Computational geometry: an introduction
Computational geometry: an introduction
Upper and lower time bounds for parallel random access machines without simultaneous writes
SIAM Journal on Computing
An introduction to parallel algorithms
An introduction to parallel algorithms
A fast algorithm for Euclidean distance maps of a 2-D binary image
Information Processing Letters
Efficient Geometric Algorithms on the EREW PRAM
IEEE Transactions on Parallel and Distributed Systems
An optimal parallel algorithm for the Euclidean distance maps of 2-D binary images
Information Processing Letters
Designing systolic architectures for complete Euclidean distance transform
Journal of VLSI Signal Processing Systems
Time- and VLSI-optimal convex hull computation on meshes with multiple broadcasting
Information Processing Letters
Parallel computation of exact Euclidean distance transform
Parallel Computing
A unified linear-time algorithm for computing distance maps
Information Processing Letters
On the Generation of Skeletons from Discrete Euclidean Distance Maps
IEEE Transactions on Pattern Analysis and Machine Intelligence
Wireless Communications: Principles and Practice
Wireless Communications: Principles and Practice
Computer Vision
Linear Time Euclidean Distance Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
Euclidean distance transform on Polymorphic Processor Array
CAMP '95 Proceedings of the Computer Architectures for Machine Perception
Efficient parallel geometric algorithms on a mesh of trees
ACM-SE 33 Proceedings of the 33rd annual on Southeast regional conference
Fast parallel algorithm for distance transforms
IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
Fast and Scalable Algorithms for Euclidean Distance Transform on the LARPBS
IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
Journal of Parallel and Distributed Computing
Scalable and Efficient Parallel Algorithms for Euclidean Distance Transform on the LARPBS Model
IEEE Transactions on Parallel and Distributed Systems
Parallel Banding Algorithm to compute exact distance transform with the GPU
Proceedings of the 2010 ACM SIGGRAPH symposium on Interactive 3D Graphics and Games
Slime mould computes planar shapes
International Journal of Bio-Inspired Computation
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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脳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.