Autonomous Data Collection from Underwater Sensor Networksusing Acoustic Communication

Geoffrey A. Hollinger, Urbashi Mitra, and Gaurav S. Sukhatme

Abstract— We examine the problem of planning paths foran autonomous underwater vehicle (AUV) to collect data froman underwater sensor network. The sensors in the networkare equipped with acoustic modems that provide noisy, range-limited communication. The AUV must plan a path thatmaximizes the information collected while minimizing traveltime or fuel expenditure. This problem is closely related to theclassical Traveling Salesperson Problem (TSP), but differs inthat data from a particular sensor has a probability of being col-lected depending on the quality of communication. We proposemethods for solving this problem by extending approximationalgorithms for variants of TSP, and we compare our proposedalgorithms to baseline strategies through simulated experimentswith varying levels of communication quality. Our simulationsutilize a realistic model of acoustic communication to determinethe probability of acquiring data from each sensor. The resultsdemonstrate that planning the tour for the entire network whileexploiting the communication model during planning improvesperformance versus myopic methods.

I. INTRODUCTION

The use of sensor fields to monitor phenomena in under-water environments is of growing interest. Examples includealgal blooms [1], seismic activity, and intrusion of enemysubmarines [2]. In underwater monitoring scenarios, manystandard methods of communication are no longer feasible(e.g., WiFi, cellular, satellite). Acoustic modems can providecommunication underwater, but they suffer from severe rangelimitations and channel variations [3].

Without reliable communication, collecting data from un-derwater sensor networks becomes a challenging problem. Apotential solution is the use of a mobile autonomous under-water vehicle (AUV) equipped with an acoustic modem togather data from the sensors [4]. The problem now becomesone of planning the AUV’s path to minimize its travel timeand maximize information gathered. We will refer to thisas the Communication-Constrained Data Collection Problem(CC-DCP). The CC-DCP is closely related to the classicalTraveling Salesperson Problem (TSP) [5]. The key differenceis that information is gathered from sensors through a noisychannel, whose reliability decreases with distance and can bemodeled probabilistically. Thus, we are dealing with a newproblem of TSP with probabilistic neighborhoods.

G. Hollinger and G. Sukhatme are with the Dept. of Computer Science,Viterbi School of Engineering, University of Southern California, LosAngeles, CA 90089 USA, {gahollin,gaurav}@usc.edu

U. Mitra is with the Dept. of Electrical Engineering, Viterbi School ofEngineering, University of Southern California, Los Angeles, CA 90089USA, ubli@usc.edu

This research has been funded in part by the following grants: ONRN00014-09-1-0700, ONR N00014-07-1-00738, NSF 0831728, NSF CCR-0120778 and NSF CNS-1035866.

Fig. 1. Example sensor deployment on the ocean floor to monitorenvironmental conditions. Such sensors remain in place for many months.Retrieving data from the sensors during deployment is challenging due tocommunication limitations underwater. A mobile AUV equipped with anacoustic modem can act as a data collection device in this scenario bytraversing a path that minimizes travel time and maximizes informationgathered.

In this paper, we design algorithms for an AUV gath-ering data from an underwater sensor network. The nov-elties of this paper include: (1) the formulation of theCommunication-Constrained Data Collection Problem (CC-DCP) as a Traveling Salesperson Problem (TSP) with prob-abilistic neighborhoods, (2) the extension of approximationalgorithms for the deterministic Prize-Collecting TSP andTSP with Neighborhoods for use in the CC-DCP, and (3)the validation of proposed solutions to the CC-DCP throughsimulated experiments at varying levels of communication.

The remainder of this paper is organized as follows. Wefirst examine related work (Section II) and then formulatethe CC-DCP (Section III). We next propose a number ofalgorithms for solving the CC-DCP approximately (Sec-tion IV), and we provide theoretical analysis demonstratingperformance guarantees in certain instances (Section V). Wethen validate our approach through simulated deployments(Section VI) before concluding and discussing avenues forfuture work (Section VII).

II. RELATED WORKThe underwater data collection problem is closely re-

lated to the Traveling Salesperson Problem (TSP). In TSP,a mobile agent must visit a number of locations in theminimal amount of time. This problem is known to be NP-complete and has a long history of both approximate andoptimal solutions. With current methods, it is possible tosolve regular TSP instances optimally with more than 10,000locations [5]. The two key differences between our work andthe classical TSP are: (1) the locations are associated withsensor measurements that may provide different amounts of

information regarding the phenomena of interest, and (2) in-formation is gathered from locations based on a probabilisticcommunication model.

