A linear time approximation scheme for computing geometric maximum k-star

Size: px

Start display at page:

Download "A linear time approximation scheme for computing geometric maximum k-star"

Nathan Alfred Fleming
6 years ago
Views:

1 DOI /s A linear time approximation scheme for computing geometric maximum k-star Jia Wang Shiyan Hu Received: 11 April 2011 / Accepted: 1 February 2012 Springer Science+Business Media, LLC Abstract Facility dispersion seeks to locate the facilities as far away from each other as possible, which has attracted a multitude of research attention recently due to the pressing need on dispersing facilities in various scenarios. In this paper, as a facility dispersion problem, the geometric maximum k-star problem is considered. Given a set P of n points in the Euclidean plane, a k-star is defined as selecting k points from P and linking k 1 points to the center point. The maximum k-star problem asks to compute a k-star on P with the maximum total length over its k 1 edges. A linear time approximation scheme is proposed for this problem. It approximates the maximum k-star within a factor of (1 + ɛ) in O(n + 1/ɛ 4 log 1/ɛ) time for any ɛ>0. To the best of the authors knowledge, this work presents the first linear time approximation scheme on the facility dispersion problems. Keywords Geometric maximum k-star Approximation Linear time approximation scheme Facility dispersion 1 Introduction Facility location has been an important research topic in network design with various applications [16]. Facility dispersion seeks to locate the facilities as far away from each other as possible, which has critical applications in many scenarios. For example, it is desired to locate oil storage tanks at a large distance from each other to mitigate the impact due to terrorist attacks, to locate nuclear power plants to be remote from residence area for the quality of life, and to locate fire stations to be sparsely distributed to maximize the service range [7,15,21]. J. Wang S. Hu (B) Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, MI 49931, USA shiyan@mtu.edu J. Wang jiaw@mtu.edu

2 In the previous work [3,7 11,14,18,21,22,24,25], facility dispersion problems, such as remote clique, remote star and remote tree, have been extensively studied under various names. Given a graph, a facility dispersion problem usually asks to compute an induced subgraph such that certain distance measure is maximized. For example, given a graph of n vertices, remote k-star problem seeks to compute a complete subgraph induced from a set of k vertices such that the total edge length of the star is maximized. Many disperse facility location problems are NP-hard and most previous works are focused on proposing heuristics or approximation algorithms on them (refer to [5,19] for surveys). For example, an O(n 2 + k 2 log k)-time 2-approximation algorithm for the remote k-star problem is proposed in [7] based on maximum weight k-matching in [12]. More optimization developments in telecommunication can be found in [20]. In this paper, a closely related problem, namely, the geometric maximum k-star problem, is considered. Given a set P of n points in the Euclidean plane, a k-star is defined as selecting k points from P and linking the k 1 points to the center point, which forms k 1 edges. The maximum k-star problem asks to compute a k-star on P with maximum total edge length over the k 1 edges. This problem is closely related to the remote k-star problem studied in [7], which seeks to find k vertices such that the total length of the minimum star (i.e., the star with the smallest total length) on them is maximized. In contrast, one can treat our maximum k-star problem as computing k vertices such that the total length of the maximum star (i.e., the star with the largest total length) on them is maximized. While remote k-star problem is of its own interest, the maximum k-star problem is a more natural problem since one eventually needs to construct a solution for dispersing the locations of facilities and our algorithm computes it. For example, given n candidate locations, the solution to the maximum k-star problem would allow us to nicely locate k fire stations, with one coordinating centralized station surrounded by the others, such that the service area is maximized in the sum-of-length sense. In contrast to the fact that most facility dispersion problems are NP-hard, the geometric maximum k-star problem can be solved in quadratic time. However, for a large data set, the quadratic algorithm could be still very time consuming in practice. This motivates us to design a linear time approximation scheme for the problem. Given a positive number ρ, an algorithm is said to approximate a maximization problem within the factor of ρ if the algorithm is guaranteed to always produce a solution whose objective function value is at least 1/ρ times the optimal solution. For a maximization problem, a linear time approximation scheme is an algorithm which approximates the optimal solution within a factor of (1 + ɛ) for any ɛ>0 in time linear in the problem instance size and polynomial in 1/ɛ. Dueto its practicality, there are various research works focused on designing linear time approximation scheme. Recent works include the linear time approximation schemes for TSP in undirected planar graph [13], for maximum weight triangulation on a convex polygon [17], for minimum-radius enclosing cylinder, and for minimum-volume bounding box [1,2,6]. In this paper, a linear time approximation scheme is proposed for the geometric maximum k-star problem on a planar point set based on the idea of input structuring [23]. It approximates the optimal solution within a factor of (1 + ɛ) in O(n + 1/ɛ 4 log 1/ɛ) time for any ɛ>0, where n is the number of points. To the best of the authors knowledge, this work is the first linear time approximation scheme on the facility dispersion problems. The rest of the paper is organized as follows: Sect. 2 formulates the geometric maximum k-star problem and presents the quadratic time exact algorithm. Section 3 describes the linear time approximation scheme for the geometric maximum k-star problem. A summary of work isgiveninsect.4.

