IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 37 Target Assignent in Robotic Networks: Distance Optiality Guarantees and Hierarchical Strategies Jingjin Yu, Meber, IEEE, Soon-Jo Chung, Senior Meber, IEEE, and Petros G. Voulgaris, Fellow, IEEE Abstract We study the proble of ulti-robot target assignent to iniize the total distance traveled by the robots until they all reach an equal nuber of static targets. In the first half of the paper, we present a necessary and sufficient condition under which true distance optiality can be achieved for robots with liited counication and target-sensing ranges. Moreover, we provide an explicit, non-asyptotic forula for coputing the nuber of robots needed to achieve distance optiality in ters of the robots counication and target-sensing ranges with arbitrary guaranteed probabilities. The sae bounds are also shown to be asyptotically tight. In the second half of the paper, we present suboptial strategies for use when the nuber of robots cannot be chosen freely. Assuing first that all targets are known to all robots, we eploy a hierarchical counication odel in which robots counicate only with other robots in the sae partitioned region. This hierarchical counication odel leads to constant approxiations of true distance-optial solutions under ild assuptions. We then revisit the liited counication and sensing odels. By cobining siple rendezvous-based strategies with a hierarchical counication odel, we obtain decentralized hierarchical strategies that achieve constant approxiation ratios with respect to true distance optiality. Results of siulation show that the approxiation ratio is as low as 1.4. Index Ters Networked robots, optiality. I. INTRODUCTION IN this paper, we study the perutation-invariant assignent of a set of networked robots to a set of targets of equal cardinality. Focusing on iniizing the total distance traveled by the robots in a planar setting, we seek optiality guarantees and near-optial strategies. For robot-to-robot counication, we investigate both a siple circular range-based odel and a region-based odel in which all robots within the sae region can counicate with each other. When we consider the liited target-sensing capability of the robots, a circular range sensing odel is used. Manuscript received July 17, 013; revised January 8, 014; accepted July 5, 014. Date of publication July 30, 014; date of current version January 1, 015. This work was supported in part by AFOSR Grant FA9550110193, NSF Grant IIS-153758, and NSF Grant ECCS 10-7437. Recoended by Associate Editor E. Frazzoli. J. Yu was with the University of Illinois at Urbana-Chapaign, Chapaign, IL 6180-5711 USA. He is now with the Coputer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cabridge, MA 0139 USA e-ail: jingjin@csail.it.edu. S.-J. Chung and P. G. Voulgaris are with the Coordinated Science Lab and the Departent of Aerospace Engineering, University of Illinois at Urbana- Chapaign, Chapaign, IL 6180-5711 USA e-ail: sjchung@illinois.edu; voulgari@illinois.edu. Color versions of one or ore of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.014.34491 The class of probles that we study is denoted as target assignent in robotic networks as it shares any siilarities with the probles studied in [1]. In [1], the authors characterized the perforance of ETSP 1 ASSGMT and GRID ASSGMT algoriths strategies in achieving tie optiality i.e., iniizing the tie until every target is occupied. In contrast, we focus on iniizing the total distance traveled by all robots with significantly different assuptions on the counication and sensing odels of the robots. The total distance serves as a proper proxy to quantities such as the total energy consuption of all the robots. Note that a distance-optial solution for the target assignent proble generally does not iply tie optiality and vice versa []. As its nae iplies, the proble of target assignent in robotic networks requires solving an assignent or atching proble. The assignent proble is extensively studied in the area of cobinatorial optiization, with polynoial tie algoriths available for solving any of its variations [3] [8]. If we instead put ore ephasis on ulti-robot systes, the probles of robotic task allocation [9] [1], swar reconfiguration [13], ulti-robot path planning [14] [16], and ulti-agent consensus [17] [0] are relevant. For a ore coprehensive review on these topics, see [1]. Our work is also closely related to the study of the connectivity of wireless networks. An interesting result [] showed that, if n robots are uniforly randoly scattered in a unit square, then each robot needs to counicate with k =Θlogn nearest neighbors for the entire robotic network to be asyptotically connected as n approaches infinity. In particular, the authors of [] showed that k<0.074 log n leads to an asyptotically disconnected network whereas k>5.1774 log n guarantees asyptotic connectivity. This pair of bounds was subsequently iproved and extended in [3]. These nearest neighbor based connectivity odels were further studied in [4] [6], to list a few. In any of these settings, a geoetric graph structure is used [7]. This research effort brings forth three contributions. First, for robots with liited range-based target-sensing and counication capabilities the ranges are captured by radii r sense and, respectively, we derive necessary and sufficient conditions for ensuring a distance-optial solution. In particular, we provide a probabilistic estiate of the nuber of robots denoted by n sufficient for all robots to for a connected network for a fixed counication radius. In contrast to the asyptotic connectivity results fro [], [8], we give 1 ETSP stands for the Euclidean traveling salesan proble. 0018-986 014 IEEE. Personal use is peritted, but republication/redistribution requires IEEE perission. See http://www.ieee.org/publications_standards/publications/rights/index.htl for ore inforation.
38 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 n as an explicit function of that is also non-asyptotic. Therefore, our bounds hold without requiring n. We further show that the sae bounds are asyptotically tight when a high probability guarantee is required. Second, allowing the robots to have global target-sensing capabilities coupled with a region-based counication odel, we show that an infinite faily of hierarchical strategies can lead to decentralized target assignents while ensuring that the total expected distance traveled by the robots is asyptotically within a constant ultiple of the optial distance. Our siulation results show that this bound can often be saller than two. Moreover, because hierarchical strategies avoid running a centralized assignent algorith, significant savings on coputation tie in certain cases, a speedup of 1000 ties or ore are achieved. Third, for robots with global target-sensing capabilities and a range-based counication odel, hierarchical strategies for assignent and rendezvous-based strategies for copensating for the lack of global counication are cobined to obtain decentralized suboptial algoriths. These hybrid strategies, under ild assuptions, preserve the constant approxiation ratios on distance optiality achieved by the pure hierarchical strategies. We further show that the global target-sensing assuption can be reoved without affecting asyptotic optiality. Portions of this work were presented in [9] and [30] for the early disseination of results. Copared with [9] and [30], this paper provides a ore coprehensive view of the results along with coplete proofs for all theores. Many of the proofs have been significantly iproved to illustrate ore clearly proof techniques that ay be of interest on their own. In addition, the current paper discusses extensively generalizations of the stochastic target assignent proble to isatching nuber of robots and targets, and to higher diensions. The rest of the paper is organized as follows. In Section II, we present notations and well-known results fro other branches of research needed for the developent of our results. After stating the proble forally in Section III, we then elaborate on the three stated contributions in Sections IV VI. We present results of siulation studies in Section VII to validate our theoretical results and conclude in Section VIII. II. PRELIMINARIES In this section, we review