Algorithms for Disk Covering Problems with the Most Points

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1 Algorithms for Disk Coverig Problems with the Most Poits Bi Xiao Departmet of Computig Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Qigfeg Zhuge, Yi He, Zili Shao, Edwi H.-M. Sha Departmet of Computer Sciece Uiversity of Texas at Dallas Richardso, Texas 7, USA {qfzhuge, yxh, zxs, ABSTRACT Usually the coverig problem requires all elemets i a system to be covered. I some situatios, it is very difficult to figure out a solutio, or uable to cover all give elemets because of resource costraits. I this paper, we study the issue of the partial coverig problem. This problem is also referred to the robust k-ceter problem ad ca be applied to may fields. The partial coverig problem becomes eve more harder whe we eed to determie the subset of the group of all available elemets to share resources. Several approximatio algorithms are proposed to cover the most elemets i this paper. For some real time systems, such as the battlefield commuicatio system, the algorithm preseted with polyomial-time complexity ca be efficietly applied. The algorithm complexity aalysis illustrates the improvemet made by our algorithms, which are compared with other papers for the partial coverig problem i the literature. The experimetal results show that the performace of our algorithms is much better tha other existig -approximatio algorithm for the robust k-ceter problem. KEY WORDS Approximatio algorithms, partial coverig, k-ceter problem. Itroductio I the battlefield, there are a lot of commuicatio uits ad some commad ceters. The commad ceters are resposible for the successful iformatio exchage amog all mobile uits. Because of the mobility for both mobile uits ad ceters, we try to cover the most mobile uits by a limited umber of cotrol ceters with trasmissio rage costraits. This kid of problem is aalyzed as the clusterig problem, which is clusterig a set of poits ito a few groups. Clusterig algorithms have bee explored ad deployed i may fields [, ], such as data compressio, iformatio retrieval, databases applicatios, image processig, facility locatio, clusterig odes i ad hoc etworks etc. Give that the group umber is k, the clusterig problem is also referred to the k-ceter problem. We defie the k-ceter problem as follows: Let S be a set of objects, This work is partially supported by TI Uiversity Program, NSF EIA- 7 ad Texas ARP 7--, USA. geerally represeted as poits i a d-dimesioal metric space. Give a iteger k, compute a k-clusterig of S of the smallest possible size. I other words, the k- ceter problem is formulated by coverig S by k cogruet disks of the smallest possible size. We always assume that k disks have the same radius r. I this paper, we oly pay attetio to the metric space by a plae (d =. However, all algorithms ca be easily exteded to arbitrary metrics. May heuristic algorithms [,, ] have bee studied well for the k-ceter problem whe k. There are two major directios. Oe directio is for the umber of disks is fixed to k ad heuristic algorithms try to miimize the radius r of disks. The other focuses o the radius of each disk is fixed to r while heuristic algorithms explore miimum umber of disks to cover all poits. For the first approach, Gozalez [6] gave a -approximatio algorithm for the k- ceter problem i ay metric space with time complexity O(k ). I the same paper, Gozalez proved that there is o polyomial-time algorithm for a approximatio factor smaller tha uless P=NP. Feder ad Greee improved the -approximatio algorithm with complexity O( log k) [7]. Some (+ɛ) approximatio algorithms [7, ] are studied with o-polyomial ruig time. For the secod approach, a polyomial-time approximatio scheme ca be achieved withi approximatio factors arbitrarily close to [7]. Gozalez proposed a -approximatio algorithm for the fixed-size disks coverig problem i []. Huag devised a 7-approximatio algorithm for the -dimesio metric space, ad a -approximatio algorithm for the - dimesio metric space []. I [], Fraceschetti summarized the best kow results i a table. However, to achieve