Combined Resource Allocation and Route Optimization in MultiagentNetworks: A Scalable Approach

Amber Srivastava1 and Srinivasa M. Salapaka2

Abstract— This paper presents an algorithm to solve the

simultaneous resource allocation and route optimization prob-

lem first presented in [1]. This NP hard problem entails

finding simultaneously the locations of resources (or service

or communication exchanges) in a multi-agent network as

well as determining multihop routes from individual agents to

a common destination through a network of resource nodes

in such a way that the total cost of communication from

all agents to the destination center is minimized. The main

contribution of this article is that it develops a solution approach

that scales better than the existing algorithm in [1]. The

number of design variables in the algorithm presented in [1]

grows exponentially O(2

M) with the number of resources M ;

whereas in the algorithm proposed in this paper, the number

of design variables are only of the order O(M). The proposed

algorithm incorporates certain constraints that result from

the law of optimality, which results in the reduction of the

design parameter space. This algorithm, which is based on

Maximum Entropy Principle (MEP), guarantees local minima

and is heuristically designed to seek the global minimum.

I. INTRODUCTION

Many applications such as vehicle routing, wireless com-munication, building management systems and other wire-less sensor networks such as found in battlefield surveil-lance, industrial process monitoring and machine control andhealth monitoring pose similar combinatorial optimizationproblems. These problems simultaneously require an op-timal placement of communication resource units (CRUs)that covers a large set of underlying sensors as well asspecification of an information transfer route from eachsensor to a common destination (central processing center)through the network of CRUs. For instance consider areconnaissance mission of an unknown terrain described inFigure 1, where each agent (sensor) has to communicateto an UAV (a common destination) through a network ofintermediate communication sets (CRUs) via single hop atsingle communication set or multiple hops through a groupcommunication sets. The underlying optimization problemis to determine the locations of the communication setsand the route (possibly multihop) from each agent to theUAV such that the total cost of information transfer from allthe agents to the UAV is minimized. Alternatively considersensor network for fire-safety in a large intelligent buildingsystem. Here the information has to be transferred fromeach sensor element (that detects fire or smoke) to a centralcomputer (to activate the sprinklers in an area or not). This

1Amber Srivastava is with the Department of MechanicalEngineering, University of Illinois, Urbana-Champaign, USAasrvstv6@illinois.edu

2Srinivasa M. Salapaka is with the Faculty of MechanicalEngineering, University of Illinois, Urbana-Champaign, USAsalapaka@illinois.edu

information has to be transferred through a network of localprocessors (CRUs), where information packets can reach thecentral computer in a single hop via one local processor orthrough multiple hops via multiple local processors. Againthe problem of interest becomes that of optimal placementof local processors (CRUs) so that total length of routes(information transfer cost; for instance the cost of layingcables and communication costs) over all the routes isthe minimum. This paper proposes a framework for suchproblems that require simultaneous resource allocation androute optimization.

Both the optimal placement of CRUs to cover the sensorsand the routing problems, even when viewed separately,are computationally complex. For instance, placement of MCRUs to cover N sensors is typically posed as a coveragecost minimization problem (or clustering problem). Theseproblems are known to be non-convex with multiple localminima. Similarly, the routes through which informationwill flow to the destination center (UAV) need to considerdirectionality of the message flow and possibility of multihop transfer which results in the number of possible routesto be of the order of 2M . There is extensive literature thataddresses the optimal resource placement problem and therouting problem individually but there is scant research thataddresses both these problems together; also when theseproblems arise simultaneously in applications (such as incertain wireless network applications), they are addressedsequentially.

The results of the proposed algorithm in [1] demonstratethe benefit of solving the above two problems simultaneouslyrather than as two separate problems. However, the algorithmpresented in [1] has the number of decision (design) variablesthat increase combinatorially; more precisely for a problemwith M resource nodes the number of design variables areof the order O(2M ). The algorithm presented in this paperreduces the number of decision variables to the order of Mwhich makes it more scalable when compared to [1].

