## On the two-dimensional Davenport-Schinzel problem.(English)Zbl 0717.68050

Summary: We analyse the combinatorial complexity $$\kappa$$ ($${\mathbb{F}})$$ of the minimum M(x,y) of a collection $${\mathbb{F}}$$ 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($${\mathbb{F}})$$ is at most 0(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 $$\kappa ({\mathbb{F}})=0(n^ 2).$$
(3) If the intersection curve of each pair of functions intersects every plane $$x=const$$ in at most two points, then $$\kappa$$ ($${\mathbb{F}})$$ is at most $$0(n\lambda_{s+2}(n))$$, where the constant of proportionality depends on s and t, and where $$\lambda_ r(q)$$ 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 $$0(n\lambda_{s+2}(n)\log n).$$
(4) Finally, we present some geometric applications of these results.

### MSC:

 68Q25 Analysis of algorithms and problem complexity 68U05 Computer graphics; computational geometry (digital and algorithmic aspects)

### Keywords:

Davenport-Schinzel problem; upper bounds; lower bounds
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### References:

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