ICALP 1987: Automata, Languages and Programming pp 357-363

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

• Jacob T. Schwartz
• Micha Sharir
Algorithms And Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)

## 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.

## Keywords

Voronoi Diagram Piecewise Linear Function Intersection Curve Lower Envelope Courant Institute
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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