results on the balls and bins proble, linear assignent, and rando geoetric graphs. The sybols R, R +, N denote the set of real nubers, the set of positive reals, and the set of positive integers, respectively. For a positive real nuber x, log x denotes the natural logarith of x; the function x resp., x denotes the sallest resp., largest integer that is larger resp., saller than or equal to x. denotes the cardinality for a set and the absolute value for a real nuber. We use v to denote the Euclidean -nor of a vector v. The unit square [0, 1] R is denoted as Q. The expectation of a rando variable X is denoted as E[X].Weuse E to represent a probabilistic event and the probability with which an event e occurs is denoted as Pe. Given two functions f,g : R + R +, fx =Ogx if and only if there exist M O,x O R + such that x>x O, fx M O gx. Siilarly, fx=ωgx if and only if there exist M Ω,x Ω R + such that x>x Ω, fx M Ω gx. If fx=ogx and fx =Ωgx, then we say fx = Θgx. Finally, fx =ogx resp., fx =ωgx if and only if fx=ogx resp., fx=ωgx and fx= Θgx does not hold. A. Balls and Bins The well-studied proble in probability theory known as the urns-proble, or the proble of balls and bins, considers the distribution generated as a nuber of balls are randoly tossed into a set of bins. The following classical result on the ball and bins proble is due to Erdős and Rényi. Theore 1 Balls and Bins [31]: Suppose that a nuber of balls are tossed uniforly randoly into bins, one ball per tie step. Let T k denote the first tie such that k balls are collected in every bin. Then for any real nuber c li P T k <log +k 1 log log + c = e e c k 1!. 1 It is worth noting that the proof of Theore 1 is fairly short and elegant, eploying only basic tools fro analysis and cobinatorics. A useful corollary for k =1follows readily. Corollary : For an arbitrary real nuber c, suppose that no fewer than log + c balls are tossed uniforly randoly into bins. As, every bin contains at least one ball with probability e e c. Proof: In 1, letting k =1yields li PT 1 <log + c =e e c. The corollary directly follows recall that T 1 is the nuber of tosses needed so that every bin has at least one ball. Corollary says that T 1 = log is a sharp threshold. Letting c =5 in yields that the probability of every bin being occupied by at least one ball is greater than 0.99 when at least log +5 balls are tossed. On the other hand, the sae probability is no ore than 0.001 when no ore than log balls are tossed. B. Linear Assignent Proble The linear assignent proble, as a fundaental cobinatorial optiization proble, can be defined as follows. Proble 1 Linear Assignent: Given two finite sets X and Y with X = Y, together with a weight function C : X Y R, find a bijection f : X Y that iniizes the cost C x, fx. 3 x X
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 39 Proble 1 is also called the perfect weighted bipartite atching proble. Here, the apping C is essentially a square atrix that can be used to represent a variety of weights, such as the Euclidean distance when X and Y represent physical locations. The Hungarian ethod for the assignent proble, proposed by Kuhn [7], has an On 4 running tie, which was subsequently iproved to On 3 by Edonds and Karp [6]. Many other algoriths for the assignent proble exist, including other prial-dual linear prograing ethods [5], auction based ethods [3], and parallel algoriths [4], [8]. Nevertheless, the strongly polynoial On 3 Hungarian ethod reains as the fastest exact sequential algorith, which we use in our siulations. When X and Y are restricted to points on the plane with the weight function C being the Euclidean distances between the points, the special linear assignent proble is known as the Euclidean bipartite atching proble, which can be solved exactly using an On.5 log n prial-dual algorith [3]. Alternatively, near linear tie approxiation algoriths can yield near optial solutions with high probability [33]. 3 C. Rando Geoetric Graphs Let X = {x 1,...,x n } be a set of n points in the unit square Q. Forafixedcounication radius,thegeoetric graph G over this set of points is fored by taking each point as a vertex and connecting any two vertices whose underlying points x 1 and x satisfy x 1 x. When X is selected randoly following soe distribution, the resulting graph is called a rando geoetric graph. Properties of rando geoetric graphs have been studied extensively; see [7] for a thorough coverage. One such property is the connectivity of these graphs, which is of particular interest to wireless counication and robotic networks. Theore 3 Rando Geoetric Graphs [8]: Let G be a rando geoetric graph obtained following the unifor distribution over the unit square for soe n and. Then for any real nuber c, asn PG isconnected πnr co log n c =e e c. 4 Fro 4, it is possible to estiate the nuber of robots required to guarantee a connected geoetric graph G. III. TARGET ASSIGNMENT IN ROBOTIC NETWORKS In this section, we forally define the proble of target assignent in robotic networks and the optiality objective. A. Proble Stateent Let X 0 = {x 0 1,...,x 0 n} and Y 0 = {y1,...,y 0 n} 0 be two sets of points the superscript ephasizes that these points are A polynoial tie algorith runs in strongly polynoial tie only if its running tie does not depend on the size of the input paraeters. Note that n is the nuber of input paraeters in this case. 3 Although algoriths fro [33], [3] have theoretically faster running ties than the Hungarian ethod and apply to the proble that we study, they are ore difficult to ipleent and slower in practice unless X is very large because they are not strongly polynoial tie algoriths like the Hungarian ethod. Fig. 1. a The counication graph solid blue nodes and edges for a set of robots under Counication Model 1 with a counication radius of. Robots blue dots in the sae connected coponent of a counication graph can freely counicate with each other. b The counication graph for a set of robots under Counication Model with = b =9. obtained at the start tie t =0 in the unit square Q, 4 selected uniforly randoly. Place n = X 0 = Y 0 point robots on the points in X 0, with robot a i occupying x 0 i. Each robot has a unique integer label e.g., i. In general, we denote robot a i s location coordinates at tie t 0 as x i t. The basic task to be forally defined is to ove the robots so that at soe final tie t f 0,everyy Y 0 is occupied by a robot. We ay assue that there is a final tie t f i for each robot a i, such that x i t x i t f i for t tf i. For convenience, we also refer to X 0 and Y 0 as the set of initial locations and the set of target locations, respectively. Motion Model: The robots are single integrators, i.e., ẋ i t =u i t with u i t being piece-wise sooth and u i t {0, 1}. We assue the size of the robots is negligible with respect to the distance they travel and ignore collisions between robots. Counication Model 1: We study two counication odels in this paper. In the first counication odel, a robot a i ay counicate with other robots within a disc of radius centered at x i t.atanygiventiet 0, we define the undirected counication graph Gt, which is a geoetric graph that changes over tie, as follows. Gt has n vertices v 1,...,v n, corresponding to robots a 1,...,a n, respectively. There is an edge between two vertices v i and v j if the corresponding robot locations x i t and x j t, respectively, satisfy x i t x j t. Fig. 1a provides an exaple of a disconnected counication graph. Given our focus on distance optiality, we ake the siplifying assuption that all robots corresponding to vertices in a connected coponent of the counication graph ay exchange inforation instantaneously. In other words, robots in a connected coponent of Gt can be treated as a single robot insofar as decision aking is concerned. Counication Model : The unit square Q is divided into = b equal-sized saller squares regions. 5 Robots within each region can counicate with one another but 4 Our results are scale-invariant because all the theores hold for squares of any size with proper scaling. Hence, a unit square environent is used throughout the paper. 5 In this paper, is frequently used to denote the nuber of sall squares in a division of the unit square Q and b = is the nuber of resulting partitions on an edge of the unit square. The value of and b ay vary.
