On the Two-Dimensional Davenport Schinzel Problem

We analyse the combinatorial complexity @k(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. The following is proved. (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 @k(F) is at most O(n), and can be calculated in time 0(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 @k(F)= 0(n^2). (3) If the intersection curve of each pair of functions intersects every plane x = const in at most two points, then @k(F) is at most O([email protected]"s"+"2(n)), where the constant of proportionality depends on s and t, and where @l,(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([email protected]"s"+"2(n) log n). (4) Finally, we present some geometric applications of these results.