3 2 Preliminaries In the geometric maximum k-star problem, a set P of n points in the Euclidean plane is given. A k-star is defined as selecting k points from P and linking the k 1 points to a center point, which forms k 1 edges. The maximum k-star problem asks to compute a k-star on P with maximum total edge length over the k 1 edges. Let denote the operation of computing the total length. We first note that the optimal k-star can be computed as follows. Tentatively treat each point p in P as a center. Find k 1 points from the remaining points in P which are furthest to the center. One can compute all n 1 distances to the center and pick the longest k 1 edges, which can be accomplished in O(n) time by selecting the (k 1)-th longest distance in O(n) time [4]. Repeat this process for each of the n possible centers, and the one with the maximum total length over all nk-star gives the optimal solution. This algorithm takes O(n 2 ) time. When k = (n), it seems difficult to design a faster algorithm. The above simple quadratic algorithm could be time consuming to perform for some large data sets. This motivates us to design a linear time approximation scheme for the geometric maximum k-star problem. 3 The algorithm 3.1 Algorithmic flow At a high level, our algorithm works as first performing a translation and scaling procedure to restrict all points to be within the unit square and the length of the longest edge to be within [1, 2]. This allows us to upper bound and lower bound the total length of the optimal k-star. One then performs a rounding based approximate transform to reduce the complexity of the point set to be independent of n but dependent on ɛ, the target approximation ratio. This allows us to compute an optimal k-star in time independent of n. Together with the fact that the approximate transform introduces only small error (controlled by ɛ),a 1 + ɛ approximation is obtained. The details are as follows. 3.2 Translation and scaling The point set P is first translated such that all the points are in the first quadrant and the minimum axis-parallel bounding box is aligned with x axis and y axis. Thus, there are points along x axis and y axis after translation. Let maxx denote the maximum coordinate of P along x dimension, and maxy denote the maximum coordinate of P along y dimension. One then scales the x-coordinate and y-coordinate of each point by max{maxx, maxy}. Clearly, all the points are within the area of [0, 1] [0, 1] after scaling. Refer to Fig. 1 for an illustration. Note that this uniform scaling will not impact the optimality of the solution. This process certainly takes O(n) time. Denote by OPT the optimal k-star. Since all the points are within a unit square and there are points along the horizontal or vertical boundary of the bounding box, the furthest pair of P has length at least 1 and at most 2. An upper bound on OPT can be obtained as OPT 2(k 1). On the other hand, OPT (k 1)/2 since one can always find a k-star with one of the furthest pair as the center and with total edge length at least (k 1)/2 as follows. Arbitrarily pick k 2 points among all n 2 remaining points. Construct two

4 Fig. 1 Scaling of the point set P Fig. 2 One of the two k-stars has length at least (k 1)/2 Fig. 3 Rounding all points in P to lattice points. Each grid has size of ɛ and each mapped lattice point has multiplicity 1 except the one with multiplicity 2 as shown explicitly stars by linking them to one point of the furthest pair, and by linking them to the other point, respectively. Due to triangle inequality, the star with the larger total edge length has length at least (k 1)/2 by noting that the furthest pair and each of the k 2 points forms a triangle. Refer to Fig. 2 for an illustration. We reach the following lemma. Lemma 3.1 After the above scaling procedure, the length of the optimal k-star OPT on a set of P satisfies (k 1)/2 OPT 2(k 1). 3.3 Rounding Subsequently, based on the idea of input structuring [23], given any positive ɛ, one roundsthe x and y coordinates of each point by ɛ. Precisely, they are rounded to the nearest multiples of ɛ. After rounding, there are at most O(1/ɛ) distinct x-coordinate and at most O(1/ɛ) distinct y-coordinate. Consequently, there are at most O(1/ɛ 2 ) distinct points. One can also treat this as laying down a lattice to the plane and rounding each point in P to a mapped lattice point. Since multiple points in P can be rounded to the same mapped lattice point, record them together with the multiplicity (which is the number of points in P rounded to a mapped lattice point) at each mapped lattice point. Refer to Fig. 3 for an example of rounding. This rounding process takes O(n) time. 3.4 Computing geometric maximum k-star The optimal maximum k-star on the transformed point set (i.e., mapped lattice points) is to be computed. The procedure is different from the quadratic algorithm for computing the

