Preferential Field Coverage Through Detour-BasedMobility Coordination

HANY MORCOS AZER BESTAVROS IBRAHIM MATTA

hmorcos@bu.edu best@bu.edu matta@bu.edu

Abstract—Controlling the mobility pattern of mobile nodes(e.g., robots) to monitor a given field is a well-studied problem insensor networks. In this setup, absolute control over the nodes’mobility is assumed. Apart from the physical ones, no otherconstraints are imposed on planning mobility of these nodes. Inthis paper, we address a more general version of the problem.Specifically, we consider a setting in which mobility of eachnode is externally constrained by a schedule consisting of alist of locations that the node must visit at particular times.Typically, such schedules exhibit some level ofslack, which couldbe leveraged to achieve a specific coverage distribution of a field.Such a distribution defines the relative importance of differentfield locations. We define the Constrained Mobility Coordinationproblem for Preferential Coverage (CMC-PC) as follows: givena field with a desired monitoring distribution, and a number ofnodesn, each with its own schedule, we need to coordinate themobility of the nodes in order to achieve the following two goals:1) satisfy the schedules of all nodes, and 2) attain the requiredcoverage of the given field. We show that the CMC-PC problemis NP-complete (by reduction to the Hamiltonian Cycle problem).Then we propose TFM, a distributed heuristic to achieve fieldcoverage that is as close as possible to the required coveragedistribution. We verify the premise of TFM using extensivesimulations, as well as taxi logs from a major metropolitan area.We compare TFM to the random mobility strategy —the latterprovides a lower bound on performance. Our results show thatTFM is very successful in matching the required field coveragedistribution, and that it provides, at least, two-fold query successratio for queries that follow the target coverage distribution ofthe field.

I. I NTRODUCTION

Controlling mobility of a number of objects (e.g., robots)in order to cover a given field is a well-studied problem inthe literature. In this model, node mobility could be used to(i) circumvent low density of nodes, (ii) navigate to hard-to-reach areas (due to natural barriers) in order to achieveuniform coverage of the field, and/or (iii) react to some changein the environment (e.g., forest fire), or address preferentialcoverage based on changing demands. In such a model, it isusually assumed that the mobile nodes are under the controlof a single authority that decides the mobility pattern of eachmobile node.

In this paper, we consider a model ofautonomousmobileusers (nodes / sensors). These autonomous mobile users areinterested in monitoring a given field according to some targetdistribution — the distribution defines the percentage of timedifferent field locations should be covered by, at least, onenode (sensor). The field monitoring distribution stems fromthe inherent interest of users to query the state of differentfield locations. We also assume that mobility of each user iscoarsely directed by anexternalschedule. A node’s scheduledefines a list of locations and a corresponding list of times(waypoints), such that, for a node to satisfy its schedule, it

has to be present at the specified locations at the indicatedtimes. An important attribute of such a schedule is howtight/relax are the consecutive journeys between waypoints.That is, if a schedule allows much more time, than the neededminimum, for a node to reach each waypoint, then it wouldbe a relaxed schedule with plenty ofslack, otherwise, it wouldbe a tight schedule. The problem is then,how to coordinatethe mobility of nodes and manage their slacks so as to achievethe requested monitoring distribution.

To see why this is the case, consider a situation wherea user moving between two points A and B may havemultiple choices of paths of almost equal expected quality(e.g., in terms of traveled distance or time). Taking any of thealternative paths leads to monitoring different field locations.Such a scenario is particularly true for paths between locationsin a dense urban setting. As an illustration, consider Figure8, which shows paths followed by cabs on the streets of theSan Francisco Bay area. The grid structure of the paths taken(underscoring the underlying city blocks in SF) demonstratesthe existence of multiple routesof indistinguishable lengths, totravel between arbitrary points A and B on the grid. In such acase, it is perceivable that one might think that all nodes wouldsatisfy their own schedules in one of the following manners:(1) Nodes would prefer paths leading to the monitoring ofhigh-demand spots in the field, or (2) nodes would takerandom routes in each journey between each two consecutivewaypoints in the schedule.

In the first scenario, if all users end up monitoring thesame (highest-demand) field locations, the rest of the fieldwould be left unmonitored, resulting in missing many ofthe users queries. On the other hand, if nodes take randompaths, as we will show in the evaluation section (SectionV), this will lead to poor coverage of the field, since the“importance” of each field location (indicated by the de-sired/target monitoring distribution) will be ignored in makingrandom mobility decisions. This accentuates the importanceof coordinating mobility of users, while ensuring that allschedules are satisfied.

Our contributions can be summarized as follows:• We apply the above mobility model of autonomous nodes

and its features (i.e., slack) to the problem of distributedfield coverage. We coin the problem of ConstrainedMobility Coordination for Preferential field Coverage(CMC-PC). We show that this problem is NP-complete(Section II), and argue that none of the existing researchefforts is adequate to solve the problem (Section III).

• We develop TFM, the first mobility coordination strategythat aims to achieve a given distribution of field coverage(Section IV). Under TFM, in steady state, nodes are in

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a dynamic (i.e., mobile) state. This salient characteristicof TFM enables it to achieve the required coveragedistribution of a spatio-temporal field with a low-densitynetwork.

• Using extensive simulations, we compare TFM to therandom mobility strategy — the latter provides a lowerbound on performance. Our results indicate the significantperformance gain attained by using TFM over randommobility (Section V-A).

