NDLI: Computing Envelopes in Four Dimensions with Applications

Please wait, while we are loading the content...

Computing Envelopes in Four Dimensions with Applications

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Aronov, Boris Agarwal, Pankaj K. Sharir, Micha
Copyright Year	1997
Abstract	Let ${\cal F}$ be a collection of nd-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of ${\cal F}$ in expected time $O(n^{d+\epsilon})$ for any $\epsilon > 0$. For d = 3, by combining this algorithm with the point-location technique of Preparata and Tamassia, we can compute, in randomized expected time $O(n^{3+\epsilon})$, for any $\epsilon > 0$, a data structure of size $O(n^{3+\epsilon})$ that, for any query point q, can determine in O(log2n) time the function(s) of ${\cal F}$ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the "biggest stick" in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in randomized expected time $O(n^{17/11+\epsilon})$, for any $\epsilon > 0$, improving previous solutions that run in time $O(n^{8/5+\epsilon})$. We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require $O(n^{3+\epsilon})$ storage and preprocessing time, for any $\epsilon > 0$, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.
Starting Page	1714
Ending Page	1732
Page Count	19
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539794265724
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	closest pair Analysis of algorithms and problem complexity Computer graphics; computational geometry ray shooting lower envelopes point location Combinatorics
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in