TY - JOUR

T1 - On the two-dimensional davenport-schinzel problem

AU - Schwartz, Jacob T.

AU - Sharir, Micha

PY - 1990

Y1 - 1990

N2 - We analyse the combinatorial complexity κ(F) of the minimum M(x,y) of a collection F of n continuous bivariate functions f1(x,y), ... , fn(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 κ(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 κ(F)= 0(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(nλs+2(n)), where the constant of proportionality depends on s and t, and where λ,(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) Finally, we present some geometric applications of these results.

AB - We analyse the combinatorial complexity κ(F) of the minimum M(x,y) of a collection F of n continuous bivariate functions f1(x,y), ... , fn(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 κ(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 κ(F)= 0(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(nλs+2(n)), where the constant of proportionality depends on s and t, and where λ,(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) Finally, we present some geometric applications of these results.

UR - http://www.scopus.com/inward/record.url?scp=84983636187&partnerID=8YFLogxK

U2 - 10.1016/S0747-7171(08)80070-3

DO - 10.1016/S0747-7171(08)80070-3

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AN - SCOPUS:84983636187

SN - 0747-7171

VL - 10

SP - 371

EP - 393

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

IS - 3-4

ER -