TY - JOUR
T1 - Kinetic and dynamic data structures for convex hulls and upper envelopes
AU - Alexandron, Giora
AU - Kaplan, Haim
AU - Sharir, Micha
PY - 2007/2
Y1 - 2007/2
N2 - Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O( n2βs+ 2(n)logn) critical events, each in O( log2n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, βs(q) = λs(q)/q, and λs(q) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1-28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.
AB - Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O( n2βs+ 2(n)logn) critical events, each in O( log2n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, βs(q) = λs(q)/q, and λs(q) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1-28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.
KW - Convex hull
KW - Dynamic data structures
KW - Kinetic data structures
KW - Lower envelope
KW - Treaps
UR - http://www.scopus.com/inward/record.url?scp=84867970623&partnerID=8YFLogxK
U2 - 10.1016/j.comgeo.2006.01.002
DO - 10.1016/j.comgeo.2006.01.002
M3 - מאמר
AN - SCOPUS:84867970623
VL - 36
SP - 144
EP - 158
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
SN - 0925-7721
IS - 2
ER -