each receiver, it creates a set ofinterfering parents based on SINR thresholds and itera-tively assigns the next available channel starting from themost interfered parent (the parent with the highest numberof interfering links). However, due to adjacent channeloverlaps, SINR values at the receivers may not alwaysbe high enough to tolerate interference, in which case thechannels are assigned according to the ability of thetransceivers to reject interference. We proved approxima-tion factors for RBCA when used with greedy scheduling in[9]. Fig. 4c shows the same tree given in Fig. 1a scheduledwith RBCA for aggregated convergecast. Initially, all nodes

are on frequency F1. RBCA starts with the most interferedparent, node 2 in this example, and assigns F2. Then, itcontinues to assign F3 to node 3 as the second mostinterfered parent. Since all interfering parents are assigneddifferent frequencies, sink can receive on F1.

6 IMPACT OF ROUTING TREES

Besides transmission power control and multiple channels,the network topology and the degree of connectivity alsoaffect the scheduling performance. In this section, wedescribe schemes to construct topologies with specificproperties that help to reduce the schedule length.

6.1 Aggregated Data Collection

We first construct balanced trees and compare theirperformance with unbalanced trees. We observe that in bothcases, the sink often creates a high-degree bottleneck. Toovercome this, we then propose a heuristic, as described inAlgorithm 3, by modifying Dijkstra’s shortest path algorithmto construct degree-constrained trees. Note that constructingsuch a degree-constrained tree is NP-hard. Each source nodei in our heuristic keeps track of the number of its children,CðiÞ, which is initialized to 0, and a hop count to the sink,HCðiÞ, which is initialized to 1. The algorithm starts withthe sink node, and adds a node i0 62 T at every iteration to thetree such that HCði0Þ is minimized. It stops when jT j ¼ jV j,or when no more nodes can be added to the tree because theneighbors of all these new nodes have reached the limit ontheir maximum degree. Consequently, in this latter situation,the heuristic might not always generate a spanning tree. Inour evaluation presented in Section 7.3, we consider onlythose instances of the topologies where spanning trees withthe specified degree constraint are produced.

Algorithm 3. DEGREE-CONSTRAINED TREES

1. Input: GðV ;EÞ, s, max degree2. T fsg3. for all i 2 V do

4. CðiÞ 0; HCðiÞ 15. end for

6. HCðsÞ 0

7. while jT j 6¼ jV j do

8. Choose i0 62 T such that:

9. (a) ði; i0Þ 2 E, for some i 2 T with CðiÞ <max degree� 1

10. (b) HCði0Þ is minimized

11. T T [ fi0g12. HCði0Þ ¼ HCðiÞ þ 1

13. CðiÞ CðiÞ þ 1

14. if 8i 2 V , CðiÞ ¼ max degree then

15. break

16. end if

17. end while

To illustrate the gains of degree-constrained trees,consider the case when all the N nodes are in range ofeach other and that of the sink. If the nodes select theirparents according to minimum hop without a degreeconstraint, then all of them will select the sink, and thiswill give a schedule length of N . However, if we limit thenumber of children per node to 2, then this will result in


Fig. 4. Scheduling with multichannels for aggregated convergecast:(a) Schedule generated with JFTSS. (b) Schedule generated withTMCP. (c) Schedule generated with RBCA.


two subtrees rooted at the sink, and if there are enoughfrequencies to eliminate interference, the network can bescheduled using only two time slots, thus achieving a factorof N=2 reduction in the schedule length.

6.2 Raw-Data Collection

As emphasized in [13], routing trees that allow more paralleltransmissions do not necessarily result in small schedulelengths. For instance, the schedule length is N for a networkconnected as a star topology, whereas it is ð2N � 1Þ for a linetopology once interference is eliminated. Theorem 1suggests that the routing tree should be constructed suchthat all the branches have a balanced number of nodes andthe constraint nk < ðN þ 1Þ=2 holds. In this section, weconstruct such routing trees.