• Furthermore, we perform a trace-driven evaluation ofTFM and random mobility. We use cab traces from cabsin the San Francisco area. Results of the trace-drivenevaluation underscore the superiority of TFM in practicalsettings (Section V-B).

II. PROBLEM DEFINITION

In this section, we define the Constrained Mobility Coordina-tion problem for Preferential Coverage (CMC-PC), then showthat it is NP-complete.

Definition 1: (Nodes): N autonomously mobile nodesmove in the target field. Mobility of each node is externallyconstrained by a schedule (Definition 2). The prime goal ofthese nodes is to satisfy their own schedules. While doing so,they also try to cooperatively cover the target field accordingto the required coverage distribution (Definition 4).

Definition 2: (ScheduleL): A schedule of nodeni is alist L(ni) of tuples of the formuij = (τij , lij), where1 ≤j ≤ |L(ni)|. To satisfy a schedule entryuij , nodeni has tobe at locationlij at timeτij . For ni to satisfy its schedule, ithas to satisfyuij for all 1 ≤ j ≤ |L(ni)|.

Definition 3: (Field G): The target field is represented asa graphG = (V,E), such that each vertexv ∈ V representsa field location, and each edgee ∈ E connects two verticesrepresenting two field locations that could be directly reachedfrom each other.

Definition 4: (Coverage Distribution D): Coverage of agiven field is defined by a target coverage distributionD, suchthat D(v) is the relative importance of field locationv ∈ V .The coverage distributionD represents the preferential interestin covering different locations in the field, and is application-specific. Practically,D could be interpreted in a number ofways. For example, we could require that more importantfield locations be covered morefrequentlythan less importantones. Another interpretation, is to require that more importantlocations be covered forlonger periodscompared to lessimportant ones. In this paper, we adopt the latter interpretation.Specifically, we interpretD(v) as the required percentage oftime, during which, field locationv ∈ V should be covered, byat least one node. We also note that, at any time, a field locationis either covered or not. Hence, covering a given location withonly one node is exactly equivalent to covering it with morethan one node.

Definition 5: (Communication Range r): Any twonodes can communicate with each other only if the distancebetween them is less than or equal to a (given) fixed commu-nication ranger.

Definition 6: (Speed of Motionηi): The maximum speedof motion of a nodeni is ηi. Without loss of generality, we

assume thatηi = ηmax, 1 ≤ i ≤ N .Definition 7: (The CMC-PC Problem P ): The Con-

strained Mobility Coordination problem for Preferential Cov-erage CMC-PC is defined by the tupleP (G,D,N,L), suchthat G is a given field to cover with a target distributionDusing a set ofN mobile nodes, each with its own scheduleL(ni). In order to solve a given instance of the CMC-PCproblem, we need to coordinate mobility of theN nodes inorder to achieve two goals: 1) satisfy schedulesL of all nodes,and 2) cover each field location,v ∈ V , the percentage of timeindicated by the target distributionD(v). Clearly, any feasiblesolution to the CMC-PC problem must satisfy the maximumspeed requirement,i.e., no node is allowed to move with aspeed higher thanηmax.

Theorem 1 shows that CMC-PC is NP-complete by re-duction to the Hamiltonian Cycle Problem. A HamiltonianCycle is a cycle in an undirected graph which visits eachvertex exactly once and also returns to the starting vertex.Determining (and finding) whether a Hamiltonian cycle existsin a given graph is NP-complete.

Theorem 1:The CMC-PC problem is NP-complete.Proof: We show that a simple instance of the CMC-

PC problem is equivalent to the Hamiltonian cycle problem.Consider the following parameterization of the CMC-PC prob-lem. The target field is represented by a connected graphG = (V,E), such that|V | = T . The monitoring distributionfunction is a uniform function. This means that each fieldlocation has the same importance as every other location, sothe goal is to spend the exact amount of time at each location.In graph terminology, this corresponds to visiting each vertexv ∈ V exactly the same number of times. Also, assume thenumber of nodes in the systemN = 1. The schedule of thisnode has only two entriess0 = {(1, v0), (T + 1, v0)}, i.e.,this node starts at some field locationv0 ∈ V at time 1 andends at the same location at timeT +1. Coupling the uniformmonitoring distribution with the given node’s schedule, weend up with the requirement that this node is required to visiteach field locationv ∈ V exactly once (recall that|V | = T ).Hence, we need to plan mobility of the given node that startsand ends at locationv0 and visits each field locationv ∈ Vexactly once. This defines an instance of the CMC-PC problemas P ′(G, uniform, 1, L(n1) = {(1, v0), (T + 1, v0)}). Solvingthe instanceP ′ amounts to finding a Hamiltonian Cycle inthe graphG starting and ending at vertexv0. Since findingHamiltonian cycles are NP-complete, then so is the CMC-PCproblem.

III. RELATED WORK

The problem we study here is related to main researchareas which received a lot of attention from the community.Specifically, these areas include sensor deployment and rede-polyment, field coverage, and motion planning.Field Coverage With Static Nodes: Studying coverageof static sensor networks is a fairly well-studied problem.Multiple research efforts [13], [7], [5], [15] concentrated oncalculating the coverage level attained by a static network. Forexample, Dhillonet al. [5] formulate the coverage problemas an optimization problem where they attempt to optimize

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placement of sensors in the field to maximize attained averagecoverage of the field. They also model preferential coverageof certain field spots. As we mentioned, all these researchefforts study the coverage properties of a static configurationof nodes in the field.Field Coverage With Mobile Nodes:Another group of re-search efforts concentrate on the effect of mobility on networkcoverage [16], [18], [6], [10], [14], [17]. Most efforts in thisgroup start from a sub-optimal deployment of nodes in the field(e.g., random), calculate an “optimal” deployment, and thenmove each node to its newly calculated location. These effortsdiffer, basically, in the way they calculate the new locationsof sensors. For example, in [6], Howardet al. introduce theidea of controlling nodes mobility using potential fields. Inthis framework, nodes are viewed as charged objects thatexert either attractive or repulsive forces on each other, basedon distance between them. Field borders also exert repulsiveforces on nodes. Mobility of any node is the result of thesum of all these forces. In [16], Wanget al. propose threedistributed algorithms to achieve uniform coverage of the field.Their algorithms calculate the current Voronoi diagram of thenetwork and move nodes to cover voids in the network.

