Kinetic Facility Location

Degener, Bastian; Gehweiler, Joachim; Lammersen, Christiane

doi:10.1007/s00453-008-9250-7

Cited by 8 publications

(10 citation statements)

References 43 publications

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“…For dynamic environments various global event-based algorithms solving clustering related problems are known: a constant factor approximation for the minimal number of centers to cover all points with a given radius [4], solutions to k-center problems with k = 1 [5] and even for the facility location problem [6]. In [7] a solution for the dynamic k-center problem that does not require any updates is proposed.…”

Section: Related Workmentioning

confidence: 99%

A local, distributed constant-factor approximation algorithm for the dynamic facility location problem

Degener

Kempkes

Pietrzyk

2010

2010 IEEE International Symposium on Parallel &Amp; Distributed Processing (IPDPS)

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We present a distributed, local solution to the dynamic facility location problem in general metrics, where each node is able to act as a facility or a client. To decide which role it should take, each node keeps up a simple invariant in its local neighborhood. This guarantees a global constant factor approximation when the invariant is satisfied at all nodes. Due to the changing distances between nodes, invariants can be violated. We show that restoring the invariants is bounded to a constant neighborhood, takes logarithmic (in the number of nodes) asynchronous rounds and affects each node at most twice per violation.

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Section: Related Workmentioning

confidence: 99%

A local, distributed constant-factor approximation algorithm for the dynamic facility location problem

Degener

Kempkes

Pietrzyk

2010

2010 IEEE International Symposium on Parallel &Amp; Distributed Processing (IPDPS)

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“…In this section we present such a fast (O(n log n log log n)-time) constant-factor approximation algorithm. The algorithm is a simple modification of an algorithm by Mettu and Plaxton [21] and its extensions in [4,6]. Let P ⊆ R 2 be a set of n points in the plane.…”

Section: Ptas (P )mentioning

confidence: 99%

“…The definition of radius is an adaptation of a similar definition in [21]. It had been extended to ∞ -balls in [6]. The definition of discrete radius is similar to a definition in [4].…”

Section: Ptas (P )mentioning

confidence: 99%

(1 + ε)-Approximation for Facility Location in Data Streams

Czumaj¹,

Lammersen²,

Monemizadeh³

et al. 2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P ⊆ R 2 and an opening cost f ∈ R + , and we want to find a set of facilities F ⊆ R 2 that minimizeswhere d(p, q) is the Euclidean distance between p and q. We obtain two main results:• A (1 + ε)-approximation algorithm with running time O(n log n(log n log log n + (log log n)which is O(n log 2 n log log n) for any constant ε.• The first (1 + ε)-approximation algorithm for the cost of the facility location problem for dynamic geometric data streams, i.e., when the stream consists of insert and delete operations of points from a discrete space {1, . . . , ∆} 2 . The streaming algorithm usesOur PTAS is significantly faster than any previously known (1 + ε)-approximation algorithm for the problem, and is also relatively simple. Our algorithm for dynamic geometric data streams is the first (1 + ε)-approximation algorithm for the cost of the facility location problem with polylogarithmic space, and it resolves an open problem in the streaming area.Both algorithms are based on a novel and simple decomposition of an input point set P into small subsets Pi, such that:• the cost of solving the facility location problem for each Pi is small (which means that for each Pi one needs to open only a small, polylogarithmic number of facilities),, where for a point set P , OPT(P ) denotes the cost of an optimal solution for P . The decomposition can be used directly to obtain the PTAS by splitting the point set in the subsets and efficiently solve the problem for each subset independently. By combining our partitioning with techniques to process dynamic data streams of sampling from the cells of the partition and estimating the cost from the sample, we obtain our data streaming algorithm.

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“…The problem in the Euclidean metric cannot be approximated to within a factor of 1.822 (assuming P = NP) [17]. Demaine et al [15] give algorithms for the k-center problem on planar graphs and map graphs that achieve time bounds exponential in the radius and k.…”

Section: Introductionmentioning

confidence: 99%

“…In such a model it is unprofitable to open a new facility to service a small number of isolated customers. Recently, Degener et al [14] gave a deterministic algorithm for the kinetic version of the non-robust facility location problem and Agarwal and Phillips [1] gave a randomized algorithm for the robust 2-center problem with an O (nk 7 log 3 n) expected execution time.…”

Section: Introductionmentioning

confidence: 99%

Approximation algorithm for the kinetic robust K-center problem

Friedler

Mount

2010

Computational Geometry

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Two complications frequently arise in real-world applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the k-center problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard k-center problem, the objective is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center, or equivalently, the smallest radius such that S can be covered by k disks of this radius. In the discrete k-center problem the disk centers are drawn from the points of S, and in the absolute k-center problem the disk centers are unrestricted. We generalize this problem in two ways. First, we assume that points are in continuous motion, and the objective is to maintain a solution over time. Second, we assume that some given robustness parameter 0 < t 1 is given, and the objective is to compute the smallest radius such that there exist k disks of this radius that cover at least tn points of S. We present a kinetic data structure (in the KDS framework) that maintains a (3 + ε)-approximation for the robust discrete k-center problem and a (4 + ε)-approximation for the robust absolute k-center problem, both under the assumption that k is a constant. We also improve on a previous 8-approximation for the non-robust discrete kinetic k-center problem, for arbitrary k, and show that our data structure achieves a (4+ε)-approximation.All these results hold in any metric space of constant doubling dimension, which includes Euclidean space of constant dimension.

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Kinetic Facility Location

Cited by 8 publications

References 43 publications

A local, distributed constant-factor approximation algorithm for the dynamic facility location problem

A local, distributed constant-factor approximation algorithm for the dynamic facility location problem

(1 + ε)-Approximation for Facility Location in Data Streams

Approximation algorithm for the kinetic robust K-center problem

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