A balanced tree satisfying the above constraint is avariant of a capacitated minimal spanning tree [24]. The CMSTproblem, which is known to be NP-complete, is todetermine a minimum-hop spanning tree in a vertexweighted graph such that the weight of every subtreelinked to the root does not exceed a prescribed capacity. Inour case, the weight of each link is 1, and the prescribedcapacity is ðN þ 1Þ=2. Here, we propose a heuristic, asdescribed in Algorithm 4, based on the greedy schemepresented by Dai and Han [25], which solves a variant of theCMST problem by searching for routing trees with an equalnumber of nodes on each branch. We augment their schemewith a new set of rules and grow the tree hop by hopoutward from the sink. We assume that the nodes knowtheir minimum-hop counts to sink.

Algorithm 4. CAPACITATED-MINIMALSPANNINGTREE

1. Input: GðV ;EÞ, s2. Initialize:3. B roots of top subtrees // the branches

4. T fsg [B5. 8i 2 V , GSðiÞ unconnected neighbors of i

at further hops

6. 8b 2 B, WðbÞ 1

7. h 2

8. while h 6¼ max hop count do

9. Nh unconnected nodes at hop distance h10. Connect nodes N 0h that have a single potential

parent: T TSN 0h

11. Update Nh Nh nN 0h12. Sort Nh in non-increasing order of jGSj13. for all i 2 Nh do

14. for all b 2 B to which i can connect do

15. Construct SSði; bÞ16. end for

17. Connect i to b for which WðbÞ þ jSSði; bÞj is

minimum

18. Update GSðiÞ and WðbÞ19. T T

SfigSSSði; bÞ

20. end for

21. h hþ 1

22. end while

Rule 1. Nodes with single potential parents are con-nected first.

Rule 2. For nodes with multiple potential parents, wefirst construct their growth sets (GS) and choose the one with

the largest cardinality for further processing, breaking tiesbased on the smallest id. We define the growth set of a nodeas the set of neighbors (potential children) that are not yetconnected to the tree and have larger hop counts.

Rule 3. Once a node is chosen based on the growth setsaccording to Rule 2, we construct search sets (SS) to decidewhich potential branch the node should be added to. Asearch set is thus branch specific and includes the nodesthat are not yet connected to the tree and are neighbors of anode that are at a higher hop count. In particular, if thechosen node has access to branch b, and has a neighbor thatcan connect to only branch b if b is selected, then thisneighbor and its potential children are included in thesearch set for b. However, if the neighbor has access to atleast one other branch even after b is selected, then it is notincluded in the search set.

The search sets guarantee that the choices for the nodesat longer hops to join a particular branch are not limited bythe decision of the joining node. This balances out thenumber of nodes on different branches and prevents one togrow faster than others. Once the search sets are con-structed, we choose the branch for which the sum of its load(W ) and the size of the search set is minimum.

To illustrate the merit of search sets, consider thesituation shown in Fig. 5. Dotted lines represent potentialcommunication links and solid lines represent alreadyincluded tree edges. At this point, node 4 is beingprocessed, and the loads on branches b1 and b2 are 2 and4, respectively, where bi denote the branch rooted at node i.The search set SSð4; b1Þ is f8; 9; 10g, because the neighbornode 8 has access to only b1 if b1 is selected by node 4.However, the search set SSð4; b2Þ is empty, because theneighbor node 8 has access to another branch b1 (via node3). Therefore, the sum of the load and the size of the searchset for b1 is 5, and that for b2 is 4. So, we attach node 4 to b2,and in the next step attach node 8 to b1. This balances outthe number of nodes over the two branches.

7 EVALUATION

In this section, we evaluate the impact of transmissionpower control, multiple channels, and routing trees on thescheduling performance for both aggregated and raw-dataconvergecast.

We deploy nodes randomly in a region whose dimen-sions are varied between 20� 20 m2 and 300� 300 m2 tosimulate different levels of density. The number of nodes iskept fixed at 100. For different parameters, we average eachpoint over 1,000 runs. We use an exponential path-lossmodel for signal propagation with the path-loss exponent �


Fig. 5. Balanced tree construction: Node 4 is attached to b2 based on thesearch sets; load on both b1 and b2 is 5.


varying between 3 and 4, which is typical for indoorenvironments. We also use the physical interference modeland simulate the behavior of CC2420 radios that are usedon Telosb and TmoteSky motes and are capable ofoperating on 16 different frequencies. The transmissionpower can be adjusted between �24 and 0 dBm over eightdifferent levels, and the SINR threshold is set to � ¼ �3 dB.1

We first evaluate the schedule length for single-channelTDMA, and then its improvement using transmissionpower control, multiple channels, and routing trees.