smaller approximatio factor (α < 7), the polyomial ruig time will be very huge. For example, the algorithm i [] for a approximatio factor α = 6 will require the ruig time to be O(k ) with k 6. A topic has bee studied durig these years for the k-ceter problem is to cover part poits [, ]. I some cases, there is o solutio for coverig poits i a plae by k disks with fixed radius. Examples are like buildig facilities to provide service withi a fixed radius to a certai fractio of populatio, or allocatig commad ceters i a battlefield to support commuicatios amog mobile uits. The problem defied i [] is called the robust k-ceter problem, which is to cover at least p poits (p ) by k disks with radius r. I this paper, we wat to cover the

2 most poits (at least p poits) for poits i a plae by k available disks. Those k disks are assumed with the same radius r. This kid of problem is NP-complete problem. The reasos is that to cover at least p poits, whe we set p =, it will be the same to the k-ceter problem. I order to cover as may poits as possible (at least p from ) with k disks, i this paper we have made the cotributios as follows: Come up with a ew -approximatio algorithm RKC to the robust k-ceter problem. I [], the authors oly preseted the best result with a - approximatio algorithm. Whe p is close to, this ew -approximatio algorithm becomes polyomialtime ruig. (Sectio Propose a greedy algorithm to cover part poits. (Sectio.) Modify the RKC algorithm to the RKCP algorithm for the purpose of coverig most poits. (Sectio. Trasfer the -approximatio algorithm for the robust k-ceter problem i [] to the RKCP algorithm, which is also applicable to the problem of coverig most poits. (Sectio.) The rest of this paper is orgaized as follows. I Sectio we itroduce the RKC algorithm for the robust k- ceter problem ad prove it to be -approximatio. Sectio shows differet heuristic algorithms to cover the most poits ad oe example is illustrated. Experimetal results are preseted i Sectio. Ad coclusios are draw i Sectio. New RKC Algorithm The robust k-ceter problem defied i this paper oly cosiders poits i a plae (the dimesio is. Let be the umber of all poits, p be a give positive iteger such that p. If we have k same disks with radius r, the robust k- ceter problem ca be defied as whether k disks ca cover at least p poits. This problem is a NP-complete problem. Suppose that poits i a plae area are the cliets we wat to serve, the ceter of k disks are the facility locatio to cover p poits amog poits, the the questio studied here is the same as the facility locatio problem with outlier defied i []. Figure illustrates how robust measures lead to better clusterig solutios. I the example, to cover all poits requires at least k = disks. However, if we eed to cover the most poits with oly disks, Figure (b) shows the result to cover p = poits. Followig the same defiitio of algorithm approximatio factor i [], we show our RKC algorithm below ad prove it to be -approximatio to the optimal cost. For some cases, we ca eve kow that it is impossible to cover p poits by k disks. Let V be the set of all poits i a plae (a) Figure. (a) To cover all poits with k=; (b) To cover p poits with k=, p=. ad V =. We suppose that the set of poits satisfies the triagular iequality. p is a give positive umber ad p. Let C be a set i our algorithm that icludes all poits to be the ceters of k disks. T i is a temporary poit set that icludes p poits from V. Each disk has the same radius r. We use Flag below to show whether there is a solutio for k disks to cover ay p poits withi the poit set V. The dist(a, b) is the Euclidea distace betwee the poit a ad b. The algorithm RKC is as follows: Flag = No For i =,..., (, do Select p differet poits from V, which geerates a ew poit set T i with T i T j (j =,..., i-) Arbitrarily select oe poit from T i, let this poit be c, C = {c } For j =,..., k, do For a poit t T i ad t / C, let d j (t) = mi[dist(t, c l ), for c l C] Let d j = max t Ti d j (t) Let c j be the poit t, which makes d j to have the maximum value C = C {c j } If d k r (b) If k disks with ceters i C by radius r ca cover at least p poits i V Retur Yes Else Flag = Yes If Flag = No It is impossible to cover p poits with k disks Retur No We explai how this algorithm works. First, Flag is reset to be No. Wheever it is possible to cover p poits with the give k disks for a particular T i by the above algorithm, Flag is set to Yes. I the outer loop, p differet poits from V are selected i every iteratio. Those p poits yield a ew poit set T i. Because there are ( ) p times of differet poit sets, this outer loop will ot ed util i = (, which meas we already check all possible

