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 f 1(x,y), ..., f n(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 (n 2). (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.
This work has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation.
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Schwartz, J.T., Sharir, M. (1987). On the bivariate function minimization problem and its applications to motion planning. In: Ottmann, T. (eds) Automata, Languages and Programming. ICALP 1987. Lecture Notes in Computer Science, vol 267. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18088-5_30
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DOI: https://doi.org/10.1007/3-540-18088-5_30
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