Abstract
We address the problem of a succinct static data structure representing points on an M × M grid (M = 2m where m is size of a word) that permits to answer the question of finding the closest point to a query point under the L ∞ or L 1 norm in constant time. Our data structure takes essentially minimum space. These results are extended to d dimensions under L ∞.
on leave from Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia
This work was done while the first author was a graduate student at the University of Waterloo and was supported in part by the NSERC of Canada, grant number A-8237, and the ITRC of Ontario.
Preview
Unable to display preview. Download preview PDF.
References
S. Albers and T. Hagerup. Improved parallel integer sorting without concurrent writting. In 3rd ACM-SIAM Symposium on Discrete Algorithms, pages 463–172, Orlando, Florida, 1992.
A. Andersson, T. Hagerup, S. Nilsson, and R. Raman. Sorting in linear time? In 27th ACM Symposium on Theory of Computing, pages 427–436, Las Vegas, Nevada, 1995.
J.L. Bentley and J.H. Friedman. Data structures for range searching. ACM Computing Surveys, 11(4):397–409, 1979.
J.L. Bentley and H.A. Maurer. Efficient worst-case data structures for range searching. Acta Informatica, 13:155–168, 1980.
J.L. Bentley, B.W. Weide, and A.C. Yao. Optimal expected-time algorithms for closest-point problems. ACM Transactions on Mathematical Software, 6(4):563–580, December 1980.
A. Brodnik. Searching in Constant Time and Minimum Space (MINIM/ARRES MAGNI MOMENTI SUNT). PhD thesis, University of Waterloo, Waterloo, Ontario, Canada, 1995. (Also published as technical report CS-95-41.).
C.-C. Chang and T.-C. Wu. A bashing-oriented nearest neighbor searching scheme. Pattern Recognition Letters, 14(8):625–630, August 1993.
B. Chazelle. An improved algorithm for the fixed-radius neighbor problem. Information Processing Letters, 16(4): 193–198, May 13th 1983.
B. Chazelle, R. Cole, F.P. Preparata, and C. Yap. New upper bounds for neighbor searching. Information and Control, 68(1–3):105–124, 1986.
Y.-J. Chiang and R. Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, 80(9):1412–1434, September 1992.
T.H. Cormen, C.E. Leiserson, and R.L. Rivest. Introduction to Algorithms. MIT Press, Cambridge, Massachusetts, 1990.
C.R. Dyer and A. Rosenfeld. Parallel image processing by memory-augmented cellular automata. IEEE Transactions on Pattern Analysis and Machine Intelligence, 3(1):29–41, January 1981.
H. Edelsbrunner. Algorithms in Combinatorial Geometry. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, Berlin, 1987.
P. van Emde Boas. Machine models and simulations. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume A: Algorithms and Complexity, chapter 1, pages 1–66. Elsevier, Amsterdam, Holland, 1990.
P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10(1):99–127, 1977.
A. Faragó, T. Linder, and G. Lugosi. Nearest neighbor search and classification in O(1) time. Problems of Control and Information Theory, 20:383–395, 1991.
M.L. Fredman and D.E. Willard. Surpassing the information theoretic bound with fusion trees. Journal of Computer and System Sciences, 47:424–436, 1993.
R.G. Karlsson. Algorithms in a Restricted Universe. PhD thesis, University of Waterloo, Waterloo, Ontario, Canada, 1984. (Also published as technical report CS-84-50.).
R.G. Karlsson, J.I. Munro, and E.L. Robertson. The nearest neighbor problem on bounded domains. In W. Brauer, editor, Proceedings 12th International Colloquium on Automata, Languages and Programming, volume 194 of Lecture Notes in Computer Science, pages 318–327. Springer-Verlag, 1985.
D.E. Knuth. The Art of Computer Programming: Sorting and Searching, volume 3. Addison-Wesley, Reading, Massachusetts, 1973.
D.T. Lee and C.K. Wong. Voronoi diagrams in L 1 (L ∞) metrics with 2-storage applications. SIAM Journal on Computing, 9(1):200–211, February 1980.
K. Mehlhorn. Data Structures and Algorithms: Multi-dimensional Searching and Computational Geometry, volume 3. Springer-Verlag, Berlin, 1984.
R. Miller and Q.F. Stout. Geometric algorithms for digitized pictures on a mesh-connected computer. IEEE Transactions on Pattern Analysis and Machine Intelligence, 7(2):216–228, March 1985.
P.B. Miltersen. Lower bounds for union-split-find related problems on random access machines. In 26th ACM Symposium on Theory of Computing, pages 625–634, Montréal, Québec, Canada, 1994.
O.J. Murphy and S.M. Selkow. The efficiency of using k-d trees for finding nearest neighbors in discrete space. Information Processing Letters, 23(4):215–218, November 8th 1986.
F.P. Preparata and M.I. Shamos. Computational Geometry. Texts and Monographs in Computer Science. Springer-Verlag, Berlin, 2nd edition, 1985.
V. Ramasubramanian and K.K. Paliwal. An efficient approximation-elimination algorithm for fast nearest-neighbour search based on a spherical distance coordinate formulation. Pattern Recognition Letters, 13(7):471–480, July 1992.
R.F. Sproull. Refinements to nearest-neighbor searching in k-dimensional trees. Algorithmica, 6:579–589, 1991.
F.F. Yao. Computational geometry. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume A: Algorithms and Complexity, chapter 7, pages 343–389. Elsevier, Amsterdam, Holland, 1990.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brodnik, A., Munro, J.I. (1996). Neighbours on a grid. In: Karlsson, R., Lingas, A. (eds) Algorithm Theory — SWAT'96. SWAT 1996. Lecture Notes in Computer Science, vol 1097. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61422-2_141
Download citation
DOI: https://doi.org/10.1007/3-540-61422-2_141
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61422-7
Online ISBN: 978-3-540-68529-6
eBook Packages: Springer Book Archive