330 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 robots fro different regions cannot exchange inforation [see, e.g., Fig. 1b]. This odel iics the natural geoetrical resource allocation process in which supplies and deands are first atched locally; the surpluses and deficits within each region then get balanced out at larger regions, giving rise to a hierarchical strategy. Target-Sensing Model: We assue that a robot is aware of a point y Y 0 if y x i t r sense,thetarget-sensing radius. The proble we consider in this paper is defined as follows. Proble Target Assignent in Robotic Networks: Given X 0, Y 0, r sense, and Counication Model 1 with or Counication Model, find a control strategy ut = [u 1 t,...,u n t], such that for soe 0 t f i < and soe perutation σ of the nubers 1,...,n, x i t f i =y0 σi for all 1 i n. Over all feasible solutions to an instance of Proble, we are interested in iniizing the total distance traveled by all robots, which can be expressed as D n := n tf i 0 ẋ i t dt. 5 As an accurate proxy to the energy consuption of the entire syste, the cost defined in 5 is an appropriate objective in practice. Unless otherwise specified, distance optiality refers to iniizing D n. Over all perutations σ of the nubers 1,...,n, and for fixed X 0 and Y 0, the iniu total distance for robots oving along continuous paths is D n := in σ n x 0 i yσi 0 6 which ay or ay not be achievable depending on the capabilities of the robots e.g, if the robots cannot follow straightline paths, then D n >D n. Let U denote the set of all possible control strategies that solve Proble given a fixed set of capabilities for the robots, we say that distance optiality is achieved if in U D n = D n. Besides distance optiality, we also briefly discuss the total task copletion tie i.e., the su of the individual task copletion ties as targets are occupied, denoted by T n. If all robots start oving toward targets and do not stop in the iddle, then T n = D n. In particular, we define T n := D n. IV. GUARANTEEING DISTANCE OPTIMALITY FOR ARBITRARY AND r sense In this section, we use Counication Model 1. In general, when r sense < or <, it is ipossible to guarantee distance optiality, since global assignent is no longer possible in general. For exaple, as r sense 0, the robots ust search for the targets before assignents can be ade; it is very unlikely that the paths taken by the robots toward the targets will be straight lines, which is required to obtain D n. This raises the following question: Given a pair of and r sense, under what conditions can we ensure distance optiality? Theore 4 answers this question. Fig.. General setup in which the two robots cannot counicate with each other at t =0and therefore cannot always decide an optial assignent at t =0. Theore 4: In a unit square, under sensing and counication constraints i.e.,,r sense <, distance optiality can be achieved with probability one if and only if at t =0: i the counication graph is connected; ii every target is within a distance of r sense to soe robot. Proof: We first prove that the conditions are necessary with two clais: 1 an optial assignent that iniizes D n is possible in general only if G0 is connected, and an optial assignent that iniizes D n is possible only if for all y Y 0, y is within a distance of r sense to soe x X 0. To see that the first clai is true, we note that distanceoptial assignents forbid robots fro oving unnecessarily, requiring at t =0a pairing between eleents of X 0 and Y 0 that iniizes D n. We now show that this is not possible in general when <.Forn =, assue that the two targets are located at y 1 and y as given in Fig. solid red dots. Assue the first robot a 1 is located at x 1 the blue solid dot at the lower left of Fig. and a 1 is of equal distance to y 1 and y. Let the second robot a take two possible locations x and x as shown, which are syetric along a diagonal of Q. Ifa is at x resp. x, then a should go to y resp. y 1, forcing a 1 to go to y 1 resp. y. Not knowing a s location because a 1 is out of a s counication radius, a 1 has at ost 50% chance of picking the distance iniizing choice at t =0. We can readily extend the locations of the robots and targets to include neighborhoods around the the dotted circles in Fig. to show that there is a non-zero probability that an optial assignent cannot be ade at t =0. This proves that that G0 cannot have ore than one connected coponent and ust be connected. The exaple can be extended to work for arbitrary n by adding additional robots and targets to close vicinities of x 1 and y 1, respectively. For the second clai, suppose that at t =0,soey Y 0 is not within a distance of r sense to any x X 0. A robot ust ove to search for that y. This will cause the robot to follow a path that is not a straight line with probability one, iplying that D n = Dn with probability zero. It is not hard to see that the necessary conditions fro the two clais are also sufficient: when G0 is connected and each target is observable by soe robot a i, the robots can decide at t =0aglobal assignent that iniizes D n.
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 331 Theore 4 suggests a siple way for ensuring distance optiality by either increasing the nuber of robots or increasing one or both of and r sense. This essentially leads to a centralized counication and control strategy Strategy 1. Note that given the assignent perutation σ, each robot a i can easily copute its straight-line path between x 0 i and y0 σi. Since every robot can carry out the coputation in Strategy 1, to resolve conflicting decisions and avoid unnecessary coputation, we ay let the highest labeled robot e.g., a n handle the entire assignent process. The rest of this section establishes how the conditions fro Theore 4 can be et. We point out that siilar conclusions can also be reached by exploring Theore 3, which yields an asyptotic relationship between the required nuber of robots for G0 to be connected and. We take a different approach and produce the required nuber of robots as an explicit function of without the asyptotic assuption. A. Guaranteeing a Connected G0 Since the robots can be anywhere in the unit square Q,given a counication radius of <, intuitively, at least Θ1/rco robots are needed for a connected G0, which requires the robots to take a lattice-like foration such as a grid. It turns out that when the robots are uniforly randoly distributed, only a logarithic factor ore robots are needed to ensure a connected G0. Lea 5: Suppose that n robots are uniforly randoly distributed in the unit square. For fixed < and 0 < ε < 1, att =0, the counication graph is connected with probability at least 1 ε if n 5 log 1 5 ε. 7 Proof: We divide the unit square Q into = b equalsized sall squares with b = 5/. Label these sall squares {q 1,...,q }. Under this division schee, a robot residing in a sall square q i can counicate with any other robot in the four squares sharing a side with q i see Fig. 3. Therefore, G0 is connected if each q i contains a robot. Let n i denote the nuber of robots in q i. Then Pn i =0= 1 1 n <e n. Fig. 3. If the sall squares have a side length of 5/ or saller, then a robot in such a square e.g., the gray square can counicate with any robot in the four neighboring sall squares sharing a side with the gray square. The inequality holds because 1 x n <e nx for 0<x<1. To see this, let fx =log1 x/x. The Taylor expansion of fx at x =0is 1 x/ x /3+ox 3 < 1 for 0 < x < 1. This shows that log1 x < x for 0 < x < 1 n log1 x < nx 1 x n <e nx. By Boole s inequality i.e., the union bound, the probability that at least one of q 1,...,q is epty can be upper bounded as P En i =0 Pn i =0<e n. Setting e n/ = ε and replacing = 5/ yields 5 1 exp n r 5 = ε co n = 5 log 1 5 ε which guarantees that each sall square contains at least one robot with probability 1 ε. The bound in Lea 5 can be further tightened; Corollary 6 below illustrates one way to achieve this. It produces n saller than that given by 7 when < 5/. Corollary 6: Suppose that n robots are uniforly randoly distributed in the unit square. For fixed < and 0 < ε < 1, att =0, the counication graph is connected with probability at least 1 ε if 5 n log 1 1 5 5 +. 8 ε Proof: If each of the shaded sall squares in Fig. 4 has at least one robot, then G0 ust be connected: any robot falling in a sall white square ust be connected to soe robot in a shaded sall square. This shows that 8 is sufficient. Reark: In coparison to Theore 3, Lea 5 provides n as an explicit function of. Moreover, our sufficient condition on n given in 7 [and 8], unlike 4, is not an asyptotic bound. Therefore, our bound applies to an arbitrary. On the other hand, if we let 0, then an asyptotic stateent can also be ade.