A common factor of these efforts is that the steady stateof the network is a static configuration of the nodes that isconsidered to be “optimal” in some way (e.g.,cover the entirefield uniformly, or equalize power consumption at differentnodes). So, the network starts from a static configuration, thennodes move once to reach another optimal static configuration.Since, the new configuration is considered to be optimal,there is no need to move nodes again, until an new factoris introduced (e.g.,a node’s battery dies) which might requireredeployment of sensors.

Another group of research efforts concentrated on theattained dynamic coverage of a mobile network. For example,in [11], the authors study the efficacy of a mobile networkin field surveillance. They gauge the ability of the networkto detect a static and a mobile intruder. They conclude that,random node mobility is the best strategy to detect anymobile intruder. A common factor in these efforts is that thesteady state of the network is a dynamic one, unlike previousresearch efforts. Our work resembles these efforts in thisregard. However, our work addresses the general problem ofconstrainedmobility coordination of nodes in order to achievesomegivenmonitoring distribution.Robotics Motion Planning: Motion planning has been studiedin the robotics field [9], [8]. Coupling robotics and sensornetworks concepts has also been studied [4], [2], [12]. Theseefforts study problems of sensors carried by robots, and therequired modifications in robots mobility planning in ordertosupport tasks of sensor networks. Our work is also differentfrom these efforts in that, we assume that sensors are embed-ded into platforms that areautonomouslymobile by nature,and whose mobility has a limited degree of freedom (i.e.,slack) that could be planned to optimize performance of theembedded sensor network.

IV. PREFERENTIAL FIELD COVERAGE MOBILITYSTRATEGY

In Section II, we showed that CMC-PC is NP-complete. Inthis section we propose the Targeted Field Monitoring (TFM)mobility strategy, a distributed heuristic to solve the problem.To execute this algorithm, each mobile nodeni needs to knowits own scheduleL(ni), and the target coverage distributionfunction D. This algorithm does not assume existence ofa centralized decision-making facility nor knowledge aboutschedules of other nodes.

TFM uses another algorithm to assign a utility value toeach field location, based on the coverage distributionD.Then, at each time unit, TFM plans node mobility by selectingthe field location to be visited at the next time unit such thatitmaximizes the utility of visited field locations, satisfying thenode’s schedule at all times.

Specifically, let us denote the current location of nodeni as vc. Let us also denote the immediately accessible fieldlocations from the current location,i.e., the set of neighboringlocations, byN(vc). N(vc) is the set of field locationsvf ∈V such that there exists a direct edge betweenvc and vf .Formally,

N(vc) = {vf ∈ V |vf 6= vc AND e = (vc, vf ) ∈ E} (1)

Figure 1 (left) shows field locationvc, and its neighboringlocationsN(vc). In order to plan its mobility, nodeni needsto decide at each time unit, which of the neighboring fieldlocations,N(vc), it will move to next. To that end,ni executesthe Two-phase Utility Assignment (2UA) Algorithm to assigna utility value to each locationvf ∈ N(vc). Node ni, then,decides on the next step greedily to maximize the utility ofvisited locations. We will now describe the two phases of the2UA algorithm.Phase One of the 2UA Algorithm: During this phase,ni

assigns aninitial utility value Ui(v) to each field locationv ∈ V . Utility of each field location is a function of thepopularity of this location (defined byD(v)), the specific nodecarrying out the calculationsni, and the time of performingthis calculation.

More specifically,ni keeps a local “view” of the fieldrepresenting the last time each field location was last visitedby any of the mobile sensors. Let us denote the local viewof nodeni as Ci, whereCi(v) is the last time field locationv was last visited byany node, according to nodeni (Figure1 center). Nodeni updates its local view of the field at twooccasions: 1) Whenever it visits a new field location, it updatesthe last time this location was visited to the current time, and2) whenever it encounters another nodenj , the two nodesexchange their views of the field. The result of this exchangeis that, each node keeps the most recent version of the twoviews.

