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 -