33 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 Fig. 4. As long as each of the shaded sall squares contains an robot, G0 ust be connected. Therefore, only b /+bsall squares need to have robots in the. Lea 7: Suppose that n robots, each with a counication radius of, are uniforly randoly distributed in the unit square. At t =0, the counication graph is asyptotically connected with arbitrarily high probability e e c for soe c>0 if 5 n log 5 + c. 9 10 holds for all >0 we use the definition 0 0 =1 here. To prove Lea 8, because all bins are initially epty, after tossing the first ball, soe bin contains exactly one ball. That is, P 1, E 1 =1. Let the bin occupied by the first ball be bin 1. As k 1 additional balls are tossed into the bins, the probability that none of these k 1 balls occupy bin 1 is 1 1/ k 1. Therefore, for 1 k,wehave k 1 P k, E 1 P 1, E 1 P 1, E 1 = 1 1 1 1 1 1 1 1 >e 1. Lea 9: Suppose that < n < log balls are tossed uniforly randoly into bins. As P n, E 1 1 e e 1 1 1 > 1 e e e 1. Proof: Given the division schee used in the proof of Lea 5, distributing robots into the unit square Q is equivalent to tossing the robots balls into the sall squares bins uniforly randoly. By Corollary, as,havingn log + c = log 5/ + c 5/ robots guarantees that all sall squares ust have at least one robot each with probability e e c. Since fx =cx grows slower than gx =x log x as x, Lea 7 says that n = Θ1/ log1/ robots can ensure that G0 is connected with probability arbitrarily close to one asyptotically. Next, we show that these any robots are also necessary for the high probability guarantee. Let P n, E denote the probability of an event E happening after tossing n balls into bins. We work with two events: E 0, the event that at least one bin is epty, and E 1, the event that at least one bin contains exactly one ball. We want to show that P n, E 1 is not too sall for n up to log, which is proven in the next two leas. Lea 8: Suppose that 1 n balls are tossed uniforly randoly into bins. Then P n, E 1 1 1 1 >e 1. Proof: First we prove a useful inequality: for N 1 1 1 >e 1. 10 To see this, note that the function log1 x 1/x 1 has a Taylor expansion of 1+x/+ox > 1 for sall x> 0, yielding that 1 x 1/x 1 >e 1 for sall x>0. Since the derivative of 1 x 1/x 1 is positive for x 0, 1, Proof: Suppose that after an experient of n tosses into bins, E 0 holds; i.e., at least one bin is epty. Without loss of generality, we assue the epty bin is bin 1. Now consider tossing an additional k balls into the bins. The probability of exactly one of these k balls falling in bin 1 is P k, exactly one ball falls in bin 1 k 1 = 1 1 1 Therefore k 1 = k 1 1 k 1. P n +k,e 1 P n,e 0 P k, exactly one ball falls in bin 1 = k 1 1 k 1 P n,e 0. 11 Letting c = 1 in Corollary, we have li PT 1 log =1 e e. 1 That is, as,for0<n <log, P n,e 0 1 e e. Plugging this into 11 and letting k =, we have that for <n<log,as P n, E 1 1 e e 1 1 1 > 1 e e e 1 in which the last inequality is by 10. Under the assuptions of Leas 8 and 9, we always have that as, P n, E 1 >in{e 1, 1 e e e 1 } > 0.34. We now show that n = Θ1/ log1/ is a tight bound on the nuber of robots for guaranteeing the connectivity of G0 with high probability.
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 333 and 0 <ε<1, every target is observable by soe robot at t =0with probability at least 1 ε if n log 1. 14 ε r sense r sense Fig. 5. A 3 3 block as defined in the proof Theore 10. Theore 10: For n uniforly randoly distributed robots in a unit square with a counication radius 1 n =Θ rco log 1 13 is necessary and sufficient to ensure that at t =0, the counication graph is asyptotically connected with arbitrarily high probability. Proof: Lea 7 covers sufficiency; we are to show that there is soe non-trivial probability that G0 is disconnected if the nuber of robots satisfies 1 n = o rco log 1. To prove the clai, we partition the unit square Q into = b equal-sized sall squares in which b = 1.1/. The factor of 1.1 in the expression akes the side of the sall square larger than. Assuing that is divisible by 3 it is always possible to truncate soe sall squares to satisfy this, we ay group the sall squares into /9 groups of 3 3 blocks see, e.g., Fig. 5. If there is a single robot in a 3 3 block, the robot cannot counicate with the rest of the robots if it falls inside the sall square in the center of the block e.g., the solid gray square in Fig. 5. By Leas 8 and 9, for less than /9 log/9= 1.1/ log 1.1/ /3/9 robots, the probability of having at least one of these 3 3 blocks containing exactly one robot is at least 0.34 as i.e., 0. If a 3 3 block has exactly one robot in it, with probability of 1/9, the robot is in the iddle square. Therefore, with probability at least 0.34/9 0.04, G0 is disconnected. B. Ensuring Target Observability With a connected counication graph G0 guaranteed by Lea 5, we can solve a single assignent proble if for each y Y 0, y x r sense for soe x X 0. Siilar techniques used in the proof of Lea 5 lead to a siilar lower bound on n. Lea 11: Suppose that n robots and n targets are uniforly randoly distributed in the unit square. For fixed r sense Proof: If we partition the unit square Q into /r sense equal-sized sall squares and there is at least one robot in each sall square, then any point of Q is within r sense distance to soe robot. Following the sae arguent used in the proof of Lea 5, the inequality fro 14 ensures that this happens with probability at least 1 ε. Putting together Leas 5 and 11, we obtain a lower bound on n that akes a distance-optial assignent possible. Theore 1: Suppose that n robots and n targets are uniforly randoly distributed in the unit square. Fixing 0 <ε<1, at t=0, the counication graph is connected and every target is observable by soe robot with probability at least 1 ε if 10 n log 1 10 15 θ ɛ θ in which θ := in{ 5r sense, }. Proof: When θ = 5r sense, 15 becoes 14, which iplies 9. Therefore, G0 is connected with probability 1 ε. When θ =, i.e., r sense 10 /5, by Lea 5, 9 iplies that G0 is connected with probability 1 ε. Moreover, there is at least one robot in each of the sall squares with a side length of at ost / 5 as specified in the proof of Lea 5. Having r sense 10 /5 guarantees that robots in a sall square observes all targes within the sae sall square. Therefore, every y Y 0 is within a distance of r sense to soe x X 0. Reark: Theore 1 is not an asyptotic result and applies to all and r sense. If a high probability asyptotic result is desirable, Lea 11 can be readily turned into a version siilar to Theore 10, by following the sae proof techniques. In view of this fact, the bounds fro Theore 1 are asyptotically tight. V. H IERARCHICAL STRATEGIES FOR r sense : OPTIMAL DISTANCE AND PERFORMANCE GUARANTEES In this section, we work with the region-based Counication Model and assue that r sense that is, every robot is aware of the entire Y 0. The study of Counication Model, besides leading to interesting conclusions on hierarchical strategies, also facilitates the analysis in Section VI as we revisit Counication Model 1. A region-based counication odel naturally leads to a hierarchical strategy for solving Proble under the optiality criterion of iniizing the cost defined by 5. Let h 1 be the nuber of hierarchies and i, 1 i h, be the nuber of equal-sized regions at hierarchy i. We ake the following assuptions that are ainly used in Theore 16: i 1 1, ii i+1 i, and iii a region at a higher nubered hierarchy is contained in a single region at a lower nubered hierarchy.