Using its current view of the fieldCi, nodeni calculatesthe utility of field locationv as

Ui(v) = D(v) × (tc − Ci(v)) (2)

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c

c

5.3 6.9 2.2 6.74.9

6.1 6.0 3.0 5.34.6

7.8 5.6 3.8 9.84.3

6.4 5.2 8.6 8.14.2

6.8 4.0 6.5 3.24.0

c

f

f c

Fig. 1. Locationvc, and its neighboring locations (left),Ci: local field view of nodeni (center), and range of PFPP (vf , h), whereh = 1.

wheretc is the actual time of performing the utility calculation(i.e., the current time). Notice that Equation 2 is a linearfunction of the popularity of the location,D(v), and the lengthof the interval since locationv was last visited(tc − Ci(v)).Equation 2 is just an example for utility calculations, whichcould take any different form (e.g.,exponential in the locationpopularity). Notice also that this equation is related to ourinterpretation ofD(v) as the required percentage of timeduring which field locationv should be covered.Phase Two of the 2UA Algorithm: In this phase, nodeni calculates acoarseutility value, Ui(vf ), for each of thedirectly neighboring locations,vf ∈ N(vc). The coarse utilityof vf is calculated as the sum of utilities of field locationscomprising the highest-utility path of lengthh that starts fromvf . More specifically, for eachvf ∈ N(vc), we find all pathsof length h that start fromvf , called Potential Future Paths(PFP’s). The utility of each PFP is the sum of initial utilities(calculated in Phase 1 of 2UA) of field locations comprisingthis path. The coarse utility ofvf is the highest utility of allsuch PFP’s starting atvf . Formally,

Ui(vf ) = Up(Pbest(vf , h)) =∑

vx∈Pbest(vf ,h)

Ui(vx) (3)

where functionUp(P ) calculates the sum of utilities of loca-tions in PFPP , andPbest(vf , h) is the PFP with the highestutility and is defined as,

Pbest(vf , h) ={P ′(vf , h) : Up(P ′(vf , h)) ≥ Up(P (vf , h)),

∀P (vf , h) ∈ P(vf , h)}(4)

whereP(vf , h) is the set of all PFP’s of lengthh that startfrom locationvf , andP (vf , h) ∈ P(vf , h) is defined as a se-ries ofh connected field locations, the first of which isvf . Fig-ure 1 (right) shows the current locationvc = (3,3), its directlyneighboring locationsN(vc) = {(2, 3), (4, 3), (3, 2), (3, 4)},and the range of PFP’s from the neighboring locations of

length h = 1. For example, forvf = (2, 3), P (vf , 1) ={{(2, 3), (1, 3)}, {(2, 3), (2, 2)}, {(2, 3), (2, 4)}}.

Scope of PFP’s:In the model we described, PFP’s constitutesome form of lookahead in order to optimize performance,and h is a tunable parameter that defines the exact amountof lookahead to perform. Hence, we need to answer thequestion: “What is the optimum range of PFP’s to consider?Consequently, what is the best value forh?”. It is natural tothink that the higher the value ofh, the longer the range ofconsidered PFP’s, the longer it takes to plan mobility, and themore optimal the mobility decisions are. There is, however,a dynamic restriction on the value ofh to be used at anyneighboring locationvf . Theorem 2 states this restriction, thenwe give an example to illustrate it.

Theorem 2:In a field coverage system with a single node,while determining the PFP’sP (vf , h) of a locationvf , onlyfield locations that could be actually reached by the node (dueto scheduling constraints) should be included in PFP’s. Hence,the optimum value of the locale radiush is the amount of slackk available to the node at the current time. Using any othervalues ofh could lead to sub-optimal decisions.

Proof:

The crux of the theorem is that: in order to optimize themobility planning process, all reachable field locations, andonly these locations, should be included in PFP’s. Hence weneed to show that, in determining locations to be included inPFP’s, the following two alternatives may lead to sub-optimaldecisions: 1) not including all reachable field locations, and2) including field locations that are not reachable.

1) First, we show that not including reachable field loca-tions could lead to sub-optimal decisions,i.e., using avalue of h < k. To show that, we define two sets ofPFP’s for each neighboring field locationvf ∈ N(vc),the first,P(vf , h), is of lengthh < k. This set leavesout PFP’s that consume more thanh units of slack. Thesecond set,P(vf , k), includes all PFP’s that consume upto k units of slack. Let us also denote all field locationsthat belong to PFP’s inP(vf , h) by A(vf , h). Formally

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A(vf , h) = {v : v ∈ P AND P ∈ P(vf , h)} (5)

Similarly, we denote all field locations that belong toPFP’s inP(vf , k) by A(vf , k).

A(vf , k) = {v : v ∈ P AND P ∈ P(vf , k)} (6)

As a result, the following relation is true by definition:

A(vf , h) ⊂ A(vf , k) (7)

We refer to the set of locations inA(vf , k) but not inA(vf , h) as set of exclusive locationsE . Formally

E = P(vf , k) − P(vf , h) (8)

Notice that, by definition, all exclusive locationsv ∈ Eare within reach, subject to scheduling constraints, ofthe mobile node (since we include only locations thatconsume up to the maximum available slackk, and notmore).Now, consider the case when all field locationsv ∈A(vf , k) are (almost) of close values of initial utilitiesUi(v), except one locationvx ∈ E whose initial utilityis substantially higher. In this case, optimal mobilityplanning would make sure to visit locationvx. In thiscase, all PFP’sP ∈ P(vf , h), would have similarutilities (sinceA(vf , h) ⊂ A(vf , k)). Hence, dependingon PFP’sP ∈ P(vf , h) could miss visiting locationvx

(since all of these PFP’s would have similar utilities).This is not the case withP(vf , k), since some PFP’s inthe latter set would visitvx, and would have substantiallyhigher utility, effectively guiding mobility to visitvx.Hence, using a value ofh < k could lead to sub-optimalperformance.

