NDLI: On counting pairs of intersecting segments and off-line triangle range searching

Content Provider	Springer Nature Link
Author	Pellegrini, M.
Copyright Year	1997
Abstract	We describe a method for decomposing planar sets of segments and points. Using this method we obtain new efficientdeterministic algorithms for counting pairs of intersecting segments, and for answering off-line triangle range queries. In particular we obtain the following results: (1) Givenn segments in the plane, the number of pairs of intersecting segments is counted in timeO(n 1+ɛ+K 1/3 n 2/3+ɛ), whereK is the number of intersection points among the segments, and ɛ>0 is an arbitrarily small constant. (2) Givenn segments in the plane which are colored with two colors, the number of pairs ofbichromatic intersecting segments is counted in timeO(n 1+ɛ+K m 1/3 n 2/3+ɛ), whereK m is the number ofmonochromatic intersection points, and ɛ>0 is an arbitrarily small constant. (3) Givenn weighted points andn triangles on a plane, the sum of weights of points in each triangle is computed in timeO(n 1+ε+ϰ1/3 n 2/3+ε), where ϰ is the number of vertices in the arrangement of the triangles, and ɛ>0 is an arbitrarily small constant. The above bounds depend sublinearly on the number of intersections among input segmentsK (resp.K m , ϰ), which is desirable sinceK (resp.K m , ϰ) can range from zero toO(n 2). All of the above algorithms use optimal Θ(n) storage. The constants of proportionality in the big-Oh notation increase as ɛ decreases. These results are based on properties of the sparse nets introduced by Chazelle [Cha3].
Starting Page	380
Ending Page	398
Page Count	19
File Format	PDF
ISSN	01784617
Journal	Algorithmica
Volume Number	17
Issue Number	4
e-ISSN	14320541
Language	English
Publisher	Springer-Verlag
Publisher Date	1997-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computational geometry Intersection of segments Triangle range searching Sparse nets Derandomization Computer Systems Organization and Communication Networks Data Structures, Cryptology and Information Theory Theory of Computation Algorithm Analysis and Problem Complexity Mathematics of Computing Algorithms
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Science Applications

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