On the bivariate function minimization problem and its applications to motion planning

Jacob T. Schwartz, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

An important technical problem which arises in many geometric contexts including robot motion planning is to analyze the combinatorial complexity κ(F) of the minimum M (x,y) of a collection F of n continuous bivariate functions f1(x,y),.., fn(x,y), such that each triple of function graphs intersect in at most s points, and each pair of functions intersect in a curve having at most t singular points. We have obtained the following results for this problem: (1) If the intersection curve of each pair of functions intersects each plane x = const in exactly one point and s=1 (but not if s=2) then κ(F) is at most O (n), and can be calculated in time O (n log n) by a method extending Shamos' algorithm for the calculation of planar Voronoi diagrams. (2) If s=2 and the intersection of each pair of functions is connected then κ(F)=O (n2). (3) If the intersection curve of each pair of functions intersects every plane x = const in at most two points, then κ(F) is at most O (nλs+2(n)), where the constant of proportionality depends on s and t, and where λr(q) is the (almost linear) maximum length of a (q,r) Davenport-Schinzel sequence. We also present an algorithm for calculating M in this case, running in time O (nλs+2(n) log n). (4) Various new geometric applications of these results have also been derived.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 14th International Colloquium, Proceedings
EditorsThomas Ottmann
PublisherSpringer Verlag
Pages357-363
Number of pages7
ISBN (Print)9783540180883
DOIs
StatePublished - 1987
Event14th International Colloquium on Automata, Languages, and Programming, ALP 1987 - Karlsruhe, Germany
Duration: 13 Jul 198717 Jul 1987

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume267 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th International Colloquium on Automata, Languages, and Programming, ALP 1987
Country/TerritoryGermany
CityKarlsruhe
Period13/07/8717/07/87

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