The case where locations are associated with different“prizes” has been studied by a number of researchers. ThePrize-Collecting TSP was originally introduced by Balas [6]and has been extended to a number of related variants [7],depending on the type of path required, restrictions on theprizes, and whether new locations appear during the tour [8].Goemans and Williamson proposed an approximation al-gorithm for Prize-Collecting TSP using a LP primal/dualschema that achieves a factor of two guarantee [9]. Slightlybetter approximation guarantees are also possible at the costof computation and implementation complexity [10].

Researchers have also studied the TSP with deterministicneighborhoods. In this formulation, the locations are vis-ited when the agent moves within a fixed radius aroundthe location. Constant factor approximation algorithms areavailable for TSP with neighborhoods for both disjoint andoverlapping regions of the same size [11] as well as regionsof varying sizes [12]. To our knowledge, the combinationof TSP with Neighborhoods and Prize-Collecting TSP hasnot been studied, and the use of probabilistic communicationmodels to describe the neighborhoods has also not previouslybeen examined.

Related problems have been studied in the context ofrobotic data mules. Bhadauria and Isler derived approxi-mation algorithms for multiple data mules that must tra-verse a sensor field and download data [13]. In their work,downloading time is considered as part of the tour, andthe communication radii are assumed to be uniform, fixed,and deterministic (i.e., data from a sensor is known to beaccessible at a given location). In the present paper, we utilizea probabilistic acoustic communication model that degradeswith distance [14]. Such models, developed by researchersin acoustic communication, have been shown to provideaccurate predictions for experimental data [3].

Vasilescu et al. developed a system of mobile and sta-tionary nodes for underwater data collection based on theuse of both optical and acoustic communication [4]. Theydescribed the networking architecture and sensor specifi-cations necessary for underwater data collection, and theypresented experiments in the field on a mobile network.These experiments showed the feasibility of utilizing AUVsfor underwater data collection, but the authors left openthe problem of path planning for the mobile nodes in largenetworks.

III. PROBLEM SETUP

We are given a pre-deployed network of N sensors locatedin <d. For this paper, we limit analysis to d ∈ {2, 3}, whichyields the 2D and 3D problems respectively. We assume thatthe location xn ∈ <d is given for each sensor n ∈ [N ],where [N ] := {1, . . . , N}. Each sensor n contains data forretrieval, which we denote as Yn. We define the informationquality of the data as I(Yn). I(x) corresponds to the expectedvalue of information received at location x (e.g., information

gain in an inference problem [15], or variance reduction in aregression problem [16]). Generally, I(x) incorporates bothinformation received from the stationary sensors and alsoinformation sensed by a vehicle at location x. For this paper,we will not consider sensing actions taken by the vehicle,although these can be incorporated.

In the general case, coupling between the sensor measure-ments can lead to information being subadditive or super-additive. In the context of data collection, we will assumethat information is additive (i.e., I(Yi, Yj) = I(Yi) + I(Yj)for all i 6= j) and that multiple observations of any Yn donot provide additional information. Relaxing either of theseassumptions leads to interesting extensions (see Section VII).

The sensors are assumed to be low-cost with limitedcapabilities. Each sensor is capable of transmitting packetsof data over a noisy channel. A single mobile vehicle hasthe capability to communicate with the sensors. The locationxv ∈ <d of the vehicle is controlled and may be subject toconstraints, such as obstacles or vehicle kinematics. Basedon these constraints, a traversal cost c(x1, x2) is defined forall pairs of points x1, x2 ∈ <d. We assume the traversal costis metric and that the location of the AUV is known. Thecommunication quality of the vehicle degrades with distance,C(xv, xn) = f(D(xv, xn)), where D(xv, xn) = |xn − xv|and f is some relationship function. We will assume that fdecreases monotonically with distance.