5 optimal k-star since the multiplicity needs to be handled. Treat each of mapped lattice points as a possible center point. For each center, denoted by p c, one first computes the total length of an approximate k-star without explicitly constructing it. This allows to design an algorithm only having additive terms to n in time complexity. Set a counter c to record the size of the current partial k-star and is initialized to 1 due to the center. Set a length l to record the total length of the current partial k-star and is initialized to 0. Note that if the multiplicity of the center is greater than 1, the points there will be treated as those closest to the center. First sort in the non-increasing order the O(1/ɛ 2 ) mapped lattice points according to the distance to the center p c to obtain a candidate list, denoted by L(p c ). The main part of the algorithm is an iterative procedure. Each time, imagine that the original points corresponding to the current furthest mapped lattice point p are included into the partial k-star. Denote the distance from p to the center p c by l(p,p c ), and the multiplicity of p by m(p). Subsequently, p is removed from the candidate list L(p c ), l is updated by adding l(p,p c ) m(p), andc is updated by adding m(p). This process is iterated until the the counter c reaches k. For example, supposethat in Fig. 3, the point corresponding to the multiplicity of 2 is picked and the distance between this point and the center is 1. If currently l = 10 and c = 5, they will be updated as l = = 12 and c = = 7. Note that in the last iteration, if the counter c would be greater than k by adding m(p), one can update l to l + (k c) l(p, p c ) and then update c to k. For this, one can treat it as arbitrarily picking k c original points from P corresponding to the mapped lattice point p in forming the k-star. Given a center p c, the time complexity of the whole process is bounded by sorting O(1/ɛ 2 ) mapped lattice points according to the distance to the center, which takes O(1/ɛ 2 log 1/ɛ) time. The above process is performed with each mapped lattice point as a center and find the center which gives the maximum total length. This takes O(1/ɛ 4 log 1/ɛ) time. Note that during the above process, only the multiplicities and edge lengths are involved, and the original points corresponding to the picked mapped lattice points are not computed. This is due to the fact that computing corresponding original points needs O(k) time per center and O(k/ɛ 2 ) time for all centers, which will introduce the term of O(n/ɛ 2 ) to the total time complexity when k = (n). In contrast, after finding the center which gives the maximum total length, one can compute the corresponding k-star of P similar to the above procedure. Initialize c = 1. First sort O(1/ɛ 2 ) mapped lattice points according to the distance to the computed center p c to obtain L(p c ). Each time, the original points corresponding to the current furthest mapped lattice point p are included into the partial k-star. Subsequently, p is removed from L(p c ) and c is updated by m(p). This process is iterated until c reaches k. In the last iteration, if c would be greater than k by adding m(p), one can arbitrarily pick k c original points corresponding to p in forming the k-star. This takes (k + 1/ɛ 2 log 1/ɛ) time. Including the time for scaling and rounding, the total time complexity is O(n + 1/ɛ 4 log 1/ɛ). Refer to Table 1 for the pseudocode of the algorithm. 3.5 Approximation ratio and time complexity Denote by ALG the k-star computed by our algorithm. The difference between OPT and ALG, which is due to the rounding in the approximate transform, is to be bounded. For each edge formed by the points in P, its length is changed by at most 2ɛ by the approximate transform since the distance between each point in P and its mapped lattice point is at most (ɛ/2) 2 + (ɛ/2) 2 = ɛ/ 2. This gives the maximum rounding error for

6 Table 1 A linear time approximation scheme for the geometric maximum k-star problem Geometric Maximum k-star Input: P : a set of points, k: a positive number no greater than P Output: a (1 + ɛ) approximation for the maximum k-star on P 1. translate and scale P to be within the unit square such that the length of the longest edge to be within [1, 2] 2. round the x and y coordinates of each point to the nearest multiples of ɛ 3. for each mapped lattice point p c 4. treat p c as the center 5. initialize c = 1andl = 0 6. sort all other mapped lattice points in decreasing order according to the distance to p c to obtain L(p c ) 7. while c<k 8. update l by l(m(l(p c )[1]), p c ) m(p), update c by m(l(p c )[1]), and remove L(p c )[1] from L(p c ).Ifc would be greater than k, update l to l + (k c) l(l(p c )[1],p c ) and then update c to k 9. find the maximum total length, i.e., l and its center, denoted by p c 10. initialize c = 1andK = 11. sort all other mapped lattice points in decreasing order according to the distance to p c to obtain L(p c ) 12. while c<k 13. add points in P corresponding to L(p c )[1] to K, update c by multiplicity of L(p c )[1], and remove L(p c )[1] from L(p c ). If c would be greater than k, arbitrarily add k c points in P corresponding to L(p c )[1] to K and update c to k 14. return K each edge. Summing over all k 1 edges, the rounding error on the total edge length is bounded by 2(k 1)ɛ. Thus, for every possible k-star on P (denote its length by T ), there exists a corresponding k-star on P with rounded locations (i.e., when the points are rounded to mapped lattice points) and with total length at least T 2(k 1)ɛ. This applies to the optimal k-star on P. On the other hand, the computed k-star is optimal on P with rounded locations. These lead to that the total length of the computed k-star is at most 2(k 1)ɛ away from that of the optimal k-star on P without rounding. Due to Lemma 3.1, (k 1)/2 OPT which means that 2(k 1)ɛ 2 2ɛ OPT. Thus, ALG OPT 2 2ɛ OPT. This means that the obtained k-star is a ( ɛ) approximation for the optimal solution. Substituting 2 2ɛ by ɛ, there is an algorithm which approximates the optimal k-star within the factor of 1+ɛ and runs in O(n+1/ɛ 4 log 1/ɛ ) time. We reach Theorem 3.2. Theorem 3.2 The maximum k-star on a planar point set can be approximated within a factor of (1 + ɛ) in O(n + 1/ɛ 4 log 1/ɛ) time for any ɛ>0, wheren is the number of points.