2) Second, we show that including field locations that couldnot be reached due to scheduling constraints (i.e., usinga value of h > k) leads to sub-optimal decisions.Similar to Case 1, we define two sets of PFP’s for eachneighboring field locationvf : the first,P(vf , h), is oflength h > k, and the secondP(vf , k). SetsA(vf , h),andA(vf , k) are also defined as in Equations 5 and 6,respectively. Notice that, unlike the previous case, sinceh > k, thenA(vf , k) ⊂ A(vf , h). We refer to the set oflocations inA(vf , h) but not inA(vf , k) as exclusivelocationsE , and is defined as

E = A(vf , h) −A(vf , k) (9)

Notice that, by definition, locationsv ∈ E are notwithin reach of the mobile node (since these locationsare beyond locations that consume up to the maximumavailable slackk). Following a similar argument to theone we used in Case 1, considering PFP’sP ∈ P(vf , h)would guide mobility to visit a field locationvx ∈ Eof high initial utility, however, unlike Case 1,vx isnot reachable. The end effect is that includingvx insome PFP’sP ∈ P(vf , h) would falsely boost utilityof these PFP’s, resulting in a falsely boosted coarseutility of some neighboring locationvf , clearly leadingto suboptimal mobility planning.

TABLE ISCHEDULE OFn1 .

Time Location1 (2,4)5 (4,2)

We state two remarks about Theorem 2:

• Theorem 2 addresses the case of a field coverage systemcomprised of a single node. In the case of multiple nodes,mobility planning decisions made by one node in the past,could be “invalidated” in the future due to mobility ofother nodes in the system. For example, a field locationvx of high initial utility according to nodeni in the past,could be visited by another nodenj causing its currentutility to drop when actually visited byni. We argue,however, that Theorem 2 still holds in systems of sparsedeployment, and we give evidence to this insight in ourtrace-driven evaluation in Section V-B.

• Theorem 2 does not address optimality of PFP’s of anylength other thank. In a general multi-peaked coveragedistribution D, it could be the case that in some situ-ations increasingh leads to better performance, and insome other situations, the same increase leads to worseperformance, depending on the actual location of thenode performing these calculations. In a “simple” single-peaked coverage distributionD, it is generally desirableto increaseh as much as possible provided that is lessthan k. We show an evidence of this observation in ourtrace-driven evaluation in Section V-B.

The following example illustrates the idea of Theorem2. It concentrates on Case 1 of Theorem 2 showing that notincluding all reachable field locations in deciding the PFP’s ofeachvf results in sub-optimal coverage of the field.

Table I gives the schedule of noden1, while Figure 2(left) showsvc the current location of noden1, along with fieldlocations directly accessible fromvc, N(vc) = {(2, 3), (3, 4)}.It also shows the initialutility of visiting each location, theresult of phase one of the 2UA algorithm. Notice that locations(1, 4) and (2, 5) are not members of the setN(vc), becausethe schedule ofn1 does not allow enough slack to visit any ofthese locations. Figure 2 (left) also highlights the set of fieldlocations that could be reached given the schedule ofn1.

Node n1 needs to assign a coarse utility valueUi(vf )to each of the neighboring locations. Figure 2 (center)shows the PFP with the highest utility for eachvf suchthat h = 3, which matches the time needed to get tothe destination.Pbest((3, 4), 3) = {(3, 4), (4, 4), (4, 3), (4, 2)},while Pbest((2, 3), 3) = {(2, 3), (3, 3), (4, 3), (4, 2)}. In thiscase, both PFP’s visit the high-utility location(4, 3).Up(Pbest((3, 4), 3)) = 4.3, while Up(Pbest((2, 3), 3)) = 4.1.Based on these calculations,ni moves to(3, 4) as its nextstep.

Figure 2 (right) shows the same paths whenh = 1. Noticethat in this case, not all reachable field locations are includedin the range of PFP’s.Pbest((3, 4), 1) = {(3, 4), (4, 4)}, whilePbest((2, 3), 1) = {(2, 3), (2, 2)}, and Up(Pbest((3, 4), 1)) =0.9, while Up(Pbest((2, 3), 1)) = 1.0. Based on these calcula-

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tions, ni moves to(2, 3) as its next step, which is clearly asub-optimal decision.

V. PERFORMANCEEVALUATION

In order to evaluate the efficacy of TFM in achieving aspecific coverage distribution of a given field, we developeda mobility simulator. Our simulator models the mobility ofnodes by keeping track of the location of each node at eachtime unit. It also models the exchange of local views betweennodes upon an encounter. Since our goal is to evaluate thesynthesized mobility of our detour-based techniques, we makesimplifying assumptions about the communication model aswe assume that nodes within a certain communication rangecould successfully exchange data. We assume that the size ofexchanged messages is small with respect to the bandwidthin a single contact between two nodes. We also, willingly,overlook storage limitations. We do this motivated by currentadvances in storage technology that make memory devices oftens of gigabytes available off-the-shelf.Performance Metrics: The performance metrics we use arethe Kullback-Leibler (KL) distance, and the query successratio (QSR). The KL distance is a measure of distancebetween distributions [3]. Having a true distributionP , andan approximated oneQ, the KL-distance betweenP and Q,KL(P‖Q), is calculated as:

KL(P‖Q) =∑

i

P (i)logP (i)

Q(i)(10)

Mobility of nodes over the field induces a distributionQof the length of periods during which each location is cov-ered. We measure the distance between the required coveragedistribution, D, and the induced distributionQ, KL(D‖Q).Lower values of the KL distance indicate that the induceddistribution is close to the required distributionD, which isthe prime requirement in field coverage.