3 solutios. For every selected poit set T i, it oly icludes p poits. Subsequetly, oe ode is arbitrarily chose from T i ad this poit becomes the ceter of oe disk. We wat to fid the other k ceter positios for disks. Such k poits costruct the ceter set C. The ext ceter icluded to C is the poit i T i that has the maximum distace from all the poits already i C. Whe there are k poits i C, we already fid k ceters by the RKC algorithm for the p poits arbitrarily chose to T i. d k meas the maximum distace from c k (the kth ceter) to every other ceters (c,..., c k ). I other words, if k disks have the diameter by d k, it is eough for them to cover all poits i T i. If d > r, it is impossible for k disks with radius r to cover all poits i T i, which will be proved by Lemma.. Otherwise, Flag should be set to Yes to show the existece for all poits i T i covered oly by k disks. Furthermore, if k disks with ceters i C by radius r ca cover p poits i V, the RKC algorithm already reach a solutio to the robust k-ceter problem ad is eded by the retur of Yes. The Theorem. will prove it. After ( cases are tested ad Flag is still No, it guaratees that there is o way to cover p poits from V by k disks with radius r. Sice we still ca ot fid a method to cover p poits, the algorithm will retur No. I the RKC algorithm, the process to geerate a ceter set C with give p poits is similar to the greedy algorithm i [6], which has bee proved to be a - approximatio algorithm. The idea is to execute the mai loop aother iteratio, ad let d k+ be the resultig distace to the k + ceter to be added. Also the ew ceter poit set C becomes C {c k+ } that has k + poits i it. By the defiitio of d k+, we ca see that the distace betwee ay two poits i C is at least d k+. Furthermore, ay k-clusterig must have two poits of C i oe same cluster for C = k +. Thus, ay k-clusterig of T i must have radius o less tha d k+ /. However for the RKC algorithm, the radius of the k-clusterig with ceters at C is exactly d k+. The we have the lemma below: Lemma.. Give p poits i T i, the RKC algorithm provides a factor approximatio to the miimum k-clusterig of T i. I other words, the radius by disks with ceters at poits i C is o larger tha times the radius of ay k clusterig disks of T i. Theorem.. Give a set V of poits from a arbitrary metric, a iteger k, ad a iteger p, the RKC algorithm is a -approximatio algorithm for the robust k- ceter problem. Proof. From Lemma., it is obvious that the RKC algorithm geerates a -approximatio solutio whe d k r happeed durig the algorithm executio. This is because whe d k r, the k disks with radius r ca cover at least p poits. For the case that every time d k > r durig p poits selectio ( ( times) from V, it is impossible for k disks to cover the etire p poits with radius r. That meas o solutio exists. Thus the RKC algorithm ca ot fid a ceter set C with radius r to cover p poits. a O * O * *O b Figure. To cover poits with a disk by the greedy algorithm. Give p poits i T i, to geerate k ceters requires O(p k) time. A better way to reduce the time complexity to O(p log k) is show i [7]. Sice the RKC algorithm will execute ( times for the differet poit set Ti, the time complexity should be O( ( p log k). Whe we wat to cover the most poits, p should be the same as ad the time complexity for the robust k-ceter problem becomes O( log k). It is a polyomial-time algorithm. Furthermore, whe p is close to, the RKC algorithm ca still remai polyomial-time ruig. Differet Algorithms to Cover the Most Poits Give poits i a plae, sometimes it is impossible