NDLI: Shooting permanent rays among disjoint polygons in the plane

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Shooting permanent rays among disjoint polygons in the plane

Content Provider	ACM Digital Library
Author	Ishaque, Mashhood Speckmann, Bettina Tóth, Csaba D.
Abstract	We present a data structure for ray shooting-and-insertion in the free space among disjoint polygonal obstacles with a total of \$n\$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting-and-insertion queries in O(n $log^{2}$ n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of auto-partitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n $log^{2}n+m$ log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments to O(n $log^{2}$ n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semi-dynamic ray shooting data structure) requires $O(n^{3/2-ε/2})$ time using $O(n^{1+ε})$ space. Our data structure improves the runtime to O(n $log^{2}$ n).
Starting Page	51
Ending Page	60
Page Count	10
File Format	PDF
ISBN	9781605585017
DOI	10.1145/1542362.1542372
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2009-06-08
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Convex partition Segment insertion Ray shooting Dynamic data structure Binary space partition
Content Type	Text
Resource Type	Article

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