334 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 For exaple, dividing Q into 4 i 1 squares at hierarchy i satisfies these requireents. Fig. 6. The ratio of D n / n log n. Each data point is an average of 5 runs. We call the associated strategy under these assuptions the hierarchical divide-and-conquer strategy, the details of which are described in Strategy. Note that for each region in Strategy, the robots can again let the highest labeled robot within the region carry out the strategy locally. It is clear that Strategy is correct by construction because X 0 = Y 0. The rest of this section is devoted to analyzing the strategy. We begin with a single hierarchy h =1. Since r sense iplies that all robots are aware of the entire set Y 0, the robots ay for a consensus of which robot should go to which target at t =0by finding an optial assignent σ that yields D n as defined by 6. This assignent proble can be solved using a bipartite atching algorith such as the Hungarian ethod. Ajtai, Kolós, and Tusnády proved the following about D n. Theore 13 Optial Matching [34]: Assuing that n points are i.i.d. following the unifor distribution over a unit square, then, with probability 1 o1 C 1 n log n D n C n log n 16 in which C 1 and C are positive constants. Reark: The second inequality in 16 reains true in expectation and also for arbitrary probability easures on [0, 1], albeit with a different universal constant than C, by a result of Talagrand [35]. Therefore, D n =Θ n log n in expectation. Although no forulas for C 1 and C fro 16 were given in [34], a siulation study suggests that C 1 <C < 1 and C /C 1 1 as n. As an exaple, for 00 n 10000, 0.4 n log n D n 0.5 n log n on average see Fig. 6. Next, we look at the general case with h>1 hierarchies. To bound D n, at each hierarchy i, we need to know the nuber of robots that cannot be atched locally. We derive this nuber in Lea 14. Note that Lea 14 does not depend on and n being large. Lea 14: Suppose that n robots and n targets are uniforly randoly distributed in the unit square Q, and Q is divided into equal-sized regions. Within each of these regions, the robots are atched one-to-one with the targets until no ore atchings can be ade. The total nuber of robots that are left unatched is no ore than n 1/ in expectation. Proof: Restricting to one of the equal-sized regions, say q i, we know for x 0 j X0 and y 0 j Y 0 and P x 0 j q i = P y 0 j q i = 1 P x 0 j q i,yj 0 q i = P x 0 j q i,yj 0 1 q i = in which the event x 0 j q i,yj 0 q i represents a surplus of one robot in q i and the event x 0 j q i,yj 0 q i a deficit in q i. Thus, we ay view the experient of picking x 0 j and y0 j as a one step walk on the real line starting at the origin, with 1/ probability of oving ±1. The entire process of picking X 0 and Y 0 can then be treated as a rando walk of n such steps. Under this rando walk analogy, we ay use a rando variable Z j {0, ±1} to represent the outcoe of picking x 0 j,y0 j. We iediately have that E[Zj ]= 1/. Letting S n = Z 1 + + Z n, we can copute the variance of S n as E [ Sn] [ = E Z1 + + Z n ] = E [ Z1 + + Zn ] = ne [ Zj ] n 1 =. Applying Jensen s inequality to the concave function x with x = S n = Sn,wehave [ ] E [ S n ]=E S n E [Sn] n 1 E [ S n ]. Because, in expectation, an equal nuber of the regions have surpluses ore robots than targets and deficits fewer robots than targets, and soe of the regions ay have neither, no ore than half of the regions should have a surplus of robots on average. The total nuber of unatched robots in expectation is then no ore than / E[ S n ] n 1/.
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 335 The distance traveled by the atched robots at the botto hierarchy with regions can be bounded easily. For siplicity, we now assue that these regions are equal-sized squares. Lea 15: Suppose that n robots and n targets are uniforly randoly distributed in the unit square Q, and Q is divided into equal-sized sall squares. Within each of these sall squares, the robots are atched one-to-one with the targets until no ore atchings can be ade. The iniu total distance of atchings ade between the robots and the targets within the sall squares is no ore than C n log n in expectation, for soe positive constant C. Proof: Since Q is divided into squares, these squares all have a side length of 1/. Let one such square be q i with n i robots note that n i =n. Since a unifor distribution restricted to q i is again unifor, we can apply Theore 13 to q i.if we let these n i robots atch only with targets inside q i, then the total distance incurred locally will not exceed C n i log n i / in expectation. Here C is soe positive constant. Note that it is not necessarily the case that all n i robots will be atched locally in q i. This does not affect the current proof. For soe 1 i, it ay be the case that no local atchings can be ade because either n i =0or there is no target in q i. Let denote the nuber of these squares in which local atchings can be ade. The total distance incurred by local atchings is then upper bounded by note that n i is now indexed with respect to these squares C ni log n i = C 1 ni log n i. Here we assue that > 0, otherwise the local atchings would have a distance cost of zero. Since the function ϕx = x log x is concave, by Jensen s inequality, E[ x log x] E[x]logE[x]. Letting x = ni and the expectation be carried out over the discrete unifor distribution with 1/ probability each, we have C 1 ni log n i C n i log = C n i log C n log n. n i n i log Reark: With inor odifications, Lea 15 can be applied to regions with shapes other than squares. Defining the diaeter of a two-diensional region as the diaeter of the region s sallest enclosing circle, the ain requireent for the odification to work is that the axiu diaeter of these regions is O1/. We now give an upper bound on D n, in expectation, for general hierarchical strategies. Theore 16: Suppose that n robots and n targets are uniforly randoly distributed in the unit square Q, and Q is divided into i equal-sized sall squares at hierarchy i with a total of h hierarchies. For all applicable i 1, assue that i+1 i and any sall square at hierarchy i +1falls within a single square at hierarchy i. Then Strategy yields E[D n ] C h 1 ni+1 n log n +. 17 i Proof: The C n log n ter on the RHS of 17 is due to Lea 15. Then at each hierarchy i with 1 i<h, each of the i squares contains i+1 / i saller squares fro hierarchy i +1. Here we use the assuption that a region at a higher nubered hierarchy falls copletely within a single region at a lower nubered hierarchy. This eans that a robot that gets atched at hierarchy i needs to travel at ost a distance of /i. Since there are no ore than n i+1 1/ < i+1 n/ unatched robots at hierarchy i in expectation by Lea 14, the distance incurred at hierarchy i is no ore than ni+1 / i for 1 i<h. Suing up all the distances then gives us the inequality 17. Theore 16 allows us to upper bound the perforances of different hierarchical strategies depending on the choices of h and { i }. We observe that for fixed h and { i } independent of n, the first ter C n log n doinates the other ters in 17 as n. This iplies that Strategy yields assignents whose total distance is at ost a constant ultiple of the optial distance. This observation is suarized in Corollary 17. Recall that Dn is the iniu possible distance defined by 6. Corollary 17: For fixed h and 1,..., h that do not depend on n, asn, Strategy yields target assignents with D n /Dn = O1 in expectation. For exaple, with h and i =4 i 1 at hierarchy i, we have E[D n ] C h 1 n log n + 4n = C n log n +h 1 n. 18 For any fixed h, as n, D n /D n C/C 1 + o1 = O1. A constant approxiation ratio can also be achieved when h and { i } depend on n. For exaple, letting h =3, =logn, and 3 =log n,wehave E[D n ] C n log n + n log n =C + n log n. 19 Since hierarchical strategies need not run centralized assignent algoriths for all robots, the coputational part of these strategies can be significantly faster. We will coe back to this point in the next section. Reark: Before concluding this section, it is worth entioning that the results of this section continue to hold in only slightly weaker fors when the point sets X 0,Y 0 are drawn i.i.d. fro the sae arbitrary distribution over [0, 1] based