for all poits to be covered withi k disks by radius r. Eve if such cases exist, to fid a solutio util ow will require the computatio time to be expoetial, sice k-ceter problem is NP-complete. I the real-time system, such as coverig all commuicatio uites i a battlefield, this is too costly. To cover at least p poits withi k disks is also NPcomplete. The reaso is that whe we set p =, the robust k-ceter problem becomes the stadard k-ceter problem. I this sectio, we show some polyomial-time algorithms to cover poits as may as possible.. Greedy Algorithm For poits i a plae to be covered by disks with radius r, there are at most ( differet cases for disk coverig. This is because for ay two poit a, b, there are at most two ceter positios, from which the distace to both poits (a ad b) are exact r. I Figure, there are differet disk coverig for the poit a ad b, which ca be disk o ad o. Because disk o covers the same poits as o, it is ot ecessary for us to pay attetio to it ay more. I other words, we always ca move a disk to uiquely represet two poits with those poits o its circle edge. For the poit c ad d i Figure, there is oly oe case for disk coverig sice the distace betwee c ad d is r (the diameter). Oe disk coverig is said to be uique oly if there is at least oe poit differet from ay other disk coverig. Let D be the set with differet disks coverig ad G be the disk set retured by the greedy algorithm that oly has k uits. Below is the greedy algorithm to cover the most poits. O *

4 G = For ay two poits, we geerate two ew disk uits to cover them if possible. Suppose there are m such uique disk coverig with m (. These disks come up with a disk set D = {D, D,..., D m }. For i =,..., k, do Select the disk that covers the maximum umber of poits from D. Let this disk be D i, D = D {D i }, G = G {D i }. Remove the poits covered by D i from all disk i D. Retur G. There are at most ( differet disk coverig give ( poits. Costructig such disks will cosume time O( ) ). I oe iteratio, the time to select the disk that covers the maximum umber of poits eeds time O( ( ). After icludig D i to be i disk set G, the greedy algorithm will refresh the umber of poits covered by all other remaiig disks i D. Thus it requires a extra time O( ( ). I the greedy algorithm, we have k iteratios. The computatio complexity for the greedy algorithm should be O( ( )+ k O( ( + ( ) = O( ( ( )+O(k ) = O(k ).. RKCP Algorithm The algorithm for the robust k-ceter problem does t request to cover all poits with give k disks. With a little chage to the RKC algorithm to solve the robust k-ceter problem i Sectio, we ca apply it to the case of coverig the most poits. Here we assume p =, which meas the algorithm tries to cover all available poits. All poits are i the poit set V ad C is a poit set icludig k uits, which are the positios for the ceter of k disks. Below is the RKCP algorithm procedure to cover the most poits from the -approximatio algorithm RKC for the robust k-ceter problem. Arbitrarily select oe poit from V, let this poit be c, C = {c } For i =,..., k, do For a poit t V ad t / C, let d i (t) = mi[dist(t, c l ), for c l C] Let d i = max t V d i (t) Let c i be the poit t, which makes d i to have the maximum value C = C {c i } Retur C The time complexity for the above RKCP algorithm ca be O( log k) [7] (a) 6 (b) 7 6 Figure. (a) poits ca be covered by 6 disks with radius m; (b) The result by the greedy algorithm.. RKCP Algorithm I [], the authors illustrate a -approximatio algorithm for the robust k-ceter problem, which ca also be adopted to solve the problem of coverig the maximum umber of poits. That -approximatio algorithm has the iput with a radius r for k disks ad a set of poits V from a arbitrary metric space. It will figure out a solutio S such that cost(s) r. For the poits V, it assumes that there exists a optimal solutio O such that cost(o) = r. For each poit v i V, G i (E i, resp.) is deoted as the set of poits that are withi distace r (r, resp.) from v i. The set G i is referred as disks of radius r ad the set E i as the correspodig expaded disks of radius r. The RKCP algorithm procedure to cover the most poits ca be described as follows with a little modificatio to the origial -approximatio algorithm i []. Costruct all disks ad correspodig expaded disks. For i =,..., k, do Let G i be the heaviest disk, i.e. cotais the most ucovered poits. Mark as covered all poits i the correspodig expaded disk E i. Update all the disks ad expaded disks, i.e., remove from them all covered poits. Retur {G, G,..., G k }. The algorithm above has bee proved to be a - approximatio algorithm for the robust k-ceter problem 6

