On minima of function, intersection patterns of curves, and davenport-schinzel sequences

Abstract

We present several results related to the problem of estimating the complexity M(f1, ..., fn) of the pointwise minimum of n continuous univariate or bivariate functions f1, ..., fn under the assumption that no pair (resp. triple) of these functions intersect in more than some fixed number s of points. Our main result is that in the one-dimensional case M(f1, ..., fn) - O(nα(n)O(α(n)s-3)) (α(n) is the functional inverse of Ackermann's function). In the twodimensional case the problem is substantially harder, and we have only some initial estimates on M, including a tight bound Θ(n2) if s = 2, and a worst-case lower bound Ω(n2α(n)) for s ≥ 6. The treatment of the twodimensional problem is based on certain properties of the intersection patterns of a collection of planar Jordan curves, which we also develop and prove here.