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Optimal Parallel Algorithms for Finding Proximate Points, with Applications
December 1998 (vol. 9 no. 12)
pp. 1153-1166
ASCII Text
x 
 
Tatsuya Hayashi, Koji Nakano, Stephan Olariu, "Optimal Parallel Algorithms for Finding Proximate Points, with Applications," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 12, pp. 1153-1166, December, 1998.
 
 
BibTex
x
 
@article{ 10.1109/71.737693,
author = {Tatsuya Hayashi and Koji Nakano and Stephan Olariu},
title = {Optimal Parallel Algorithms for Finding Proximate Points, with Applications},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {9},
number = {12},
issn = {1045-9219},
year = {1998},
pages = {1153-1166},
doi = {http://doi.ieeecomputersociety.org/10.1109/71.737693},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - Optimal Parallel Algorithms for Finding Proximate Points, with Applications
IS - 12
SN - 1045-9219
SP1153
EP1166
EPD - 1153-1166
A1 - Tatsuya Hayashi,
A1 - Koji Nakano,
A1 - Stephan Olariu,
PY - 1998
KW - Proximate points
KW - convex hulls
KW - parallel algorithms
KW - digital geometry
KW - image analysis
KW - pattern recognition
KW - largest empty circles
KW - cellular systems.
VL - 9
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
 

Abstract—Consider a set P of points in the plane sorted by x-coordinate. A point p in P is said to be a proximate point if there exists a point q on the x-axis such that p is the closest point to q over all points in P. The proximate point problem is to determine all the proximate points in P. Our main contribution is to propose optimal parallel algorithms for solving instances of size n of the proximate points problem. We begin by developing a work-time optimal algorithm running in O(log log n) time and using ${{n \over {\log \log n}}}$ Common-CRCW processors. We then go on to show that this algorithm can be implemented to run in O(log n) time using ${{n \over {\log n}}}$ EREW processors. In addition to being work-time optimal, our EREW algorithm turns out to also be time-optimal. Our second main contribution is to show that the proximate points problem finds interesting, and quite unexpected, applications to digital geometry and image processing. As a first application, we present a work-time optimal parallel algorithm for finding the convex hull of a set of n points in the plane sorted by x-coordinate; this algorithm runs in O(log log n) time using ${{n \over {\log \log n}}}$ Common-CRCW processors. We then show that this algorithm can be implemented to run in O(log n) time using ${{n \over {\log n}}}$ EREW processors. Next, we show that the proximate points algorithms afford us work-time optimal (resp. time-optimal) parallel algorithms for various fundamental digital geometry and image processing problems. Specifically, we show that the Voronoi map, the Euclidean distance map, the maximal empty circles, the largest empty circles, and other related problems involving a binary image of size n×n can be solved in O(log log n) time using

$${{{n^2} \over {\log \log n}}}$$Common-CRCW processors or in O(log n) time using ${{{n^2} \over {\log n}}}$ EREW processors.

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Index Terms:
Proximate points, convex hulls, parallel algorithms, digital geometry, image analysis, pattern recognition, largest empty circles, cellular systems.
Citation:
Tatsuya Hayashi, Koji Nakano, Stephan Olariu, "Optimal Parallel Algorithms for Finding Proximate Points, with Applications," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 12, pp. 1153-1166, Dec. 1998, doi:10.1109/71.737693
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