7.1 Impact of Transmission Power Control

We investigate two cases: 1) when nodes transmit atmaximum power, and 2) when nodes adjust theirtransmission power according to the algorithm describedin Section 5.1. In both cases, nodes communicate on thesame channel and use minimum-hop routing trees. In thefirst case, time slots are assigned according to BFS-TIMESLOTASSIGNMENT for aggregated data, and accord-ing to LOCAL-TIMESLOTASSIGNMENT for raw data. In thesecond case, we follow the scheduling rules in [6].

7.1.1 Aggregated Convergecast

Fig. 6a shows the variation of schedule length with densityfor different values of � on minimum-hop trees. Weobserve that the schedule length decreases as the deploy-ment gets sparser. This happens because at low densities,the interference is less, and so more concurrent transmis-sions can take place. In the densest deployment (L ¼ 20)when all the nodes are within the range of each other, thesink is the only parent, and the network is scheduled in99 time slots regardless of power control. However, insparser scenarios, using power control, the network can bescheduled with fewer time slots as the level of interferencegoes down. We achieve a 10-20 percent reduction inschedule length for the best case.

We also observe that power control is more effective inreducing the schedule length for denser deployments thanin sparser ones where the results tend to be similar. This isdue to the discrete power levels and limited power range.Moreover, due to the �95 dBm threshold for the transcei-vers to be able to decode a signal successfully, furtherpower reduction is limited.

7.1.2 Raw-Data Convergecast

For raw-data convergecast, we observe in Fig. 6b that theschedule length increases as the network gets sparser onminimum-hop trees. This is counterintuitive because insparse networks, the reuse of slots should be higher whichwould reduce the schedule length. However, as the networkgets sparser, the number of nodes that can directly reach thesink decreases and packets have to be relayed over morehops. Thus, more packets need to be scheduled than in asingle hop. We see that the number of packets to be scheduledincreases faster than the reuse ratio. In the densest settingwhere all the nodes can directly reach the sink, the schedulelength is 99, which is equal to the number of sources.

With power control, we observe a reduction in theschedule length in Fig. 6b as some of the interfering linksare eliminated, thus increasing slot reusability. When� ¼ 3:0, most of the interference can be eliminated bypower control, and beyond which the structure of therouting tree, especially the number of nodes nk on thelargest branch with ð2nk � 1Þ > N , becomes the bottleneck.However, for � � 3:5, power control cannot always elim-inate interference as networks get sparser and nodes tend totransmit at their maximum power.

7.2 Impact of Multichannel Scheduling

In this section, we analyze the performance of the channelassignment methods discussed in Section 5.2. We useCC2420 radios that have 16 channels in the 2.4 GHz range,with adjacent channels overlapping according to therejection and blocking values given in the data sheet.We assume that the nodes transmit at maximum power anduse minimum-hop trees. In TMCP and RBCA, time slots areassigned according to BFS-TIMESLOTASSIGNMENT foraggregated convergecast and LOCAL-TIMESLOTASSIGN-

MENT for raw-data convergecast. The path-loss exponentis taken as 3.5.

7.2.1 Aggregated Convergecast

Comparing the plots in Figs. 7a and 6a, we observe that thechannel assignment methods achieve schedule lengths thatare shorter than those achieved by power control. While it’strue that power control helps in reducing the effects ofinterference, this gain is limited due to the discrete levels andlimited range of transmission power (e.g., CC2420 has eightdifferent power levels between 0 and �24 dBm). In sparsedeployments, nodes cannot reduce their transmit power


Fig. 6. Scheduling on minimum-hop trees with and without powercontrol: (a) Aggregated convergecast. (b) Raw-data convergecast.

