TY - GEN

T1 - On the power of the semi-separated pair decomposition

AU - Abam, Mohammad Ali

AU - Carmi, Paz

AU - Farshi, Mohammad

AU - Smid, Michiel

PY - 2009/9/14

Y1 - 2009/9/14

N2 - A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A i ,B i )} of pairs of subsets of S such that for each i, there are balls and containing A i and B i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A i and q B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.

AB - A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A i ,B i )} of pairs of subsets of S such that for each i, there are balls and containing A i and B i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A i and q B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.

UR - http://www.scopus.com/inward/record.url?scp=69949154324&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-03367-4_1

DO - 10.1007/978-3-642-03367-4_1

M3 - Conference contribution

AN - SCOPUS:69949154324

SN - 3642033660

SN - 9783642033667

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 12

BT - Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings

T2 - 11th International Symposium on Algorithms and Data Structures, WADS 2009

Y2 - 21 August 2009 through 23 August 2009

ER -