The main conceptual contribution is the inclusion of thelaw of optimality, which results in the reduction of thedesign parameter space. More precisely, we note that ifany two optimal information transfer routes (originatingfrom different sensors) have a common CRU (that is bothroutes intersect), then both routes should have identical pathsafter that CRU. This is evident from the law of optimalitythat if an information packet reaches a certain CRU, thenits subsequent path to the common destination should beindependent of the path prior to reaching the CRU. Inclusionof the law of optimality allows the decision space to be justthe space of the CRU locations and length of the route from

Fig. 1: A reconnaissance mission scenario : The agents xi want tocommunicate with the UAV (destination center) located at r. Theoptimization problem here is to place resources yj in the terrain ⌦

and find routes for information flow from the agents to the UAVsuch that the total communication cost is minimized.

each CRU (a total of 2M decision variables) as opposed todecision space in [1] which apart from having M variablesfor CRU location, also has 2M decision variables to includeall possible routes. Another important result of the algorithmproposed in this article is that it allows for number of hopsto be a decision variable, and therefore provides the optimalnumber of hops required to transfer information from thesensors to the common destination. The paper also elaborateson the flexibility of the proposed algorithm by demonstratingits ability to incorporate various practical application specificconstraints.

As in [1], the main underlying concept of the proposedalgorithm is based on developing a homotopy from an appro-priate convex function to the non-convex cost function, whichis analogous to the DA algorithm in [12]. More precisely,the algorithm poses a minimization problem with the costfunction of the form D� 1

�H , where � is increased from 0to a large value. Here D (distortion) is the non-convex costfunction that reflects the objectives of simultaneous resourceplacement and routing and �H is an appropriately chosenconvex function. The solution approach incorporates a prob-abilty distribution on the space of distribution variables, andin fact H is the Shannon entropy of this distribution. Sinceinitial iterations with � = 0 is a convex function �H whoseminimization yields a global minimum, also the procedurepresented here is independent of the initialization. The mainheuristic here is that the global minimum of the non-convexcost function D is tracked by the algorithm as the convexcost function changes to desired non-convex cost functionvia homotopy.

II. PROBLEM FORMULATION

We consider N sensors whose spatial coordinates are givenby xi 2 ⌦ ⇢ Rd, where d is the dimension of the spatialcoordinates (for ease of illustration, we will consider d = 2in this paper). The objective is to transmit information fromthe sensors xi to the common destination at r through thenetwork of CRUs yj , where information packet can takemultihop route in the network of CRUs. The associatedresource allocation problem is to place the CRUs {yj} whichyields the minimum communication cost. The related routeoptimization problem is to find a route zl (a vector in R2q)which can be expressed as a sequence of CRUs yj given by

zl := [y0l1 , y0l2 , ...., y

0lq

]0, 1 q M (1)

Here, [(·)]0 represents the transpose operation. A communi-cation route zl taken by an information packet originating atxi to reach the destination can be understood as sequence ofmultiple hops along the nodes such that :

xi ! yl1 ! yl2 ! · · · ! ylq

! r. (2)

The maximum length (or the number of hops) that a routecan have is M . We will denote the cost of informationflow from xi to r through a path zl as d(xi, zl). Hencethe corresponding combinatorial optimization problem canbe written as:

D0

= min{y

j

}⇢⌦

NX

i=1

⇥pi min

zl

(d(xi, zl))] (3)

where pi represents the relative importance of the ith sensornode; it is assumed to be a known quantity and also withoutloss of generality we choose these weights such that

Pi pi =

1. The cost metric d(xi, zl) is a generic communication cost(which may vary from one application to the other), thoughfor purpose of illustration we consider a relatively simplemodel given in [1] by:

d(xi, zl) = kxi � yl1k2+

q�1X

i=1

�ikyli

� yli+1k

2

+ �Mkylq

� rk2 (4)

where zl is the route as defined in (1),�i > 0 are the knownrelative weights and ||(·)|| represents the euclidean distance.The first term in the equation (4) denotes the cost incurredin the information hop from the element xi to the entry CRUyj , the second term denotes cost incurred due to informationhop between the CRUs and the last term denotes the costincurred due to hop taken from the last CRU yl

q

in the routeto the destination r.