The optimization problem is to plan a path for the vehiclethat retrieves data from the sensors and minimizes the traver-sal cost of the path. Let P (xv, xn) ∼ C(xv, xn) representthe probability that the observation made by the sensor at xnis received by a vehicle at xv . This represents the probabilitythat the AUV has successfully pinged the sensor and receivedthe entire data packet as a reply. We will assume that theAUV is always sending ping messages to each sensor, andthat the sensors reply with a data packet whenever a ping isreceived. More general communication architectures are alsopossible, but they are not examined in this paper. Given thismodel, the expected received information quality at point xvcan be written as

R(xv) =∑N∈2K

P (xv,N )I(YN ), (1)

where 2K is the powerset of K sensors with uncollecteddata, I(YN ) is the joint information from a set of N ∈ 2K

sensors, and P (xv,N ) is the probability of observing N .Using the independence assumption on sensors, this reducesto

R(xv) =∑n∈[N ]

P (xv, xn)I(Yn). (2)

With the independence assumption, we can also calcu-late the expected received information along a path P =[xv(1), xv(2), . . . , xv(T )] as

R(P) =∑n∈N

1−∏t∈[T ]

[1− P (xv(t), xn)]

I(Yn). (3)

The Communication-Constrained Data Collection Problem(CC-DCP) can now be written formally.

Problem 1: Given path costs c, information quality I ,communication quality C, a set of possible AUV paths Ψ,and a starting AUV location xs, find

P∗ = argminP∈Ψ

∑t∈[T ]

c(P(t− 1),P(t)) s.t. R(P) ≥ τ, (4)

where T is the index of the last location on the path, and τis a threshold on information quality.

In some cases, we may want to set the hard constraintto gather information from all sensors, which makes thetraversal distances stochastic. For instance, it may take ashort amount of time to gather data if the transmissions arereceived early or a longer time if they are received later. Forthis case, the optimal solution is a mapping 2N ×<d → <dfrom the 2N possible received and not received states of theN sensors and the current location of the AUV to the nextbest location of the AUV. This solution can be encoded asa policy for an underlying Markov Decision Process (seeSection IV).

IV. ALGORITHMS

We now present algorithms for solving the CC-DCP bothoptimally, at the cost of high computation, and heuristically,based on existing approximation algorithms for TSP variants.In the following section, we will demonstrate that our algo-rithms exhibit performance guarantees for some instances ofthe CC-DCP.

A. Optimal MDP Solution

The optimal solution to the CC-DCP can be encoded asa policy mapping from states to actions. To see this, notethat the problem can be formulated as a Markov DecisionProcess (MDP) [17]. In the 3D problem, the states in theMDP are defined as S = X ×Y×Z×2N , where X , Y , andZ are the coordinate spaces for the location of the AUV. The2N states represent whether or not data has been collectedfrom each of the N sensors. The reward function R(s, s′) =∑n∈F I(n)− c(s, s′), where F is the set of sensors that go

from uncollected to collected between s and s′, and c(s, s′) isthe movement cost. The state transitions T (s, s′) are definedby the communication model C(s, s′) and the motion modelof the AUV.

If the environment is discretized, the MDP above can besolved using any standard method (e.g., value iteration orpolicy iteration) [17]. However, it is important to note thatthe number of states is exponential in the number of sensorsand polynomial in the size of the environment (to the powerof 2 or 3 depending on the dimension). Thus, we can expectoptimal solutions to become infeasible for any instance ofmore than a few sensors. In addition, there may be additional

error that arises from discretizing the possible locations ofthe AUV. This formulation requires the location of the AUVto be known. Some position uncertainty can be incorporatedin a conservative manner by employing the communicationprobability associated with the worst-case location of theAUV. Otherwise, a Partially Observable Markov DecisionProcess (POMDP) is required [18], which leads to a furtherincrease in computational complexity.

B. Approximate Solutions

The CC-DCP is a generalization of the TSP, which makesit NP-hard. Due to the computational intractability of theoptimal solution on large instances, we develop heuristicsfor solving the CC-DCP approximately. The key differencebetween the CC-DCP and the TSP with neighborhoods [11]is that communication is modeled probabilistically. Ourapproach is to generate contours of equal probability aroundthe sensors and utilize these as if they were deterministicneighborhoods.

We define a probabilistic neighborhood Mn for sensor ndetermined by P (xm, xn) > ρ for all xm ∈Mn. The valueof ρ ∈ (0, 1) determines how conservative the probabilisticneighborhood is. As ρ → 1, it will be near certain thatinformation will be received from sensor n if the AUV iswithin the neighborhood. As ρ → 0, the AUV may need toquery a sensor n many times before receiving data from it.

Once the probabilistic neighborhoods are defined, we cangenerate a maximal independent set of neighborhoods bygreedily choosing sensors and removing adjacent sensorswith overlapping neighborhoods. A valid tour can then befound by circling the neighborhoods in the maximal inde-pendent set (see Figure 3) [11]. In the case of probabilisticneighborhoods, it may be necessary to wait for informationto be received from the entire set of sensors before movingto the next set.

