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

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),.., 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 (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 (nλ_s+2(n)), where the constant of proportionality depends on s and t, and where λ_r(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 (nλ_s+2(n) log n). (4) Various new geometric applications of these results have also been derived.