7 4Conclusion This paper considers a global optimization problem, namely, the planar geometric maximum k-star problem. Given a set P of n points in the Euclidean plane, a k-star is defined as selecting k points from P and linking k 1 points to a center point. The maximum k-star problem asks to compute a k-star on P with the maximum total length over its k 1 edges. In this paper, a linear time approximation scheme is proposed, which approximates the maximum k-star within a factor of (1 + ɛ) in O(n + 1/ɛ 4 log 1/ɛ) time for any ɛ>0. References 1. Agarwal, P.K., Varadarajan, K.R., Procopiuc, C.M.: Approximation algorithms for k-line center. Algorithmica 42(3), (2005) 2. Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. ACM 51(4), (2004) 3. Baur, C., Fekete, S.P.: Approximation of geometric dispersion problems. Algorithmica 30(3), (2001) 4. Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), (1973) 5. Cappanera, P.: A survey on obnoxious facility location problems. Technical Report: TR-99-11, University of Pisa (1999) 6. Chan, T.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. Theory Appl. 35, (2006) 7. Chandra, B., Halldorsson, M.M.: Approximation algorithms for dispersion problems. J. Algorithms 38(2), (2001) 8. Erkut, E.: The discrete p-dispersion problem. Eur. J Oper. Res. 46(1), (1990) 9. Erkut, E., Neuman, S.: Comparison of four models for dispersing facilities. INFOR: Inf. Syst. Oper. Res. 29(2), (1991) 10. Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), (2001) 11. Fekete, S.P., Merjer, H.: Maximum dispersion and geometric maximum weight cliques. Algorithmica 38(3), (2004) 12. Hassin, R., Rubinstein, S., Tamir, A.: Approximation algorithms for maximum dispersion. Oper. Res. Lett. 21, (1997) 13. Klein, P.N.: A linear-time approximation scheme for tsp in undirected planar graphs with edgeweights. SIAM J. Comput. 37(6), (2008) 14. Kuby, M.J.: Programming models for facility dispersion: the p-dispersion and maxisum dispersion problems. Geogr. Anal. 19(4), (1987) 15. Moon, I.D., Chaudhry, S.S.: An analysis of network location problems with distance constraints. Manag. Sci. 30(3), (1984) 16. Pardalos, P.M., Du, D.Z. (eds.): Network design: connectivity and facilities location. DIMACS Series vol. 40, American Mathematical Society (1998) 17. Qian, J., Wang, C.-A.: A linear-time approximation scheme for maximum weight triangulation of convex polygons. Algorithmica 40(3), (2004) 18. Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Heuristic and special case algorithms for dispersion problems. Oper. Res. 42(2), (1994) 19. Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Approximation algorithms for facility dispersion. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics 1st edn. Chapman and Hall/CRC (2007) 20. Resende, M.G.C., Pardalos, P.M.: The Handbook of Optimization in Telecommunications. Springer, Berlin (2006) 21. Rosenkrantz, D.J., Tayi, G.K., Ravi, S.S.: Facility dispersion problems under capacity and cost constraints. J. Comb. Optim. 4(1), 7 33 (2000) 22. Tamir, A.: Obnoxious facility location on graphs. SIAM J. Discret. Math. 4(4), (1991) 23. Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001) 24. White, D.J.: The maximal-dispersion problem. IMA J. Manag. Math. 3(2), (1991) 25. White, D.J.: The maximal dispersion problem and the first point outside the neighbourhood heuristic. Comput. Oper. Res. 18(1), (1991)

Approximating the Maximum Quadratic Assignment Problem 1

Approximating the Maximum Quadratic Assignment Problem 1 Esther M. Arkin Refael Hassin 3 Maxim Sviridenko 4 Keywords: Approximation algorithm; quadratic assignment problem 1 Introduction In the maximum