TY - JOUR

T1 - Optimal slope selection via expanders

AU - Katz, Matthew J.

AU - Sharir, Micha

N1 - Funding Information:
Correspondence to: M. Sharir, Department of Computer Science, School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel. * Work on this paper has been supported by a grant from the fund for Basic Research administered by the Israeli Academy of Sciences. Work by the second author has also been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.

PY - 1993/9/14

Y1 - 1993/9/14

N2 - Given n points in the plane and an integer k, the slope selection problem is to find the pair of points whose connecting line has the kth smallest slope. (In dual setting, given n lines in the plane, we want to find the vertex of their arrangement with the kth smallest x-coordinate.) Cole et al. have given an O(n log n) solution (which is optimal), using the parametric searching technique of Megiddo. We obtain another optimal (deterministic) solution that does not depend on parametric searching and uses expander graphs instead. Our solution is somewhat simpler than that of [6] and has a more explicit geometric interpretation.

AB - Given n points in the plane and an integer k, the slope selection problem is to find the pair of points whose connecting line has the kth smallest slope. (In dual setting, given n lines in the plane, we want to find the vertex of their arrangement with the kth smallest x-coordinate.) Cole et al. have given an O(n log n) solution (which is optimal), using the parametric searching technique of Megiddo. We obtain another optimal (deterministic) solution that does not depend on parametric searching and uses expander graphs instead. Our solution is somewhat simpler than that of [6] and has a more explicit geometric interpretation.

KW - Computational geometry

KW - algorithms

KW - design of algorithms

UR - http://www.scopus.com/inward/record.url?scp=0027666499&partnerID=8YFLogxK

U2 - 10.1016/0020-0190(93)90234-Z

DO - 10.1016/0020-0190(93)90234-Z

M3 - Article

AN - SCOPUS:0027666499

VL - 47

SP - 115

EP - 122

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 3

ER -