TY - GEN

T1 - Two approaches to building time-windowed geometric data structures

AU - Chan, Timothy M.

AU - Pratt, Simon

N1 - Publisher Copyright:
© Timothy M. Chan and Simon Pratt.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(nα(n) log n) time (where α is the inverse Ackermann function), improving Bokal et al.'s O(n log2 n) and O(n log n log log n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

AB - Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(nα(n) log n) time (where α is the inverse Ackermann function), improving Bokal et al.'s O(n log2 n) and O(n log n log log n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

KW - Dynamic convex hull

KW - Geometric data structures

KW - Range searching

KW - Time window

UR - http://www.scopus.com/inward/record.url?scp=84976888722&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976888722&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2016.28

DO - 10.4230/LIPIcs.SoCG.2016.28

M3 - Conference contribution

AN - SCOPUS:84976888722

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 28.1-28.15

BT - 32nd International Symposium on Computational Geometry, SoCG 2016

A2 - Fekete, Sandor

A2 - Lubiw, Anna

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 32nd International Symposium on Computational Geometry, SoCG 2016

Y2 - 14 June 2016 through 17 June 2016

ER -