Fig. 7. Scheduling on minimum-hop trees with multiple channels:(a) Aggregated convergecast. (b) Raw-data convergecast.

1. Due to variation in signal strength, a fading margin can be includedsuch that some of the packets can still be captured if the RSSI is slightlylower than the threshold. Such a model [26] can easily be incorporated inour experiments, in which case retransmissions of lost packets should alsobe considered in calculating the schedule length.


below a certain threshold because a transceiver cannotdecode signals below the sensitivity level �95 dBm). Asshown in Fig. 6a, for L > 200, the schedule lengths are similarwhen nodes transmit at maximum power or when theyadjust their power levels. Moreover, in mid-sparse deploy-ments (60 � L � 180, Fig. 6a), the limited range and discretepower levels restrict the nodes to adjust their transmitpowers. On the other hand, multichannel communicationeven with just two frequencies (Fig. 7a) can eliminate theinterference limitations, and beyond this, the performancegains are limited by the connectivity structure. By transmit-ting on different channels, interference is eliminated by thehigh adjacent/alternate channel rejection values of the C2420radio, and the channels behave like orthogonal.

In Fig. 7a, sparser deployments (L > 140) with multi-channel communication show a 40 percent reduction inschedule length as compared to transmitting on a singlechannel with maximum power. However, in denserdeployments, multiple channels do not help much due toincreased connectivity, with the sink as a bottleneck in thedensest setting. From Fig. 7a, we observe that JFTSS andRBCA can optimally schedule the network using 16channels, i.e., they achieve the lower bound, as shown bythe line “Lower Bound-MHST.” The advantage of RBCAover JFTSS is that it takes into account the topologicalcharacteristics: a parent node receives data on the samechannel from its children, and does not have to switchchannels in every slot. In dense deployments, TMCPperforms better due to the different routing trees con-structed, i.e., when L ¼ 20, RBCA and JFTSS construct a startopology, whereas TMCP constructs a 2-branch tree withtwo channels and a 16-branch tree with 16 channels.

7.2.2 Raw-Data Convergecast

In Fig. 7b, we observe that none of the methods caneliminate interference completely with two channels;however, JFTSS and RBCA can do so with six or morechannels at different densities (plots are not shown due tolack of space). We also see that TMCP needs 16 channels toreach a performance similar to that achieved by RBCA andJFTSS with only two channels. This is because in JFTSS andRBCA, when a node is receiving from its children, its parentcan transmit simultaneously on a different channel, whichis not possible due to intrabranch interference in TMCP.The results also verify that JFTSS and RBCA can achieve aschedule length which is bounded by maxð2nk � 1; NÞ,shown as “Lower Bound-MHST,” so long as the number ofavailable channels is sufficient to eliminate interference.Compared to the results on a single channel in sparserscenarios, we achieve a reduction of up to 40 percent on theschedule length. In very dense scenarios, the improvementis small because most of the nodes can directly reach thesink, and so the limiting factor becomes the half-duplextransceiver.

7.2.3 Required Number of Channels

In this section, for the different channel assignmentmethods, we evaluate the required number of channels tocompletely eliminate interference as a function of deploy-ment density. In our simulation results, as shown in Fig. 8a,we assume that the number of available channels isunlimited so as to show the upper bounds.

With RBCA and JFTSS, the number of channels required islow for dense networks as the number of receivers is low. Inparticular, when L ¼ 20, all the nodes can directly connect tothe sink, and so only one frequency is needed. As the networkgets sparser, the number of receivers increases, and thusmore frequencies are required to support concurrenttransmissions. However, for L � 80, the number of nodesthat are being connected to the same parent slowly dominatesthe effect of the number of receivers, and since the networkgets very sparse, the number of channels required furthergoes down as the level of interference decreases.

The trends of both RBCA and JFTSS are quite similar, andthe number of channels required is no more than the numberof available channels on CC2420 radios (16 channels). On theother hand, TMCP requires many more channels as eachbranch is on a different channel. This is expensive fordeployments where a lot of nodes can directly connect to thesink, and thus are assigned different channels because theyform different branches. Thus, one needs to optimize thechannel usage.