Then, we assume that nodes have unlimited storage inwhich they keep collected samples from the field. A nodekeeps a sample from each field location it visits. A sample isassumed to have a time-to-live (TTL), during which this sam-ple is considered to be fresh,i.e., an accurate representationof the target phenomenon at the field location where it wascollected. Only fresh samples are kept in the local storage,while expired ones are evicted. Nodes are independentlyqueried about the state of the field1 . A query is definedby a tupleq = (vq, e). vq is the query target, the locationin which the inquirer is interested, whilee is a measure oftolerable imprecision in the answer. The specific locationsofquery targets follow some spatial distribution over the field.The answer to a queryq = (vq, e) is a sample collected atlocation v, such that the distance betweenv and vq is lessthan e (i.e., |v − vq| ≤ e). We can think of a query as acircle whose center is the query targetvq, and whose radius isthe imprecisione. In this case, the query answer is a samplecollected from within this circle.

1 We can think of this as if the user/owner hosting the mobile node isinterested in the state of some location in the field, so he usesthe localdevice, to which the sensor is attached, to submit queries andreceive answersback from the distributed system.

In order to answer any query, a queried node searchesits local storage to find a sample that could be used as ananswer to the query. If found, then the query is counted asa success. Otherwise, the queried node forwards the queryto its direct neighbors only. If one of these neighbors hasan answer to the query, this neighbor sends the answer backto the queried node, and the query is counted as a success.If neither the queried node nor any of its neighbors has ananswer to the query, it is counted as a missed query. In orderto assess the efficacy of each mobility model in achieving therequired probability distribution, we matched the distributionof query targets to that of the required coverage distributionD (i.e., field locations that are required to be monitored forlonger periods of time have higher probability of being querytargets). We define the query success ratio (QSR) as the ratiobetween the number of successfully answered queries to thetotal number of queries.

The point of the two performance metrics is to gauge thedegree to which each mobility strategy can match the requiredcoverage distributionD. If a mobility strategy could closelymatch this distribution, this should be manifested in achievinga small KL distance, and high query success ratio.Competing Strategies:We compare TFM to the random mo-bility strategy (RND), in which nodes move randomly betweenevery two consecutive waypoints provided that the scheduleissatisfied. In the trace-driven evaluation, we compare TFM toWait-at-Destination (WAD), a variation of RND. Under WAD,nodes move to the destination waypoint using the shortestpath, where they wait spending all the available slack, ifany. Clearly, both RND and WAD represent lower bounds onperformance, since they do not actively attempt to coordinatenodes’ mobility to improve coverage of the field.

In the synthetic evaluations, we evaluated the performanceof TFM and RND with respect to query handling in two differ-ent setups, distributed and centralized. In the distributed setup,only nodes within communication ranger could communicateso as to cooperate in query handling. This version is denotedas “DST” in the following graphs. In the centralized setup(denoted as “CTR” in the following graphs), query handlingis done in a centralized facility. In this case, we simulate thecase where all collected samples are forwarded to a centralprocessing facility, and all queries are directed to this facility.

A. Evaluation Using Synthetic Workloads

Schedule Generation:Every node starts at time =tcurrent

(initially, tcurrent = 1) at a random location in the fieldloc1.The entry (tcurrent, loc1) is added to the schedule. Then werandomly select another locationloc2 in the field such that theminimum time to move fromloc1 to loc2 is t. For locationloc2, we assign timets

ts = tcurrent + t + (κ × ρ) (11)

whereκ is the maximum slackwe allow in any journey, andρ is a uniform random variable such thatρ ∈ [0, 1]. The entry(ts, loc2) is appended to the schedule. We repeat this processuntil the end of the simulation time is reached.Baseline Parameters:We simulated a field of 10x10 wherenodes can communicate only when they are at the samefield location. Each simulation runs for 100 seconds. In the

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(right).

following graphs, each point is the average of 20 simulationruns, with the 95% confidence intervals shown as well. Therequired field coverage distribution is assumed to be a sym-metric bivariate normal distribution centered in the center ofthe field, with variance =4× I, whereI is the identity matrixof size 2× 2. In the following experiments, unless otherwisestated, the default value of the maximum PFP’shmax = 1,number of nodesN = 30, and query precisione = 1.Effect of Number of Nodes:Figure 3 shows the KL distanceof RND and TFM as a function of the number of nodes inthe system, using two different values of the maximum slackκ = 0 (Figure 3 left), andκ = 20 (Figure 3 right). Whenthere is no slack in the schedule (κ = 0), TFM achievesbetween 37% and 42% lower KL distance than RND; thedifference between TFM and RND is noticeable, albeit nothuge. Increasing the available slack allows TFM more freedomto match the required coverage distributionD, resulting ina considerably improved (smaller) KL distance. A maximumslack of κ = 20 allows TFM to achieve KL distance that isup to 10-times lower than that of RND.

Figure 4 shows the effect of increasing the number ofnodes on the performance of the system in handling queries.TFM achieves between 2-fold to 3-fold improvement in QSRover RND. Increasing the TTL of collected samples or themaximum slackκ improves performance of both protocols,especially centralized versions, however, TFM is always supe-rior to RND. It is also interesting to notice that the distributedversion of TFM achieves close performance to that of thecentralized one. Increasing either TTL or maximum slackκdiminishes the gap between the two versions. This confirmsthe premise of TFM in a distributed practical setup. While,there is a noticeable gap between QSR of the distributed andcentralized versions of RND, as expected of a naıve approach.