For the case where all nodes must be visited and haveequal informativeness (i.e., I(Yi) = I(Yj) for all i, j), wedesign the following heuristics for planning the path of theAUV:• Myopic: Move towards closest sensor. Once data is

received, move to next sensor.• TSP Solution: Find an optimal TSP ordering of the

sensors using the Concorde solver [5]. Visit sensors inthat order. Shortcut sensor once data is received.

• TSP Solution with Neighborhoods: Find a maximalindependent set (MIS) of probabilistic neighborhoods.Find the optimal TSP ordering of the MIS using Con-corde. Visit the sets in that order. Shortcut once data isreceived.

The myopic strategy is simply to move towards the closestsensor. This is a reactive strategy and will perform well whencommunication quality is high. In such cases, the benefit oflong-term planning is negated by the homogeneity of theexpected received information across the environment. Thesecond solution is to solve the TSP of the sensors and ignorethe communication model. This technique will perform wellwhen communication is poor, since this situation requires the

AUV to move near all sensors to gain information from them.Finally, the TSP Solution with Neighborhoods incorporatesthe communication model as a probabilistic neighborhood.

For the case where the sensors provide different quantitiesof information, I(Yi) 6= I(Yj), we propose the use of a Prize-collecting TSP algorithm (PC-TSP) to improve performance.The PC-TSP assigns a penalty ζ(i) to each location based onits informativeness. The tour now has the option of neglectingsome locations and paying the required penalty. The totalcost of the tour is then the movement cost plus ζ(i) for alli not visited. In our case, ζ(i) = λI(Yi), where λ is a scalefactor. A binary search on λ can be performed if a thresholdon information quality is desired, as in Equation (4). Weemploy the following strategies for the prize-collecting case:• Myopic: Ignore the penalties and act as above. Termi-

nate if the sum of remaining benefit is less than mini-mum distance to a sensor’s probabilistic neighborhood.1

• PC-TSP Solution: Use the primal/dual algorithmfrom [9] to determine sensors to visit. Find optimalordering of this subset using the Concorde solver [5].Visit sensors in that order. Shortcut once data received.

• PC-TSP Solution with Neighborhoods: Find a maximalindependent set (MIS) of the sensors using probabilisticneighborhoods. Use the primal/dual algorithm on theMIS to determine a subset of the MIS to visit. Findthe optimal ordering of the subset of the MIS usingConcorde [5]. Visit in that order. Shortcut once data isreceived.

The non-myopic algorithms for the case of differing sen-sor information utilize an existing PC-TSP approximationalgorithm to determine which sensors (or neighborhoods) tovisit in the tour. The selected locations are then treated in asimilar fashion to the problem with equal sensor information.We will now show that this approach provides performanceguarantees relative to optimal in some cases.

V. THEORETICAL ANALYSIS

Difficult problems in NP, such as TSP variants, cannotbe solved in time polynomial in the size of the instance,under the assumption that P 6= NP. A common approachis to develop efficient algorithms with bounded solutionquality relative to optimal [19]. Such algorithms providean approximation guarantee on solution quality. A constantfactor approximation guarantee states that for any probleminstance, the quality of the approximate solution |PA| isbounded as |PA| ≤ η|P∗|, where η is a constant value(known as the approximation factor), and P∗ is the optimalsolution. Clearly a lower η is better, and η will never beless than one (the approximate solution cannot be better thanoptimal).

In the case of the TSP with Neighborhoods (TSPN),the algorithm presented above of determining a maximalindependent set of neighborhoods and then calculating a

1An alternative myopic strategy is to move to the node with highestbenefit/cost ratio. In practice, this did not perform as well as the simplenearest neighbor.

near-optimal TSP tour of this set is known to provideconstant factor approximation guarantee. This guarantee isη = (π + 8) (1 + ε), where 1 + ε is the approximationguarantee for the algorithm used to solve the TSP on themaximal independent set [11].

We now show that this approximation algorithm for theTSPN extends to the Prize-Collecting TSP with Neighbor-hoods (PC-TSPN) without a loss in approximation qualitygiven a constraint on the prizes. We will first examinethe case where the neighborhoods are deterministic (i.e.,communication quality is measured as a step function). Inaddition, we limit the theoretical analysis to the problem in<2 and the case where the neighborhoods all have diameterd (see reference [11]).