5 (a) 7 6 We evaluate the performace of differet algorithms through coverig differet umber of poits scattered i a area by disks. The performace of each algorithm is measured by the umber of poits covered by give disks ad how may disks are used for the system. I the simulatio, we compare the results for differet algorithms. The positio of all poits are radomly distributed i a area by, *, meters. The umber of poits is varied from to durig the simulatio. All disks have the same radius by meters. Durig the simulatio, the ceter positio of a disk ca be aywhere iside the square area. Differet umber of disks is assiged accordig to how may poits i all. The relatioship betwee the umber of poits ad disks follows the equatio below. D represets how may disks available i the system while P deotes the umber of poits iside the simulated square area. 6 6 (b) Figure. (a) The result by the RKCP -approximatio algorithm; (b) The result by the RKCP -approximatio algorithm. i []. The time complexity for the algorithm should be O(k ).. Example for the Above Algorithms To cover the most poits i a plae, we show a example here applied with differet algorithms metioed above. Give poits radomly scattered i a square by, meters *, meters, a optimal solutio for all poits covered by 6 disks is show i Figure (a). The results with the greedy algorithm are illustrated i Figure (b). With the RKCP ad RKCP algorithms to cover the most poits, we have the results show i Figure (a) ad Figure (b) respectively. I the greedy algorithms, the ceter positio of oe disk is determied by poits o its edge. For the RKCP algorithm to cover the most poits, we show a bigger circle with radius 7 meters for every coverig disk i Figure (b). The umber of poits covered is couted by those i the small disks (with radius meters). All poits that are betwee oe small disk ad its correspodig big circle are removed from the system by the -approximatio algorithm. Followig this property, poit 7 is ot covered by the algorithm because it is first removed from the system after the decisio made for the first disk. Simulatio Results 6 D = P + () I Table, the performace of the greedy, - approximatio (RKCP ad -approximatio (RKCP) algorithms are listed with the model defied above. Iside the square,,,, ad poits are radomly scattered for oe simulatio evet respectively. Uder Colum # p, the data shows the umber of poits covered by a special algorithm ad the followig Colum % represets the percetage of the coverig part based o all available poits. Durig the simulatio, the umber of available disks is chaged accordig to the umber of all poits i the system, which is defied i Equatio. To cover the poits as may as possible, differet algorithms will cosume differet umber of disks. The data i the Colum # d is the result of used disks by the executio of those three algorithms. For the system with poits, there are 6 disks available. However, the greedy algorithm ca cover all poits oly with disks for some cases. The RKCP algorithm from the -approximatio algorithm i [] is ot efficiet because it oly uses part of available disks. We ca see from Table that whe the system has poits, disks with radius 7 meters ( meters * ) is probably eough to cover them all. O the cotrary, the - approximatio algorithm always tries to cover poits by all disks. Uder a few circumstace, such as less umber of poits i the area ad fewer disks available, the RKCP algorithm yields a better performace tha the RKCP algorithm, which ca be see from Table for the system with poits. Most of the time, the greedy algorithm performs the best while the -approximatio algorithms geerate the worst results. We show the simulatio results for differet algorithms i Figure whe the umber of poits icreases from to. The curves i Figure represet the umber of covered poits by differet algorithms. That straight lie represets the total poits i the simulatio eviromet. Whe a algorithm has a better performace to cover more poits, the simulated curve will be closer to the straight lie. Thus, the greedy algorithm always yields a good performace, especially for a large umber

