TY - GEN

T1 - On the bivariate function minimization problem and its applications to motion planning

AU - Schwartz, Jacob T.

AU - Sharir, Micha

N1 - Publisher Copyright:
© 1987, Springer-Verlag.

PY - 1987

Y1 - 1987

N2 - 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),.., 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. 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 (nλs+2(n)), where the constant of proportionality depends on s and t, and where λr(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) Various new geometric applications of these results have also been derived.

AB - 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),.., 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. 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 (nλs+2(n)), where the constant of proportionality depends on s and t, and where λr(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) Various new geometric applications of these results have also been derived.

UR - http://www.scopus.com/inward/record.url?scp=85034762083&partnerID=8YFLogxK

U2 - 10.1007/3-540-18088-5_30

DO - 10.1007/3-540-18088-5_30

M3 - פרסום בספר כנס

AN - SCOPUS:85034762083

SN - 9783540180883

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 357

EP - 363

BT - Automata, Languages and Programming - 14th International Colloquium, Proceedings

A2 - Ottmann, Thomas

PB - Springer Verlag

Y2 - 13 July 1987 through 17 July 1987

ER -