Theorem 1: Let PA be the approximate solution to a PC-TSPN instance found by generating a maximal indepen-dent set I of N neighborhoods and then finding a PC-TSP tour that visits a subset of I . If ζ(i) ≥ 2d for alli ∈ N , the approximate solution is bounded by |PA| ≤(π + 8) (1 + ε) |P∗|, where 1 + ε is the approximation guar-antee for the PC-TSP tour.

Proof: The proof follows that of standard TSPNclosely [11]. The key difference is ensuring that choosingnot to visit neighborhoods, by paying the required penalty,does not lead to loss in approximation quality. For the case ofdisjoint disk neighborhoods (i.e., no neighborhoods overlap),let PC be the path generated by running a PC-TSP tour witha (1 + ε) approximation guarantee on the center points ofthe disks. Note that this tour does not necessarily visit allneighborhoods. Let PR be the optimal PC-TSPN tour. Forany path, let |P| be the cost of that path (i.e., traversal costsplus penalties for neighborhoods not visited). We claim that

|PC | ≤((

1 +8

π

)|PR|+ 8

)(1 + ε) . (5)

Let l = |PR|, let M ⊂ N be the neighborhoods visitedby PR, and let m = |M |. W.l.o.g. scale the distances so thatthe size of all disk neighborhoods are radius 1. Since PRvisits a subset of the disks, the area Al swept by a disc ofradius 2, whose center moves along PR is bounded as

πm ≤ Al ≤ 4l + 4π, (6)

and thus m ≤ 4 + 4l/π. A tour PM that visits the center ofthe M disks can be found by going along PR and makinga detour of length at most 2 to visit the center of each disk.This yields

|PM | ≤ l + 2m ≤ l + 2

(4 +

4l

π

)=

(1 +

8

π

)l + 8. (7)

Finally, we note that |PC | ≤ (1 + ε) |PM |, since |PM | is avalid center tour. Note that PC and PM may visit differentsubsets of the neighborhoods. Thus, PC is bounded as inEquation (5). As |PR| grows large, the approximation factorbecomes

(1 + 8

π

)(1 + ε).

Now, for the case of overlapping discs, we find a maximalindependent set I ⊂ N of the disks. Let PJ be the PC-TSPtour that visits J ⊆ I and is computed by running a (1 + ε)

approximation algorithm on I , such that the prize of anyneighborhood i ∈ I is the sum of prizes of all neighborhoodscovered by i. Note that PJ may not be a valid disc tour ofJ . Next we compute a valid disk tour PA that circles theentire neighborhoods visited by PJ . Let P∗ be the optimalPC-TSPN tour, and let PI∗ be the optimal PC-TSPN tourof I . We can derive an approximation guarantee from thefollowing inequalities:

|PA| ≤ π|PJ |+ 2π, (8)

|PJ | ≤((

1 +8

π

)|PI∗|+ 8

)(1 + ε) , (9)

|PI∗| ≤ |P∗|. (10)

Equation (10) follows from the fact that I ⊂ N and thatζ(j) ≥ 2d for all j ∈ N . The requirement on ζ(j) ensuresthat if one neighborhood covered by i ∈ I is visited in theoptimal tour, all neighborhoods covered by i are visited in theoptimal tour. Equation (9) follows from the case of disjointdisks described above. Equation (8) is found through thesame geometric analysis as in the regular TSPN [11], whichis omitted here. From these inequalities, we have that

|PA| ≤ ((π + 8) |P∗|+ 8π) (1 + ε) + 2π. (11)

As |P∗| becomes large, we have

|PA| ≤ (π + 8) (1 + ε) |P∗|. (12)

For the case of probabilistic neighborhoods, the guaranteeabove provides a bound on the solution quality of thealgorithm when run on p contours. For instance, the solutionreturned for 80% contours is guaranteed to be less than afactor of (8 + π) (1 + ε) worse than the optimal solutionthat visits the 80% contours. This guarantee holds for bothTSPN with probabilistic neighborhoods (from reference [11])and PC-TSPN with probabilistic neighborhoods (from The-orem 1). Note that this guarantee does not account forsolutions that may not visit all p contours (e.g., by sendingmany transmissions in a low communication quality area toachieve a high probability of receiving data).