6 Algo. poits poits poits poits poits # p % # d # p % # d # p % # d # p % # d # p % # d Greedy Appr Appr Table. Covered poits ad used disks. Poits covered by differets algorithms All poits Greedy Appr Appr Number of poits Figure. The umber of covered poits. of poits scattered i the square area. The umber of covered poits goes up for all three algorithms whe there are more poits available. But the icreasig rate is slow for the -approximatio algorithm. Whe poits are scarcely scattered i the simulatio area, the -approximatio algorithm sometimes performs better tha the -approximatio algorithm. However i the log ru, the -approximatio algorithm geerates much better results. Coclusio I daily life or some special circumstaces (such as the battlefield), it is very hard or impossible to cover all users with limited servers. Thus to cover most users will be a satisfyig result. This problem belogs to the robust k-ceter problem. I this paper we preseted a ew - approximatio algorithm (RKC for the robust k-ceter problem, which is polyomial-time ruig whe p is close to. Based o it, a real polyomial-time algorithm (RKCP is proposed to cover the most poits. Amog those three proposed heuristic polyomial-time algorithms (greedy, RKCP, RKCP) for the robust k-ceter problem to cover the most poits, the greedy algorithm yields a good performace idicated by the simulatio. The outcome of the -approximatio (RKCP) algorithm origially from [] is poor because the covered poits is below % for most cases. Whe more disks are available, the polyomial -approximatio algorithm (RKCP yields close results as the greedy algorithm while the ruig time is O( log k). [] D. B. Shmoys, É. Tardos, ad K. Aardal, Approximatio algorithms for facility locatio problems (exteded abstract), i Proceedigs of the th Aual ACM Symposium o Theory of Computig, pp., 7. [] P. K. Agarwal ad C. M. Procopiuc, Approximatio algorithms for projective clusterig, i Proceedigs of th ACM-SIAM Sympos. Discrete Algorithms, pp. 7,. [] D. Johso, Approximatio algorithms for combiatorial problems, Joural of Computig ad Systems Scieces, vol., pp., 7. [] H. Broiamm ad M. Goodrich, Almost optimal set covers i fiite vcdimesio, Discrete Computatioal Geometry, vol., pp. 6 7,. [6] T. Gozalez., Clusterig to miimize the maximum itercluster distace, Theoretical Computer Sciece, vol., pp. 6,. [7] T. Feder ad D. Greee, Optimal algorithms for approximate clusterig, i Proceedigs of the th Aual ACM Symposium o the Theory of Computig (STOC), pp.,. [] P. K. Agarwal ad C. M. Procopiuc, Exact ad approximatio algorithms for clusterig (exteded abstract), i Proceedigs of th ACM-SIAM Sympos. Discrete Algorithms, pp ,. [] T. Gozalez., Coverig a set of poits i multidimesioal space, Iformatio Processig Letters, vol., pp.,. [] H. Huag, A. W. Richa, ad M. Segal, Approximatio algorithms for the mobile piercig set problem with applicatios to clusterig i ad-hoc etworks, ACM Joural o Mobile Networks (MONET), pp. 6,. [] M. Fraceschetti, M. Cook, ad J. Bruck, A geometric theorem for approximate disk coverig algorithms,. [] M. Charikar, S. Khuller, D. M. Mout, ad G. Narasimha, Algorithms for facility locatio problems with outliers, i Proceedigs of the Twelfth Aual ACM-SIAM Symposium o Discrete Algorithms, pp. 6 6,. [] R. Gadhi, S. Khuller, ad A. Sriivasa, Approximatio algorithms for partial coverig problems, i Proceedigs of the Twety-Eighth Iteratioal Colloquium o Automata, Laguages, ad Programmig (ICALP), LNCS 6, pp. 6, Jul.. Refereces [] D. Z. (Editor), Facility locatio: A survey of applicatios ad methods, Spriger Series i Operatios Research, Spriger Verlag, New York,.