336 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 on Talagrand [35]. Since the topic of arbitrary probability easures diverges fro the ain focus of this paper, we only briefly discuss extending the results of this section to deal with arbitrary probability easures on [0, 1]. To adapt Lea 14 for arbitrary probability easures, assue that each region q i see the proof of Lea 14 has an overall probability of p i of receiving a robot or target. Note that p i =1. This changes the upper bound of E[ S n ] for the region q i to np i 1 p i. Then, over all regions, the total nuber of unatched robots is bounded by npi 1 p i = 1 n pi 1 p i n 1 = n 1 = n 1 p i 1 1 p i in which the inequality is obtained by applying Jensen s inequality to the concave function x1 x. Besides updating the unifor distribution of X 0 and Y 0 to an arbitrary probability easure, the stateent and proof of Lea 15 reain largely unchanged. This is because the second inequality in 16 does not change asyptotically as the underlying robot and target distribution changes. Then, the inequality 17 fro Theore 16 erely adds a ultiplicative constant of to its second ter on the RHS. Because the first inequality in 16 is not known to hold for arbitrary probability easures, we do not have a parallel of Corollary 17 for arbitrary probability easures. Nevertheless, these bounds for arbitrary probability easures suggest that the unifor distribution is aong the worst distributions for Proble under the optiality constraint of iniizing 5. This is because the unifor distribution leads to an optial assignent distance of Ω n log n, and an arbitrary distribution leads to an optial assignent distance of O n log n. Note that these updates also apply to the results in the next section with appropriate odifications. A. Near Distance-Optial Rendezvous Strategy Our first suboptial strategy uses oving robots for inforation aggregation until soe robot is aware of the locations of all robots i.e., the set X 0, at which point a centralized optial assignent can be ade. Although soe robots will ove and change their locations during this process, the oved robots nevertheless are aware of their initial locations in X 0. To carry out the strategy, the unit square Q is divided into = b disjoint, equal-sized sall squares, with b = /. These sall squares are labeled as q i,j s, in which i and j are the row nuber and colun nuber of the square, respectively see Fig. 7. Based on its initial location, each robot can identify the sall square q i,j it lies in. At t =0, the robots in the squares on row 1 and row b start oving in the direction as indicated in Fig. 7. We want to use these robot to pass the inforation of where all robots are. At ost one robot per square is required to ove since all robots in a sall square can counicate with each other by the assuption b = /. Assuing that a robot in a square q i,j is oving downwards, it keeps oving until it is within the counication radius of a robot in a square with label q i+k,j,k 1, at which point it passes over the inforation it has and stops. The robot in q i+k,j then does the sae. The procedure continues until a robot reaches the iddle of Q row b/. Then, the robots in the squares on row b/ repeat the sae process horizontally until a robot in the center of Q knows the locations of all other robots. At this point, the robot in the center of Q that knows the location of all other robots akes a global assignent so that each robot is atched with a target. The oved robots then reverse their travel directions to deliver the assignent inforation to all robots. The outline of the strategy is given in Strategy 3. VI. NEAR OPTIMAL STRATEGIES After exploring hierarchical strategies for the region-based Counication Model, we now return to the range-based Counication Model 1. If is arbitrary and the conditions specified in Theore 4 are not known to hold, the best we can do is obtain near distance-optial strategies. In this section, we show that constant ratio approxiation of distant optiality is possible for arbitrary r sense and. The basic idea behind our strategies is to ove the robots to pass around inforation about the locations of other robots. The assuption r sense is ade teporarily. At the end of this section, we show how to reove this assuption without affecting asyptotic optiality.
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 337 Fig. 7. Directions for robots to ove in the rendezvous strategy. Fig. 8. Illustration of robot oveents in a potential hierarchical strategy. The correctness of Strategy 3 as an algorith is proven by construction. Besides the distance cost fro the assignent, the robots in each colun travel at ost a total distance of two. The iddle row incurs an extra distance of at ost two. Thus, in expectation, D n <D n +b +. Since D n =Θ n log n, D n doinates b +when b = o n log n. In particular, n = Θ1/r co satisfies this requireent. Therefore, Strategy 3 can yield near distance-optial solution without requiring an n as large as 13 with respect to 1/. A drawback of Strategy 3 is that no robot can ove to the targets until the assignent phase is coplete. This yields a total task copletion tie of T n n + T n in expectation, which is undesirable since T n = O n log n asyptotically. Furtherore, Strategy 3 requires running a centralized assignent algorith for all robots. This ight be ipractical for large n. We address these issues with decentralized hierarchical strategies. B. Decentralized Hierarchical Strategies We first look at a decentralized hierarchical strategy that cobines Strategies and 3. Instead of waiting for a centralized assignent to be ade, in each of the sall square q i,j as specified in Strategy 3, we let the robots in q i,j be assigned to targets that belong to the sae square we refer to these as local assignents. The robots that are not atched to targets then carry out Strategy 3. We denote this hierarchical rendezvous strategy as Strategy 4 and oit the pseudo code. Corollary 18: For Strategy 4 -level Hierarchical Rendezvous, as n and E[D n ] C n log n + n + + 0 E[T n ]=Θ n log n + n. 1 Proof: The bound on E[D n ], given by 0, is straightforward to copute using Theore 16, in which the first two ters on the right side of 0 correspond to the first and second ters of the right side of 17, respectively, and the last two ters are due to counication overhead. For total copletion tie, all but Θ n robots can start oving to their targets at t =0.FortheΘ n robots, they need to wait no ore than two units of tie each before oving to their targets. This gives us 1. Reark: Siilar to Strategy 3, for any fixed, in expectation, D n /Dn = O1 as n. Moreover, in contrast to Strategy 3, for any fixed, T n /Tn = O1 in expectation. Suppose that a centralized algorith requires tn running tie. Using the sae centralized algorith, Strategy 4 has a running tie of Otn/+t n. Iftn =On 3 as given by the Hungarian ethod, then Strategy 4 has a running tie of On 3 / +n 3/. Taking n = 10000, =10, for exaple, we get a 1000-tie speedup. By introducing additional hierarchies, Strategy 4 can be easily extended to a ulti-hierarchy decentralized