7.2.4 Interference Models and Orthogonal Channels

We now evaluate the impact of interference and channelmodels on the schedules, in particular, the idealisticassumption of the protocol model and the orthogonalityassumption. We examine the feasibility of the schedulesbased on the adjacent and alternate channel rejection valuesof the transceivers and the SINR threshold.

Fig. 8b shows the results for JFTSS in terms of thepercentage of nodes that are incorrectly scheduled (hence-forth, referred to as errors). The top two lines show theerrors for two and 16 channels with both the assumptions,whereas the bottom line shows the errors only for theorthogonality assumption. We observe that the errorsare much higher in sparser deployments, because althoughthe interference created by an individual sender is not highenough to jam concurrent transmissions, the cumulativeeffect from multiple senders is very high, which is notcaptured in the protocol model. On the other hand, in densedeployments, a single transmitter can jam another onebecause of smaller internode distances and higher level ofinterference. In such cases, some of the nodes might selectthe next available channels for concurrent transmissions;however, interference can still be high because the channelsin reality are not perfectly orthogonal. After the peak, thenetwork gets sparser and interference reduces. We notethat, our simulations corroborate previous results [27] that


Fig. 8. (a) Bounds on the number of frequencies. (b) Percentage ofincorrectly scheduled links.


the protocol model may result in serious interference, andadjacent channel interference cannot always be ignored.

7.3 Impact of Routing Trees

In the preceding sections, we observed that althoughinterference can be substantially eliminated by using powercontrol and multiple channels, connectivity of the tree stilllimits the performance. In the following, we discuss theimprovements with routing tress.

7.3.1 Aggregated Convergecast on Degree-Constrained

Trees

Fig. 9a shows the variation of schedule length with densitywhen the maximum tree degree is 3 (in sparser scenarios,with a maximum degree of 2, it was not always possible toconstruct connected topologies). The top two lines are fornodes transmitting at maximum power, and nodes usingpower control. The bottom two lines are for JFTSS andRBCA. When nodes transmit with maximum power, weobserve a reduction in the schedule length in densedeployments as compared to non-degree-constrained treesshown in Fig. 6a. We also notice further improvement withpower control in denser deployments than in sparser ones.

When nodes are assigned channels using RBCA, we see afactor of more than 2 reduction in the schedule length indense deployments (L < 120), as compared to that usingRBCA on minimum-hop spanning trees. We also observethat the schedule lengths are much larger than themaximum degree in the routing tree for dense deploy-ments, as compared to those in sparse scenarios (L � 120).Considering deployments at different densities, routingover minimum-hop degree-constrained spanning treestogether with RBCA achieves an order of magnitudeimprovement than routing over minimum-hop spanningtrees while transmitting at maximum power. When we useJFTSS, the schedule length is close or equal to the maximumdegree since it can handle interference using multiplechannels more effectively by reusing and assigning them tothe links instead of the receivers.

7.3.2 Raw-Data Convergecast on CMST

Fig. 9b shows the variation of schedule length on CMST.The impact of such routing trees is more prominent insparser networks (L � 200) than routing over minimum-hop spanning trees (Fig. 7b). When L < 200, the length isbounded by N . Beyond this point, it is almost always notpossible to construct trees where the constraint 2nk � 1 < N

holds. In such cases, the schedule length is limited by2nk � 1. These results indicate that RBCA and JFTSScombined with a suitable tree construction mechanism canachieve a reduction of up to 50 percent in the schedulelength as compared to single-channel communication onminimum-hop spanning trees.

8 CONCLUSIONS

In this paper, we studied fast convergecast in WSN wherenodes communicate using a TDMA protocol to minimizethe schedule length. We addressed the fundamentallimitations due to interference and half-duplex transceiverson the nodes and explored techniques to overcome thesame. We found that while transmission power controlhelps in reducing the schedule length, multiple channels aremore effective. We also observed that node-based (RBCA)and link-based (JFTSS) channel assignment schemes aremore efficient in terms of eliminating interference ascompared to assigning different channels on differentbranches of the tree (TMCP).