VI. SIMULATIONS

A simulation environment was implemented in C++ run-ning on Ubuntu Linux to test our proposed CC-DCP algo-rithms. The experiments were run on a 3.2 GHz Intel i7processor with 9 GB of RAM. We now describe the acousticcommunication modeling used in the simulation

A. Acoustic Communication Modeling

We utilize a probabilistic underwater acoustic communi-cation model developed in prior work [2], [14]. This modelaccounts for noise factors in the environment, such as windand shipping activity, as well as the distance and frequencydependent attenuation of the signal in the water medium. Themodel also accounts for both thermal noise and turbulence.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Distance (km)

Packet err

or

rate

Pwr: 0.1 W

Pwr: 1 W

Pwr: 10 W

Pwr: 100 W

Fig. 2. Modeled data error rates for varying transmission power. The curvesgive the probability that a packet will not complete a successful round-tripbetween the AUV and a sensor node.

TABLE IPARAMETERS FOR UNDERWATER ACOUSTIC COMMUNICATION

Parameter ValueTransmission Frequency 13 kHz

Transmission Power 0.1 W - 10 kWBandwidth 1 kHzPacket Size 256 symbolsWind speed 5 m/s

Shipping activity 0.5 (moderate)

The following approximation is used to determine a symbolerror rate based on a signal-to-noise-ratio (SNR):

P (e) ≈ 1

4SNR. (13)

The above equation gives the probability of symbol error,P (e), as a function of SNR for the case of Rayleigh fadingwith known channel states at the receiver. Ultimately, theSNR is a function of the transmitted power, frequency oftransmission, wind speed, shipping factor, the distance andthe bandwidth used (see reference [2] for more detail). For apacket with Q symbols, the probability of transmission errorcan then be computed as

Ppacket = 1− (1− P (e))Q. (14)

Table I gives the parameters used for the communicationmodel in the simulated experiments. Figure 2 shows thepacket error rates at different levels of transmitted power.Note that the communication quality cannot be accuratelymodeled by a simple discrete neighborhood due to theintermediate communication levels.

B. Simulations with Random Deployments

We first implemented the optimal MDP solution usingvalue iteration. This method was able to solve problemswith up to 3 nodes on a 15 × 15 environment with a 1 kmgrid resolution. For these small problem sizes, the myopicnearest neighbor heuristic performed competitively with theoptimal solution. Based on this finding, we scale up the size

Sensors

AUV tour

Sensors

AUV tour circles

neighborhoods

Contour distance

Sensors

Neighborhoods

AUV tour

Contour distance

Fig. 3. Example tours using different neighborhood types. Left: Standardtraveling salesperson tour [5]. Center: Tour circling a maximal independentset of neighborhoods [11]. Right: Tour visiting the center of a covering setof neighborhoods. All tours travel within the contour distance of all nodes,but the covering set tour is shortest.

of the instances to compare the approximate methods, andwe remove the infeasible optimal method from consideration.

We now compare the myopic nearest neighbor strategyto the TSP strategy and the TSP with probabilistic neigh-borhoods (using 80% contours). We make one heuristicmodification to the algorithm with probabilistic neighbor-hoods. Instead of generating an MIS and circling the entireneighborhoods, we generate a covering set at half the contourdistance. This allows us to visit all sensors by simplyplanning a tour of the locations in the covering set. Thismodification improves performance in practice and also al-lows straightforward extension to 3D environments. Figure 3gives a visualization of planning with varying neighborhoodtypes, and the video attachment provides an animation of thesimulated AUV executing data gathering tours.

The next experiments utilize random deployments of100 sensors in 100 km × 100 km 2D environments. Thesimulated underwater vehicle moves at 5 km/hr. The nodeutilities are set to be uniform, and the requirement is tovisit all nodes. Figure 4 shows results from these trials.At low transmission power (poor communication), the valueof utilizing the neighborhoods is minimal, and the problemreduces to the classical TSP. At high transmission power(good communication), the value of planning is reduced, andthe simple myopic strategy moves closer to the quality of thenon-myopic strategies. At moderate communication levels,there is some improvement from both solving the TSP andutilizing neighborhoods.

The same experiments were run with random informationvalues from 0 to 25 were added to the 100 sensors. Thetotal cost is calculated by summing traveled distance plus thescaled information values of sensors not collected. The scalefactor was set as λ = 1 for these experiments. Modifying λwould allow for tuning the number of sensors visited. ThePC-TSP approximation algorithm was compared with andwithout neighborhoods to the nearest neighbor strategy.