III. PROBLEM SOLUTION

The optimization problem as stated in (3) incorporatesboth resource allocation and route optimization. In order tominimize this cost function we have to minimize it withrespect to all the possible routes z0ls as well as resource nodelocations y0js. The number of paths zl grow exponentially(2M ) with resources M . In order to prevent this we reformu-late our representation of a route which carries informationfrom an xi to the common destination at r. Suppose the entryCRU for a communication packet sent by sensor xi is yj , andthe number of hops taken to reach the common destinationat r from yj is lj , then a route can be represented as:

xi ! yj ! yj+1

! · · · ! yj+lj

�1

! r (5)

Here we have made use of the law of optimality, from whichwe have that all routes that have yj as one of their nodes willalso have all subsequent nodes to be same in their routes tothe destination node. This makes it possible to identify eachroute in terms of first CRU (entry CRU) and the numberof hops lj as in (5) as opposed to (2) (as used in [1]). Aparticular benefit of looking at the routes zl as shown in (5)is that now our route related decision variables lj grow justlinearly with M .

Now we reformulate the cost function D0

defined in (3)and the cost d(xi, zl) defined in (4) to remove all the decisionvariable based on routes zl and replace them with lj and yj .The cost metrics d(xi, zl) in (4) can be reformulated as costd(xi, yj , lj) as shown in (6), where yj is the entry CRU forthe communication packet sent by a sensor xi and lj is thenumber of hops taken by the communication packet to reachdestination r from the CRU yj . Note that for a given yj ,lj M + 1� j.

d(xi, yj , lj) = �1d1(xi, yj) + �2

lj

�1X

k=0

d2(yk+j , yk+j+1)

+ �3d3(ylj

+j�1, r) (6)

where �i and di are relative weights and cost functionsrespectively and depend on the application of interest; how-ever for the purpose of illustration we will consider themto be squared euclidean distance functions.1 We redefine ouroptimization problem defined in (3) as below:

D1

= minyj

NX

i=1

⇥pi min

yj

,lj

(d(xi, yj , lj))] (7)

In our maximum entropy method (MEP) based algorithm,we introduce weighting terms p(yj |xi) for every sensor xi

and CRU yj , where p(yj |xi) can be viewed as the associationprobability of yj being entry node of xi. Also, without loss ofgenerality we can assume that (

Pj p(yj |xi) = 1). Similarly

for every CRU yj and route length lj we introduce anotherweighting term p(lj |yj) which associates the resource nodeyj with a possible route length of lj . Note that lj 2{1, 2, . . . ,M + 1 � j} for a resource node yj . Again wecan assume that

Plj

p(lj |yj) = 1 . The algorithm seeks tominimize the following modified version of the cost functionD

1

in (7).

D =

NX

i=1

pi

MX

j=1

p(yj |xi)

M+1�jX

lj

=1

p(lj |yj)d(xi, yj , lj). (8)

It should be remarked that with specific choice of distri-bution that assigns p(yj |xi) = 1 and p(lj |yj) = 1 whend(xi, yj , lj) = miny

j

,lj

d(xi, yj , lj) and zero otherwise, thecost function D in (8) becomes similar to cost function D

1

in (7). However in our algorithm, we design these associa-tion weights p(lj |yj) and p(yj |xi) by using the MaximumEntropy Principle [16] and [17], whereby we choose thedistributions that maximize the Shannon entropy H in (9)for a given cost D.

H = �X

ijlj

p(xi, yj , lj)log(p(xi, yj , lj)), (9)

where p(xi, yj , lj) = pip(yj |xi)p(lj |(yj , xi)), and since ljonly depends on the yj , p(lj |(yj , xi)) = p(lj |yj). This choiceof weights p(lj |yj) and p(yj |xi) ensures that the algorithmremains maximally noncommittal towards any solution and atthe same time guarantee that D in (8) converges to D

1

in (7).The corresponding Lagrangian is given by the maximizationof H � �D or the equivalent minimization problem

F = D � 1

�H, where � is the Lagrange multiplier. (10)

1All the simulations of the proposed algorithm in this paper considerd1, d2, d3 as squared euclidean distances for the purpose of illustration.