Once interference is completely eliminated, we provedthat with half-duplex radios, the achievable schedule lengthis lower bounded by the maximum degree in the routingtree for aggregated convergecast, and by maxð2nk � 1; NÞfor raw-data convergecast. Using optimal convergecastscheduling algorithms, we showed that the lower boundsare achievable once a suitable routing scheme is used.Through extensive simulations, we demonstrated up to anorder of magnitude reduction in the schedule length foraggregated, and a 50 percent reduction for raw-dataconvergecast. In future, we will explore scenarios withvariable amounts of data and implement and evaluate thecombination of the schemes considered.

REFERENCES

[1] S. Gandham, Y. Zhang, and Q. Huang, “Distributed Time-OptimalScheduling for Convergecast in Wireless Sensor Networks,”Computer Networks, vol. 52, no. 3, pp. 610-629, 2008.

[2] K.K. Chintalapudi and L. Venkatraman, “On the Design of MACProtocols for Low-Latency Hard Real-Time Discrete ControlApplications over 802.15.4 Hardware,” Proc. Int’l Conf. InformationProcessing in Sensor Networks (IPSN ’08), pp. 356-367, 2008.

[3] I. Talzi, A. Hasler, G. Stephan, and C. Tschudin, “PermaSense:Investigating Permafrost with a WSN in the Swiss Alps,” Proc.Workshop Embedded Networked Sensors (EmNets ’07), pp. 8-12, 2007.

[4] S. Upadhyayula and S.K.S. Gupta, “Spanning Tree BasedAlgorithms for Low Latency and Energy Efficient Data Aggrega-tion Enhanced Convergecast (DAC) in Wireless Sensor Net-works,” Ad Hoc Networks, vol. 5, no. 5, pp. 626-648, 2007.

[5] T. Moscibroda, “The Worst-Case Capacity of Wireless SensorNetworks,” Proc. Int’l Conf. Information Processing in SensorNetworks (IPSN ’07), pp. 1-10, 2007.

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Fig. 9. (a) Scheduling on degree-constrained minimum-hop trees.(b) Scheduling on CMST.


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Ozlem Durmaz Incel received the BSc andMSc degrees in computer engineering from theYeditepe University, Turkey, in 2002 and 2005,respectively. She received the PhD degree incomputer science from the University ofTwente, Netherlands, in March 2009. She iscurrently a postdoctoral researcher in theNetworking Laboratory (NETLAB) of BogaziciUniversity, Turkey. Her dissertation is focusedon efficient data collection in wireless sensor

networks and was entitled “Multi-Channel Wireless Sensor Networks:Protocols, Design and Evaluation.” She was a visiting student in theAutonomous Networks Research Group of the University of SouthernCalifornia as part of her PhD studies from 2007-2008.

Amitabha Ghosh received the BS degree inphysics from the Indian Institute of TechnologyKharagpur, India, in 1996, and the ME degreein computer science and engineering from theIndian Institute of Science, Bengaluru, in 2000.He received the PhD degree in electricalengineering from the University of SouthernCalifornia in August 2010. Between 2000 and2004, he worked for Alcatel-Lucent andHoneywell on GSM/GPRS networks and wire-

less ad hoc networks. He is currently a postdoctoral researcher atPrinceton University. His research interest is in design and analysisof algorithms for scheduling, routing, and power control for cellularand wireless sensor networks.

Bhaskar Krishnamachari received the BEdegree in electrical engineering from The CooperUnion in 1998, and the MS and PhD degrees inelectrical engineering from Cornell University in1999 and 2002, respectively. He is an associateprofessor in the Ming Hsieh Electrical Engineer-ing Department at the University of SouthernCalifornia Viterbi School of Engineering. Hisresearch interests are in the design and perfor-mance analysis of protocols for next-generation

wireless networks. He received the US National Science FoundationCAREER Award in 2004.

Krishnakant Chintalapudi received the MSdegree in electrical engineering from DrexelUniversity and the PhD degree in computerscience from the University of SouthernCalifornia. He is currently working as anassociate researcher at Microsoft Research,India. His research interests include delay-sensitive and fault-tolerant wireless sensornetworks, high-data-rate wireless sensor net-work applications, hybrid sensor networks, and

wireless control and automation.

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