TY - JOUR

T1 - Computing maximally separated sets in the plane

AU - Agarwal, Pankaj K.

AU - Overmars, Mark

AU - Sharir, Micha

PY - 2006

Y1 - 2006

N2 - Let S be a set of n points in ℝ2. Given an integer 1 ≤ k ≤ n, we wish to find a maximally separated subset I ⊆ S of size k; this is a subset for which the minimum among the (2k) pairwise distances between its points is as large as possible. The decision problem associated with this problem is to determine whether there exists I ⊆ S, |I| = k, so that all (2k) pairwise distances in I are at least 2. This problem can also be formulated in terms of disk-intersection graphs: Let D be the set of unit disks centered at the points of S. The disk-intersection graph G of D has as edges all pairs of disks with nonempty intersection. Any set I with the above properties is then the set of centers of disks that form an independent set in the graph G. This problem is known to be NP-complete if k is part of the input. In this paper we first present a linear-time ε-approximation algorithm for any constant k. Next we give exact algorithms for the cases k = 3 and k = 4 that run in time O(n 4/3polylog(n)). We also present a simpler nO(√k)- time exact algorithm (as compared with the recent algorithm in [J. Alber and J. Fiala, J. Algorithms, 52 (2004), pp. 134-151]) for arbitrary values of k.

AB - Let S be a set of n points in ℝ2. Given an integer 1 ≤ k ≤ n, we wish to find a maximally separated subset I ⊆ S of size k; this is a subset for which the minimum among the (2k) pairwise distances between its points is as large as possible. The decision problem associated with this problem is to determine whether there exists I ⊆ S, |I| = k, so that all (2k) pairwise distances in I are at least 2. This problem can also be formulated in terms of disk-intersection graphs: Let D be the set of unit disks centered at the points of S. The disk-intersection graph G of D has as edges all pairs of disks with nonempty intersection. Any set I with the above properties is then the set of centers of disks that form an independent set in the graph G. This problem is known to be NP-complete if k is part of the input. In this paper we first present a linear-time ε-approximation algorithm for any constant k. Next we give exact algorithms for the cases k = 3 and k = 4 that run in time O(n 4/3polylog(n)). We also present a simpler nO(√k)- time exact algorithm (as compared with the recent algorithm in [J. Alber and J. Fiala, J. Algorithms, 52 (2004), pp. 134-151]) for arbitrary values of k.

KW - Disk-intersection graphs

KW - Geometric optimization

KW - Independent set

UR - http://www.scopus.com/inward/record.url?scp=34250828576&partnerID=8YFLogxK

U2 - 10.1137/S0097539704446591

DO - 10.1137/S0097539704446591

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AN - SCOPUS:34250828576

SN - 0097-5397

VL - 36

SP - 815

EP - 834

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 3

ER -