Figure 5 shows the numerical results for the prize-collecting case. Solving the underlying PC-TSP withoutneighborhoods does not perform well, even when comparedto the myopic strategy. Since the PC-TSP algorithm cannotaccount for the cost of the neighborhoods when determin-ing which sensors to visit, it chooses to ignore a numberof sensors that would actually improve the final cost. Incontrast, the combination of the neighborhoods and the

10−1

100

101

102

103

104

0

20

40

60

80

100

120

140

Transmission Power (W)

Avera

ge C

om

ple

tion T

ime (

hours

)

Full Collection: 100 Nodes (100 km x 100 km)

Move to Nearest Node

TSP Ignoring Contours

TSP on Contours

Fig. 4. Communication-constrained data collection experiments. Error barsare one SEM, and averages are over 100 random deployments. Solvingthe TSP provides improvement over myopic techniques, and utilizingneighborhoods provides some additional improvement.

10−1

100

101

102

103

104

0

50

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150

Transmission Power (W)

Avera

ge C

ost (h

ours

+ p

enalty)

Prize Collecting: 100 nodes (100 km x 100 km)

Move to Nearest Node

TSP Ignoring Contours

TSP on Contours

Fig. 5. Prize-collecting communication-constrained data collection ex-periments. Data is not collected from some sensors, and the requiredpenalty is paid. Error bars are one SEM, and averages are over 100random deployments. Considering neighborhoods significantly improvesperformance.

underlying PC-TSP approximation algorithms performs well.Putting Figures 4 and 5 together, we see that consideringneighborhoods helps marginally in the case without prizesand more significantly when prizes are considered.

The running time of the algorithms is dominated by thecost of calculating the TSP tour with the Concorde solver. Inthe worst-case, this computation time can grow exponentiallyin the number of nodes. In practice, typical instances of100 nodes were solved in 10-100 ms. For a more extensivediscussion of the running time of Concorde, see reference [5].When probabilistic neighborhoods are taken into account,the TSP solver uses a reduced set of nodes (those in thecovering set). Thus, utilizing neighborhoods actually reducesthe running time of the algorithm.

Additional simulations were run to determine the effectof changing the size of the probabilistic neighborhoods,determined by the parameter p. Figure 6 shows completioncost for prize-collecting problems using varying p values.

10−1

100

101

102

103

104

0

50

100

150

Transmission Power (W)

Ave

rag

e C

ost

(ho

urs

+ p

en

alty)

Prize Collecting: 100 nodes (100 km x 100 km)

Contours (p = 0.1)

Contours (p = 0.9)

Contours (p = 0.99)

Contours (p = 0.999)

Contours (p = 0.9999)

Fig. 6. Simulations with varying sizes of probabilistic neighborhoods. Eachdata point is averaged over 100 random deployments. At low transmissionpower (poor communication), the choice of neighborhood size has a largereffect on performance.

The results show that orders of magnitude changes in pare required to see significant differences in completioncost. This is an intuitive result, since the communicationmodel predicts small changes in neighborhood size forlarge changes in transmission probability (see Figure 4). Atlow transmission power (i.e., when communication qualityis poor), the value of p must be chosen more carefully.These results again demonstrate the importance of networkscale and communication quality on the consideration ofneighborhoods in planning.

VII. CONCLUSIONS AND FUTURE WORK

This paper has shown that communication-constraineddata collection is feasible with an autonomous underwatervehicle. When communication quality is high (relative tothe sparsity of the network), planning is less essential.When communication quality is low, it is important tosolve the underlying TSP, rather than using a fully reactiveapproach. Finally, at moderate levels of communication, itis beneficial to consider the communication neighborhoodsduring planning. In addition, when information quality ofthe sensors is considered and all sensors do not need to bevisited, considering neighborhoods improves performance.This analysis provides insight into the level of planningrequired to optimize information gathering at different levelsof network sparsity, communication, and information quality.Such insight motivates the use of realistic communicationmodels in the development and analysis of planning algo-rithms.

A number of interesting extensions provide avenues for fu-ture research. The case where sensor communication qualityvaries between sensors results in probabilistic neighborhoodsof different sizes. While it is possible to apply our techniquesdirectly to such cases, it is not clear if additional methods arenecessary to provide good performance. Another interestingextension is the use of non-metric and asymmetric distancesbetween sensors. For instance, in the case of ocean currents,it may be easier to travel in one direction than another. Suchcases have been examined for the classical TSP [20], but notin the case of neighborhoods. Finally, dependencies between

sensors can result from correlations between gathered data.Exploiting the locality and submodularity of the informationfunctions [16] may allow us to extend our algorithms to suchcases as well.