Thus the resource allocation and routing problem reducesto minimizing F with respect to three quantities p(lj |yj),p(yj |xi) and yj . We obtain the following explicit expressionsfor the distributions from the MEP theorem.

p(lj |yj) =e��d(x

i

,yj

,lj

)

Pl0j

e��d(xi

,yj

,l0j

)

(11)

p(yj |xi) =

Plj

e��d(xi

,yj

,lj

)

Pkl

k

e��d(xi

,yk

,lk

)

. (12)

On substituting the above into the Free energy F in (10) weobtain the following cost function:

F =� 1

�

X

i

pi log(X

jlj

e��d(xi

,yj

,lj

)) +

1

�

X

i

pi log pi (13)

The location of the CRUs can then be obtained by mini-mizing (local) F in (13) by putting @F

@Y = 0 which gives:

Y =(�1Pb + �2H2aSL + �2H2bSH + �3H3)�1

(�1Pabx+ �3cH3), (14)

where Y =⇥y1

y2

· · · yM⇤. Please refer to the

Appendix for the definitions of Pb, H2a, SL, H2b, SH , H3

and Pab.The approach that we will take in the algorithm specifies

the homotopy from the (convex) function �H to the (non-convex) cost function D, which is parameterized by theannealing parameter � . We start at a low value of � sothat the D becomes insignificant as compared to �H andthe algorithm focuses on maximizing the entropy and thusmaking itself insensitive to the initial choice of the routes.As the value of � increases, the cost function becomes moreand more significant and the association probabilities startconverging towards either 0 or 1, thereby making D convergeto D

1

. As � approaches 1, we minimize D to obtain hard(nonrandom) solution to the original problem D

1

.Thus thealgorithm steps can be listed as:

1) Initialize all the CRUs at the weighted centroid of theelements xi, i.e. yj =

Pi pixi 8j and put � to a small

value.2) Calculate the weights p(lj |yj) and p(yj |xi) using equa-

tions (11) and (12).3) Calculate the CRU locations Y using (14).4) Increase �; stop if user-specified �

max

is achieved elsego to step 2.

IV. ADDING MULTIPLE CAPABILITIES ANDCONSTRAINTS TO THE PROBLEM

An important feature of the proposed algorithm is that itis flexible and can easily accommodate practical constraintson CRUs and the hop lengths; we demonstrate this flexibilitythrough the following scenarios.A. Number of Hops Priority

A practical constraint that may arise in some applicationsis limiting the number of hops (which otherwise may resultinto time delays, extra costs or penalties) a communicationpacket can take while traveling from a sensor xi to the

(a) (b) (c)

Fig. 2: Figures (a) and (b) demonstrates the difference in solutions obtained for resource allocation and route optimization when solvingthe two as separate problems and in a simultaneous method respectively. Figure (c) illustrates the sensitivity of the ratio of single hopcost to multi hop (4 hop) cost with respect to the distance of the destination center to the centroid of all sensors. We can clearly observethat the cost of single hop increase much more (and is always more) than the cost of four hop.

common destination r. In order to limit the maximumnumber of hops we use the constraint

p(lj |yj) = 0 for all lj 2 Lj , (15)

where Lj = {lj : lj > maximum number of hops from yj}.We incorporate (15) into the maximum entropy method(MEP) to obtain the association weights p(yj |xi) andp(lj |yj). The expression for p(yj |xi) remains the same asin (12) however p(lj |yj) gets modified as below

p(lj |yj) =�l

j

e��d(xi

,yj

,lj

)

Pl0j

�l0j

e��d(xi

,yj

,l0j

), (16)

where �lj

2 {0, 1}. �lj

= 0 for all lj 2 Lj . Morespecifically �l

j

= 0 indicates that the route with entry CRUas yj can not take lj number of hops from xi to r, �l

j

= 1means otherwise. The expression for the resource placementsis same as in (14) and is be obtained by substituting the aboveassociation weights into the expression of F in (10) and thenminimizing F locally.

