An optimal-time algorithm for slope selection

Given $n$ points in the plane and an integer $k$, the problem of selecting that pair of points that determines the line with the $k$th smallest or largest slope is considered. In the restricted case, where $k$ is $O(n)$, line sweeping gives an optimal, $O(n\log n)$-time algorithm. For general $k$ th...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 18; no. 4; pp. 792 - 810
Main Authors: COLE, R, SALOWE, J. S, STEIGER, W. L, SZEMEREDI, E
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1989
Subjects:
Algorithms
Applied mathematics
Combinatorics
Combinatorics. Ordered structures
Computer science
Decision trees
Designs and configurations
Exact sciences and technology
Mathematics
Sciences and techniques of general use
Computational geometry
Configuration
Combinatorics
Time complexity
Algorithm analysis
Optimization
Slope
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:Given $n$ points in the plane and an integer $k$, the problem of selecting that pair of points that determines the line with the $k$th smallest or largest slope is considered. In the restricted case, where $k$ is $O(n)$, line sweeping gives an optimal, $O(n\log n)$-time algorithm. For general $k$ the parametric search technique of Megiddo is used to describe an $O(n(\log n)^2 )$-time algorithm. This is modified to produce a new, optimal $O(n\log n)$-time selection algorithm by incorporating an approximation idea.
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ISSN:0097-5397
1095-7111
DOI:10.1137/0218055