We conclude from Figure 4 that the distributed versionof TFM has very compatible performance to that of thecentralized version. The reason is that: to achieve the requestedcoverage distribution, different nodes could visit the same fieldlocation(s) over a period of time, which requires coordinationbetween nodes, over some period of time. At finer time

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Fig. 3. KL distance of TFM and RND as a function of the number of nodesin the system for different slack values.

intervals, only a small group of nodes could visit field locationsin a specific area of the field (i.e.,only nodes that are currentlyphysically close to this area). Hence, in order to coordinatemobility of this group of nodes, only these nodes need tocommunicate. In a distributed setting (where only nodes atthe same intersection could communicate), exactly such groupof nodes could communicate and successfully coordinate theirmobility, due to their proximity. Communication (and coordi-nation) between this group of nodes and different groups thatare physically distant (as is the case in a centralized setting),has a limited effect on performance, since there is no potentialoverlap,on the short run, between field locations that could bevisited by the different groups, due to their physical distance.Effect of Maximum Slack κ: Figure 5 shows the KL distanceof RND and TFM as a function of the maximum slackκ ofthe schedule, in a system of 5 nodes (Figure 5 left), and 15nodes (Figure 5 right). As we pointed out earlier, increasingκ allows TFM to match the required distribution resulting ina smaller value of the KL distance. This is true for systems ofboth high and low densities. Increasing the number of nodeshas a more pronounced positive effect on the KL distance ofRND compared to TFM.

Figure 6 shows the effect ofκ on the QSR. Similar tothe KL distance, increasingκ improves QSR of TFM. Thisis not always the case for RND, since, nodes under RNDdo not coordinate usage of their slack to optimize coverageof the field. TFM achieves more than 2-fold improvement inQSR over RND. Also, increasing TTL improves QSR for both

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Fig. 6. Query success ratio of TFM and RND as a function of the Maximumslack for different TTL values

mobility models. For values ofκ > 0, the distributed versionof TFM achieves QSR close to that of the centralized version,again, underscoring the usefulness of TFM in distributedsettings.Effect of Partially Following Detours: The goal of thisexperiment is to measure the effectiveness of TFM when onlya given percentage of the nodes follow detour hints providedby TFM. This scenario is motivated by the observation thatsome nodes may not be willing to participate in the fieldcoverage application, and opt to spend their slack in a differentway. Table II shows the resulting KL distance. Obviously,as the percentage of nodes following TFM hints increasesthe resulting KL distance decreases. When 60% of the nodesfollow TFM hints, the resulting KL distance decreases by upto 67% (κ = 100).

Figure 7 shows the QSR as a function of thedensityofnodes that follow TFM hints in a distributed setting. Densityis defined as the number of nodesN over monitored areaA(i.e., density =N/A). We experimented with three values ofκ: 0 (left), 20 (center), and 100 (right). In this experimentthe total number of nodes is fixed at 30. Similar to the KLdistance, increasing the percentage of nodes that follow TFMhints causes a linear increase in the QSR of the system. Figure

TABLE IIKL DISTANCE OF RESULTING DISTRIBUTION WHEN ONLY A PERCENTAGE

OF NODES FOLLOW THE DETOURS OFTFM.

percentage 0 20 40 60 80 100κ = 0 0.8781 0.8537 0.8221 0.7934 0.7774 0.7472κ = 20 0.7385 0.5319 0.4018 0.2984 0.2124 0.1407κ = 40 0.7706 0.5133 0.3588 0.2726 0.1731 0.1175κ = 60 0.7804 0.5209 0.3560 0.2647 0.1834 0.1368κ = 80 0.7326 0.4679 0.3437 0.2523 0.1944 0.1434κ = 100 0.7415 0.4780 0.3424 0.2424 0.1858 0.1586

7 reveals two interesting facts: 1) the “height” (i.e., successlevel) of each curve is a function of the TTL of samples.Higher values of TTL lead to higher QSR (we have seensimilar effects in previous experiments), and 2) the slope ofthe linear increase is a function of the slack. Intuitively,theimpact resulting from an additional percentage of nodes thatfollow TFM hints is a function of how much slack these nodeshave. The more slack the higher the impact (i.e., the higher theQSR). This effect is manifested in the increased slope of theQSR curves. We conclude that, using more nodes to cover thefield enables the distributed service to attain better coverageof the field. However, it is clear that, with enough scheduleslack, a density of TFM-compliant nodes as low as 0.3 couldachieve QSR approaching 100%.

B. Trace-Driven Evaluation

Following our motivating application, and to present evenmore realistic evaluation of the protocol we propose, we usedtaxi traces [1] for cabs in the San Francisco area as input toour models. The goal is to show that, with little coordinationbetween cabs, they could function as an effective distributedfield coverage system.Methodology: For each cab, the traces show location updatesof the cab. Each update is composed of latitude and longitudeof the cab location, the time of the location update, alongwith the cab status: metered (hailed) or not (not hailed). Wegathered more than a full day’s worth of data for more than450 cabs. In the traces we collected, some cabs have as manyas 400 location updates, while others have as few as 5 updates.We used all location updates for all cabs to construct a “map”of the San Francisco area. We represented the map as anundirected graphG = (V,E). V is the set of all legitimatelocations any cab can be in at any time, where a locationis defined by its latitude and longitude coordinates. In thedata we collected, the total number of locations is 40399,

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Fig. 7. Effect of partially following hints on query successratio for three different values of slack= 0, 20, and 100.