B. Capacity constraints on communication resource unitsIn this case we present a formulation to limit the use of

a CRU yj as an entry resource unit for the sensors xi. Theobjective is to allocate resources and optimize route such thata CRU yj is the entry resource to a prescribed fraction ⌘j ofsensors. The quantity

Pi pip(yj |xi) denotes the fraction of

sensors for which CRU yj is the entry CRU. We define pj =Pi pip(yj |xi) as the relative usage of a CRU at yj . Hence

the constraint now becomes that pj = ⌘j for all j 2 [1,M ]This constraint can be easily incorporated by minimizing(locally) the following lagrangian (free energy) with respectto p(lj |xi), p(yj |xi) and Y .

F = D � 1

�H +

X

j

⌫j(pj � ⌘j) (17)

We get the similar expression for p(lj |yj) and Y as shownin (11) and (14) respectively. The expression for p(yj |xi)changes as below

p(yj |xi) =�j

Plj

e��d(xi

,yj

,lj

)

Pkl

k

�ke��d(xi

,yk

,lk

)

, (18)

where �j = e(��⌫j

) and ⌫j is the lagrange multiplier to theconstraint as shown in (17). As pj =

Pi pip(yj |xi) and

pj = ⌘j , the update rule for �j is formulated as:

�j =⌘j

Pi pi

Pl

j

e��d(xi

,y

j

,l

j

)

Pkl

k

�k

e��d(xi

,y

k

,l

k

)

(19)

Note that the update equation (14) for the CRU locationsY have an implicit dependence on the parameter �j , whichagain is coupled to the CRU locations yj , annealing param-eter � and cluster probabilities pj . In this framework, wedeterministically optimize the free energy F at successivelydecreasing values of annealing parameter � by alternatingbetween the equations (14) and (19).

V. SIMULATIONS AND RESULTS

In this section we consider the application of the algorithmpresented above along with its proposed adaptations forthe constraints stated in section IV. For our simulations,we use randomly generated sensor and common destinationposition. Figure 2(a), 2(b) compares the solutions obtainedwhen resource allocation and route optimization problemsare solved individually as two separate problems, and whensolved simultaneously. In the first case the CRUs are allo-cated at the centroid of the sensor clusters due to which thecost of communication between the CRUs yj’s and commondestination r is high (and is not compensated by the lowercost between elements xi and their respective CRUs yj)leading to higher value of the cost function in (13). However,in the latter case the CRUs are allocated at some optimalposition which results into lower cost as compared to theformer case.

If the resource allocation and route optimization are solvedas two separate problems then the CRUs will be at thecluster centroids and the communication routes will have atendency to contain just a single hop. In Figure 2(c) we showthrough simulation the benefit of multihop routes over singlehop scenarios which is one of the gains of solving the twoproblems simultaneously rather than as two separate ones.

Figure 3, illustrates the case where the maximum numberof hops that from an entry CRU yj can take is constrained

(a) (b)

(c) (d)

Fig. 3: Figures here demonstrate the effect of limiting the maximum number of hops an information flow can take, on the resourceallocation and route optimization. In (a) no constraints on maximum number of hops, (b) only single hop is allowed, (c) maximumnumber of hops from any CRU is restricted to three, and (d) Y 1 is allowed maximum of 4 hops, all other CRUs can take only upto 2hops.