VIII. ACKNOWLEDGMENTSThe authors gratefully acknowledge Jonathan Binney,

Arvind Pereira, Hordur Heidarsson, and Srinivas Yerramalliat USC for their insightful comments.

REFERENCES

[1] R. N. Smith, Y. Chao, P. P. Li, D. A. Caron, B. H. Jones, and G. S.Sukhatme, “Planning and implementing trajectories for autonomousunderwater vehicles to track evolving ocean processes based onpredictions from a regional ocean model,” Int. J. Robotics Research,vol. 26, no. 12, 2010.

[2] G. Hollinger, S. Yerramalli, S. Singh, U. Mitra, and G. S. Sukhatme,“Distributed coordination and data fusion for underwater search,” inProc. IEEE Conf. Robotics and Automation, 2011.

[3] M. Stojanovic, “On the relationship between capacity and distance inan underwater acoustic communication channel,” in Proc. ACM Int.Workshop on Underwater Networks, 2006.

[4] I. Vasilescu, K. Kotay, D. Rus, M. Dunbabin, and P. Corke, “Datacollection, storage, and retrieval with an underwater sensor network,”in Proc. Int. Conf. Embedded Networked Sensor Systems, 2005.

[5] D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook, TheTraveling Salesman Problem: A Computational Study. PrincetonUniv. Press, 2006.

[6] E. Balas, “The Prize Collecting Traveling Salesman Problem,” Net-works, vol. 19, pp. 621–636, 1989.

[7] G. Ausiello, V. Bonifaci, S. Leonardi, and A. Marchetti-Spaccamela,“Prize-Collecting Traveling Salesman and related problems,” in Hand-book of Approximation Algorithms and Metaheuristics, T. F. Gonzalez,Ed. CRC Press, 2007.

[8] G. Ausiello, V. Bonifaci, and L. Laura, “The Online Prize-CollectingTraveling Salesman Problem,” Information Processing Letters, vol.107, no. 6, pp. 199–204, 2008.

[9] M. Goemans and D. P. Williamson, “A general approximation tech-nique for constrained forest problems,” SIAM J. Computing, vol. 24,no. 2, pp. 296–317, 1995.

[10] A. Archer, M. H. Bateni, M. T. Hajiaghayi, and H. Karloff, “Improvedapproximation algorithms for Prize-Collecting Steiner Tree and TSP,”in Proc. IEEE Symp. Foundations of Computer Science, 2009.

[11] A. Dumitrescu and J. Mitchell, “Approximation algorithms for TSPwith neighborhoods in the plane,” J. Algorithms, vol. 48, no. 1, pp.135–159, 2003.

[12] M. de Berga, J. Gudmundssonb, M. J. Katzc, C. Levcopoulosd, M. H.Overmarse, and A. F. van der Stappen, “TSP with neighborhoods ofvarying size,” J. Algorithms, vol. 57, no. 1, pp. 22–36, 2005.

[13] D. Bhadauria and V. Isler, “Data gathering tours for mobile robots,”in IEEE Int. Conf. Intelligent Robots and Systems, 2009.

[14] F. Arrichiello, D. N. Liu, S. Yerramalli, A. Pereira, J. Das, U. Mitra,and G. S. Sukhatme, “Effects of underwater communication con-straints on the control of marine robot teams,” in Proc. Int. Conf.Robot Communication and Coordination, 2009.

[15] A. Krause and C. Guestrin, “Near-optimal nonmyopic value of in-formation in graphical models,” in Proc. Uncertainty in ArtificialIntelligence, 2005.

[16] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg, “Near-optimalsensor placements: Maximizing information while minimizing com-munication cost,” in Proc. Information Processing in Sensor Networks,2006.

[17] M. L. Puterman, Markov Decision Processes: Discrete StochasticOptimization. Wiley, 1994.

[18] L. Kaelbling, M. Littman, and A. Cassandra, “Planning and acting inpartially observable stochastic domains,” Artificial Intelligence, vol.101, pp. 99–134, 1998.

[19] V. V. Vazirani, Approximation Algorithms. Berlin, Germany: Springer-Verlag, 2001.

[20] H. Kaplan, M. Lewenstein, N. Shafir, and M. Sviridenko, “Approx-imation algorithms for asymmetric TSP by decomposing directedregular multidigraphs,” in Proc. IEEE Symp. Foundations of ComputerScience, 2003.