strategy. Depending on how the subdivisions are ade, any such strategies are possible. For exaple, using h hierarchies with each hierarchy i having 4 i 1 sall squares, we get a quad-erging strategy as illustrated in Fig. 8, in which up to four representatives in four adjacent squares eet to decide a local assignent of the robots in these squares at a given hierarchy level. Although these suboptial strategies vary in detail, they can be easily analyzed with Theore 16. For exaple, we look at an extension to Strategy 4 with three hierarchies; let us call this strategy, Strategy 5. After partitioning the botto or third hierarchy to squares, the iddle or second hierarchy is partitioned into k = sall squares. At either the third or the second hierarchy, local assignents are ade, followed by applying the rendezvous strategy as given in Strategy 3. It is again straightforward to derive the following. Corollary 19: For Strategy 5 3-level Hierarchical Rendezvous, as n E[D n ] C n log n + n +4 +. Reark: Again, D n /D n = O1 as n for a fixed. Introducing ore hierarchy levels extends the total copletion tie T n, which is increased by approxiately. Thus, the total copletion tie of Strategy 5 is also given by 1. Following siilar analysis, the overall running tie required by Strategy 5 is Otn/+ t n+t n given a centralized assignent algorith that runs in tn tie. C. Handling Arbitrary r sense Because there can be targets anywhere in Q, to carry out the algoriths stated in this section, each robot ust be aware of all target locations. For this to happen for arbitrary r sense, Q ust be swept through in a worst scenario. To do this, we partition Q
338 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 TABLE I COMPARISON BETWEEN 4 AND 7 Fig. 9. Effects of n on the connectivity of G0 for different values of. into 1/r sense sall squares and let a robot in the top-left sall square zig-zag through Q i.e., following a Boustrophedon path [36] until it covers the botto side of Q. If there is no robot in the top-left sall square, then a robot in a square along the Boustrophedon path is used; iplicit tiing can be used to deterine this. Once the end of the path is reached, the robot then reverses its course until it gets back to the topleft sall square. At this point, this robot is aware of all target locations. It can then repeat a siilar path to counicate that inforation to all other robots. This procedure ensures that all robots are aware of all target locations. The total distance cost of the procedure is about 1/r sense + 1/. Taking this penalty, which does not depend on n and therefore has no ipact on the asyptotic optiality, we can then effectively assue r sense. Fig. 10. Distance optiality of Strategy 3 over varying n and. VII. SIMULATION STUDIES A. Nuber of Required Robots for a Connected G0 In this subsection, we show a result of siulation to verify our theoretical findings in Section IV. Since the bounds over and r sense are siilar, we focus on and verify the requireent for the connectivity of G0 for several s ranging fro 0.01 to 0.. For each fixed, various nubers of robots are used starting fro n = log /rco the nuber of robots goes as high as 3 10 5 for the case of =0.01. 1000 trials were run for each fixed cobination of and n. The percentage of the runs with a connected G0 is reported in Fig. 9. The siulation suggests that the bounds on n fro Theore 10 are fairly tight. To copare to 4, which also allows for estiation of n in ters of with a specified probability for obtaining a connected G0, we coputed n based on 4 and 7 for a range of -probability pairs. We then use these n s to estiate the actual probability of having a connected G0. We list the result in Table I. Each ain entry of the table has two probability nubers separated by a coa, obtained using 4 and 7, respectively. As we can see, 4 gives underestiates due to its asyptotic nature and cannot be used to provide probabilistic guarantees. On the other hand, 7 provides overestiates that guarantee the desired probability. B. Perforance of Near Optial Strategies Next, we siulate Strategies 5 and evaluate D n, T n, and running tie for these strategies over various values of Fig. 11. Distance optiality of Strategy 4 over varying n and. n and, assuing r sense. Due to our choice of k = in Strategy 5, we pick specific s so that = / is always a perfect square. These values are =0.16, 0.09, 0.057, and 0.04, which correspond to = 81, 56, 65, and 196, respectively. The nuber of robots used in each siulation ranges fro 100 to 10000. For each n, 10 assignent proble instances are randoly generated. These proble instances are then used to test all strategies. We test Strategy using the sae two-hierarchy and three-hierarchy partitions that are used with Strategies 4 and 5. Distance Optiality: The ratios D n /D n for Strategy 3 over different n and are plotted in Fig. 10. We observe that the overhead for establishing global counication aong the robots becoes insignificant as n increases, driving D n /D n to close to one. For Strategy 4, the ratios were plotted siilarly in Fig. 11 but with sall error bars. The error bars display the standard deviation over the 10 runs we oitted these fro a figure, such as Fig. 10, when they are too sall to see. They can be better seen in Fig. 1, which is a zooed-in version of the = 0.16 line fro Fig. 11. The siilarities between Fig. 10 and Fig. 11 for sall n are not surprising since both strategies spend ost of their effort distance traveled in establishing
YU et al.: TARGET ASSIGNMENT IN ROBOTIC NETWORKS 339 Fig. 1. The effect of varying n on the distance optiality of Strategy 4 with =0.16 = 81. Fig. 15. The assignent cost of a three-level pure hierarchical strategy. TABLE II RUNNING TIME FOR STRATEGIES 3 5 Fig. 13. Distance optiality of Strategy 5 over varying n and. Fig. 14. Assignent cost of a two-level pure hierarchical strategy. counication. As this extra counication cost diinishes as n grows, the actual assignent cost doinates. Strategy 3, with assignent being done in a centralized anner, becoes better than the decentralized Strategy 4. As expected, for a fixed, D n /Dn decreases as n increases. For n = 10000, the approxiation ratios for our choices of are around 1.4 due to the slow growing nature of Dn n log n; fixing any, this ratio should be close to one for large n. On the other hand, for a fixed n, as the partition of the unit square Q gets finer, D n /Dn increases, iplying that decreasing the counication radius has a negative effect on distance optiality. We observe siilar results on the distance optiality of Strategy 5 see Fig. 13. If we reove the rendezvous part fro Strategies 4 and 5, they becoe siilar to Strategy. The distance optiality perforance of these two particular versions of Strategy is shown in Figs. 14 and 15, respectively. For all partitions ade =81, 56, 65, 196, D n /D n ratios of less than two are achieved and can go as low as 1.06, showing that hierarchical strategies can provide very good optiality guarantees. Coputational Perforance: We list the running tie, in seconds, for Strategies 3 5 in Table II. The standard On 3 Hungarian ethod is used as the baseline assignent algorith. Each ain entry of the table lists three nubers corresponding to the running tie of Strategies 3, 4, and 5, respectively, for the given cobination of and n. Note that any version of Strategy has the sae aount of coputation as a corresponding rendezvous-based strategy. As expected, a hierarchical assignent greatly reduces the coputation tie, often by a factor over 10 3. The coputation was perfored on a Intel Core-i7 3970K PC under a 8 GB Java virtual achine. Tie Optiality: Since Strategies 3 5 sacrifice distance and therefore, tie to copensate for liited counication, we do not expect the total copletion tie T n of these strategies to atch T n closely. For exaple, in 1, although T n T n as n for fixed = /, it requires a very large n for log n to doinate. Thus, we only copare T n aong