(discussed in section IV(A)). We consider the following fourcases (a) No constraints on Maximum Number of Hops, (b)Only single hop allowed from every CRU, (c) Maximumthree hops allowed from every CRU, (d) For entry CRUY1 maximum of 4 hops allowed, rest all entry CRU cantake upto 2 hops. Figure 3 demonstrates how optimal CRUlocations and routes change as number of hops get restricted.Observe that in figure 3(c) where maximum of three hops areallowed from any entry CRU, route taken by the informationpacket with entry CRU Y 1 is Y 1 ! Y 2 ! Y 3 ! r(shown in red dots) while route taken by information packetwith entry CRU Y 2 is Y 2 ! Y 3 ! Y 4 ! r (shownin black dots) and not Y 2 ! Y 3 ! r since the formerone contains more number of hops (hence lesser cost) thanthe latter one. Similarly observe the figure 3(d) here theroute taken by an information packet with entry CRU Y 1is Y 1 ! Y 2 ! Y 3 ! Y 4 ! r (shown in red dots) whilethe route taken by an information packet with entry CRUY 2 is Y 2 ! Y 3 ! r (shown in black dots) instead ofthe route Y 2 ! Y 3 ! Y 4 ! r (which contains morenumber of hops) since the latter route is not permitted bythe constraint. Table (1) elaborates on the routes taken invarious cases above.

Figure 4 studies the case where CRU yj is constrainedto be the entry resource unit for ⌘j fraction of the sensors

Start Node Route for Case 1 Route for Case 2Y 1 Y 1 ! Y 2 ! Y 3 ! Y 4 ! r Y 1 ! rY 2 Y 2 ! Y 3 ! Y 4 ! r Y 2 ! rY 3 Y 3 ! Y 4 ! r Y 3 ! rY 4 Y 4 ! r Y 4 ! r

Start Node Route for Case 3 Route for Case 4Y 1 Y 1 ! Y 2 ! Y 3 ! r Y 1 ! Y 2 ! Y 3 ! Y 4 ! rY 2 Y 2 ! Y 3 ! Y 4 ! r Y 2 ! Y 3 ! rY 3 Y 3 ! Y 4 ! r Y 3 ! Y 4 ! rY 4 Y 4 ! r Y 4 ! r

TABLE I: Constrained maximum number of hop

xi (constraint discussed in section V(B)). Fig 4(a) showsthe simulation result for the case when no constraint hasbeen put on the usage of the CRUs yj . Fig 4(b) showsthe simulation result for the case when the usage of theCRU yj is constrained such that ⌘

1

= 0.3, ⌘2

= 0.3, ⌘3

=0.3, ⌘

4

= 0.1. The final results obtained from the simulationare p(y

1

) = 0.28, p(y2

) = 0.28, p(y3

) = 0.33 and p(y4

) =0.10. As can be seen pj do not converge entirely on ⌘j (i.e.pj ⇡ ⌘j) however are very close to the constraints put onthem. This is because of some numerical issues and it is partof our further work.

VI. CONCLUSION

In this paper, we have used the law of optimality to reducethe space of design variables for the problem presented in [1]

(a) (b)

Fig. 4: (a) has no constraint on the usage of the CRUs yj , p(y1) = 0.57, p(y2) = 0.14, p(y3) = 0.28, p(y4) = 0, (b) demonstratesthe case where use of yj as an entry node is constrained by pj = ⌘j . Here ⌘1 = 0.3, ⌘2 = 0.3, ⌘3 = 0.3, ⌘4 = 0.1. We obtainp(y1) = 0.28, p(y2) = 0.28, p(y3) = 0.33, p(y4) = 0.10.

from O(2M ) to O(M) which makes the algorithm proposedhere more scalable. This framework also gives us usefulextra information in terms of the length of optimal hopfrom any CRU yj . The algorithm proposed here is flexiblein incorporating various practical constraints as illucidatedin section IV and V. Also the framework we present isindependent of the distance cost function which may changefrom one area of application to other.

VII. ACKNOWLEDGEMENT

The authors would like to acknowledge NSF grants ECCS15-09302, CMMI 14-63239 and CNS 15-44635 for support-ing this work.