Fig. 8. Paths followed by cabs in the SF Bay area, from traces used toevaluate mobility coordination approaches

and the number of unique locations,|V | = 39, 103 locations.To determine the relation between different locations (i.e., theedges,E), we used a threshold-based neighborhood algorithmwith a threshold valuerth. This means that, for any twolocations a and b, such that the distance between them isDist(a, b), if Dist(a, b) ≤ rth, then we add an edge betweena and b whose cost =Dist(a, b). We usedrth = 200 meters(≈ 0.12 miles = 656 feet). This value ofrth partitioned theunique field locations into different partitions, with the largestpartition consisting of 36,368 unique locations. We used thispartition as a representative of the map. A depiction of thismap is given in Figure 8.

Finally, out of the 450 cabs, we selected the 50 cabswith the highest number of location updates. We mapped thelocation updates of the cabs to the map we generated, andused the map to “fill” in the gaps due to missing locationupdates, for the first 150 minutes. This is done by mappingeach two consecutive updates to the map, and finding theshortest route between them. Next, we interpolate a numberof locations along this route that is equal to the number ofminutes between the location updates. This process allowsus to infer the locations of cabs at one-minute granularities.The cab status for those interpolated locations is set to be itsreported status in the last location update.

Based on each cab’s mobility profile (obtained as de-scribed above), we defined the schedule of the cab as follows:every time the cab is metered, its location is added to theschedule of the cab. This means that, if the cab is hailed(according to location updates), then it has to be in the

indicated location at the indicated time. In other words, wecan not change the trajectory of a hailed cab. This leaves roomfor offering mobility hints (detours) to the cab only when itis not hailed.

We compared TFM and Wait At Destination (WAD), avariant of the RND protocol. Under WAD, when not hailed,a cab moves to the next location where it picks up its nextcustomer, as early as possible, and spends its slack time therewaiting for the customer. Throughout the trace evaluation,weassumed that cabs do not exceed speed of 30 mph, which isquite conservative.Target Distribution D: We assume that the goal of thesecabs is to cover the city to track a specific phenomenon thatbreaks out at random locations. For example, an Amber alertis issued specifying the break out location (the center of thephenomenon) which is given the highest level of attention.Attention awarded to neighboring locations is a function oftheir distance from the center of the phenomenon. To modelthis application, we define target coverage distributionD asfollows: We start with a maximum utility valueM and a utilitydecrement value per hopδ. Then we randomly select a fieldlocation, v1, to be the center of the distribution. We assignvirtual utilities uv as follows:

uv(vi) = M − δ × |v1 − vi| (12)

whereuv(vi) is the virtual utility assigned to field locationvi,and |v1 − vi| is the number of hops betweenv1 (the centerof the distribution) andvi. This function assigns tov1 themaximum value of utility,M . The utility value of field locationvi drops as a linear function of the number of hops betweenvi and the center of the distribution,v1. To get the requireddistribution D, which specifies the percentage of timeD(vi)that field locationvi should be covered, we use the followingequation:

D(vi) = uv(vi)/∑

vx∈V

uv(vx) (13)

Figure 9 illustrates an example distribution over a compactversion of the map.Results: In our comments on Theorem 2, we argued that incase of systems with single nodes, the longer the PFP’s usedto estimate the coarse utility of directly neighboring locations,the better the performance, under two conditions: 1) we limit

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Fig. 9. Example of the distribution we used with the San Francisco cabtraces. The lighter the area the higher the coverage percentages (D(v)). v1in this case is marked by the red (dark) dot in the light area

TABLE IIIKL DISTANCE RESULTING FROM APPLYINGTFM ON CAB TRACES WITH

DIFFERENT VALUES OF MAXIMUM LENGTH OF PFP’S hmax .

hmax 0 1 2 3KL distance 1.0986 0.8975 0.7441 0.6528

our consideration to field locations that are reachable under thescheduling constraints of the node, and 2) the desired coveragedistribution D is simple single-peaked (as the one we usedin the trace-driven evaluation). In this experiment we aim toevaluate this Theorem in systems with multiple nodes, but lownode density. Towards this end, we calculated the KL distanceof TFM using different values of the maximum length ofPFP’shmax. For each value, we run 20 simulations using theinferred schedule and the generated map. Each simulation hasa different distribution centerv1. Table III shows the averageKL distance for different values ofh.

It is clear that the performance improves by increasingh, provided that we only consider reachable field locations,confirming our expectations. Sinceh = 3 yields the lowestvalue of KL distance, we use this value in the query-basedperformance evaluation. The KL value of WAD = 2.1, threetimes that of TFM withh = 3.

Figure 10 shows the query performance of the two mo-bility models. TFM achieves from 30% to 120% improvementin QSR over WAD. Increasing the communication range im-proves the performance of both protocols. However, we foundout in another experiment in which we measured performanceas a function of the communication range (results not shownhere) that this improvement reaches a plateau very fast.

VI. CONCLUSION

We considered a new mobility model whereby each nodeis coarsely constrained by an external schedule. These nodeschedules have slack which can be leveraged to coordinatemobility between nodes so as to satisfy a field monitoringobjective. We coin the problem of Constrained Mobility Coor-dination for Preferential field Coverage (CMC-PC). We showthat this problem is NP-complete and propose a distributedheuristic (TFM) that provides nodes with mobility hints (de-

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Fig. 10. QSR of TFM and WAD as a function of TTL of samples (TTLmeasured in minutes). The graph show results when using two values of thecommunication range: 600 and 1200 meters.

tours) so as to achieve field monitoring that is as close aspossible to the required monitoring distribution. We verify thepremise of our mobility planning technique using extensivesimulations, as well as taxi logs from the San Francisco area.Our results underscore the evident performance improvementattained by TFM.

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