340 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 Fig. 16. Ratio of total copletion tie between Strategies 3 and 4. finding the nearest robot for each y Y 0. When X 0 Y 0, the proble becoes a ultiple salesen version of the traveling salesan proble we have a standard traveling salesan proble when X 0 =1, which is an NP-hard proble. It reains an interesting open question to investigate the iddle ground, i.e., X 0 = C Y 0 for soe constant C for exaple C [0.1, 10]. Distribution of Initial and Target Locations: Although it is beyond the scope of this paper, it would be interesting to establish a lower bound on the optial assignent distance for arbitrary probability easures. Also, it would be interesting to investigate the case in which the robots and the targets assue different distributions. Another iportant aspect not covered in this paper is the issue of targets distributed soewhat randoly over tie. Miniizing Over Other Powers of the -Nor: On the side of optiality easures, we note that Theore 13 generalizes to arbitrary powers of the Euclidean -nor [34]. That is, for D n,p := in σ n x 0 i yσi 0 p 3 it holds true that D n,p nlog n/n p. 4 Fig. 17. Ratio of total copletion tie between Strategies 3 and 5. Strategies 3 5. Using T n i to denote the T n for Strategy i, T n 4/T n 3 and T n 5/T n 3 are plotted in Figs. 16 and 17. As n increases, Strategies 4 and 5 both take uch less total copletion tie on average. VIII. CONCLUSION Focusing on the distance optiality for the target assignent proble in a robotic network setting, we have characterized a necessary and sufficient condition under which optiality can be achieved. We also provided a direct forula for coputing the nuber of robots sufficient for probabilistically guaranteeing such an optial solution. Then, we took a different angle; we looked at suboptial strategies and their asyptotic perforance as the nuber of robots goes to infinity. We showed that these strategies yield a constant approxiation ratio when copared with the true distance optial solution. Many of these decentralized strategies also provide coputational advantages over a centralized one. We conclude the paper by discussing our choice on certain eleents that can be generalized in a future work. Equal Nuber of Initial and Target Locations: In the proble stateent we assue that X 0 = Y 0.If X 0 > Y 0, soe robots do not need to ove and if X 0 < Y 0,soe robots ay need to reach ultiple targets, assuing that the ain goal is to serve the targets. Our result readily generalizes to the case in which X 0 / Y 0 is close to 1. When X 0 Y 0, it is likely that for a y i Y 0, there is a unique x i X 0 that is closest to y i [1]. Moreover, for two different y i,y j, x i x j. The spatial assignent proble then degenerates to Theore 13 corresponds to the special case of p =1.Asp, 3 iniizes the longest distance traveled by any robot. This is true because for fixed X 0, Y 0, and a sufficiently large p, the largest x 0 i y0 σi p becoes the doinating ter in the su n x0 i y0 σi p. Although we restrict our attention to p =1in this paper, our results readily extend to other values of p i.e., other optiality criteria with 4. Note that this eans the D n definition given by 5 needs to be updated accordingly to an appropriately defined D n,p. ACKNOWLEDGMENT The authors wish to thank the anonyous reviewers and S. Har-Peled and A. Nayyeri for their constructive coents. REFERENCES [1] S. L. Sith and F. Bullo, Monotonic target assignent for robotic networks, IEEE Trans. Auto. Control, vol. 54, no. 9, pp. 04 057, Sep. 009. [] J. Yu and S. M. LaValle, Distance optial foration control on graphs with a tight convergence tie guarantee, in Proc. IEEE Conf. Decision Control, 01, pp. 403 408. [3] D. P. Bertsekas, The auction algorith: A distributed relaxation ethod for the assignent proble, Annals Oper. Res., vol. 14, pp. 105 13, 1988. [4] D. P. Bertsekas and D. A. Castañon, Parallel synchronous and asynchronous ipleentations of the auction algorith, Parallel Cop., vol. 17, pp. 707 73, 1991. [5] R. Burkard, M. Dell Aico, and S. Martello, Assignent Probles. Philadelphia, PA: SIAM, 01. [6] J. Edonds and R. M. Karp, Theoretical iproveents in algorithic efficiency for network flow probles, J. ACM, vol. 19, no., pp. 48 64, 197. [7] H. W. Kuhn, The Hungarian ethod for the assignent proble, Naval Res. Logistics Quart., vol., pp. 83 97, 1955.
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Jingjin Yu S 11 M 13 received the B.S. degree in aterials science fro the University of Science and Technology of China, Hefei, China, in 1998, the M.S. degree in cheistry fro the University of Chicago, Chicago, IL, USA, in 000, the M.S. degree in atheatics fro the University of Illinois at Chicago, Chicago, IL, USA, in 001, and the M.S. degree in coputer science and Ph.D. degree in electrical engineering fro the University of Illinois at Urbana-Chapaign, Urbana, IL, USA, in 010 and 013, respectively. He is currently with the Coputer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cabridge, MA, USA. Soon-Jo Chung M 06 SM 1 received the B.S. degree with highest honors fro the Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 1998 and the S.M. degree in aeronautics and astronautics and the Sc.D. degree in estiation and control fro the Massachusetts Institute of Technology, Cabridge, MA, USA, in 00 and 007, respectively. He is currently an Assistant Professor in the Departent of Aerospace Engineering and the Coordinated Science Laboratory, the University of Illinois at Urbana-Chapaign UIUC, Urbana, IL, USA. He was a Suer Faculty Fellow with the Jet Propulsion Laboratory, working on guidance and control of spacecraft swars during the suers of 010 014. His research areas include nonlinear control and estiation theory and flight guidance and control, with application to aerial robotics, distributed spacecraft systes, and visionbased navigation. Dr. Chung received the UIUC Engineering Dean s Award for Excellence in Research, the Beckan Fellowship of the UIUC Center for Advanced Study, the U.S. Air Force Office of Scientific Research Young Investigator Award, the National Science Foundation Faculty Early Career Developent Award, and two best conference paper awards fro the IEEE and The Aerican Institute of Aeronautics and Astronautics. Petros G. Voulgaris F 11 received the Diploa in echanical engineering fro the National Technical University, Athens, Greece, in 1986, and the S.M. and Ph.D. degrees in aeronautics and astronautics fro the Massachusetts Institute of Technology, Cabridge, in 1988 and 1991, respectively. Since August 1991, he has been with the Departent of Aerospace Engineering, University of Illinois at Urbana Chapaign, where he is currently a Professor. He also holds joint appointents with the Coordinated Science Laboratory, and the departent of Electrical and Coputer Engineering at the sae university. His research interests include robust and optial control and estiation, counications and control, networks and control, and applications of advanced control ethods to engineering practice including flight control, nano-scale control, robotics, and structural control systes. Dr. Voulgaris received the National Science Foundation Research Initiation Award 1993, the Office of Naval Research Young Investigator Award 1995 and the UIUC Xerox Award for research. He has been an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the ASME Journal of Dynaic Systes, Measureent and Control.