APPENDIX

�ba(x) =

⇢1 if a� ✏ x b+ ✏0 else

(20)

For simplification of notation, consider p(yj |xi) = pbj|i,p(lj |yj) = pcl

j

|j , pi = pai and pj = pbj .

h2a(m) ,

X

jlj

pbjpclj

/j�j+l

j

�1

j (m� 1)�M�j0

(lj)�M2

(m)

h2b(m) ,

X

jlj

pbjpclj

/j�j+l

j

�1

j (m)�M�j0

(lj)�M�1

1

(m)

h3

(m) ,X

jlj

pbjpclj

/j�j+l

j

j+lj

(m)�M�j0

(lj)

H2x = diag

�h2x(1), h

2x(2), · · · , h2x(M)

�, x={a,b}

H3

= diag�h3(1)

, h3(2)

, · · · , h3(M)

�

Pb = diag�pb1

, pb2

, · · · , pbM�

Pab = [(Pab)]mi where (Pab)mi = pbm/ipai

x =⇥x1

· · ·xN ]0 y =⇥y1

· · · yM ]0 J = [1 · · · 1]0

SL =

"0 0 .. .. ..�1 1 .. .. ..0 �1 1 0 .... .. 0 �1 1

#SH =

"1 �1 0 .. ..0 1 �1 0 .... 0 1 �1 .... .. 0 1 �1

#

REFERENCES

[1] N. V. Kale and S. M. Salapaka, “Maximum entropy principle-basedalgorithm for simultaneous resource location and multihop routingin multiagent networks,” IEEE Transactions on Mobile Computing,vol. 11, no. 4, pp. 591–602, 2012.

[2] S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions,and the bayesian restoration of images,” IEEE Transactions on patternanalysis and machine intelligence, no. 6, pp. 721–741, 1984.

[3] A. Gersho and R. M. Gray, Vector quantization and signal compres-sion. Springer Science & Business Media, 2012, vol. 159.

[4] Z. Drezner and H. W. Hamacher, Facility location: applications andtheory. Springer Science & Business Media, 2001.

[5] Q. Du, V. Faber, and M. Gunzburger, “Centroidal voronoi tessellations:applications and algorithms,” SIAM review, vol. 41, no. 4, pp. 637–676, 1999.

[6] C. W. Therrien, “An estimation-theoretic approach to terrain imagesegmentation,” Computer Vision, Graphics, and Image Processing,vol. 22, no. 3, pp. 313–326, 1983.

[7] P. Sharma, S. Salapaka, and C. Beck, “A scalable approach tocombinatorial library design for drug discovery,” Journal of chemicalinformation and modeling, vol. 48, no. 1, pp. 27–41, 2008.

[8] Y. Xu, S. M. Salapaka, and C. L. Beck, “Aggregation of graph modelsand markov chains by deterministic annealing,” IEEE Transactions onAutomatic Control, vol. 59, no. 10, pp. 2807–2812, 2014.

[9] J. Cortes, S. Martinez, T. Karatas, and F. Bullo, “Coverage controlfor mobile sensing networks,” in Robotics and Automation, 2002.Proceedings. ICRA’02. IEEE International Conference on, vol. 2.IEEE, 2002, pp. 1327–1332.

[10] S. Lloyd, “Least squares quantization in pcm,” IEEE transactions oninformation theory, vol. 28, no. 2, pp. 129–137, 1982.

[11] C.-R. Hwang, “Simulated annealing: theory and applications,” ActaApplicandae Mathematicae, vol. 12, no. 1, pp. 108–111, 1988.

[12] K. Rose, “Deterministic annealing for clustering, compression, classi-fication, regression, and related optimization problems,” Proceedingsof the IEEE, vol. 86, no. 11, pp. 2210–2239, 1998.

[13] C. Perkins, E. Belding-Royer, and S. Das, “Ad hoc on-demand distancevector (aodv) routing,” Tech. Rep., 2003.

[14] J. L. Sobrinho, “Network routing with path vector protocols: The-ory and applications,” in Proceedings of the 2003 conference onApplications, technologies, architectures, and protocols for computercommunications. ACM, 2003, pp. 49–60.

[15] C. Liu and J. Wu, “Efficient adaptive routing in delay tolerant net-works,” in 2009 IEEE International Conference on Communications.IEEE, 2009, pp. 1–5.

[16] E. T. Jaynes, “Information theory and statistical mechanics,” Physicalreview, vol. 106, no. 4, p. 620, 1957.

[17] ——, Probability theory: The logic of science. Cambridge universitypress, 2003.