Abstract
In this contribution, a simple and efficient algorithm for the closest-pair problem in \(E^1\) is described using the preprocessing based on exponent bucketing and exponent windowing respecting accuracy of the floating point representation. The preprocessing is of the O(N) complexity. Experiments made for the uniform distribution proved significant speedup. The proposed approach is applicable for the \(E^2\) case.
Supported by the University of West Bohemia - Institutional research support Martinez, D.E., Martinez, A.E., Moreno, F.H.—students contributed during their Erasmus ACG course at the University of West Bohemia.
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Notes
- 1.
Data set, where \(N \ggg 10^6\). Note that \(2~ 147~483~648 \doteq 2.147~10^9\) unsigned distinct values only can be represented in single precision.
- 2.
It actually saves \(O(N^2)\) computations of the \(\sqrt{*}\) function.
- 3.
Let us consider a sorted sequence \(1.01, 1.05, \ldots , 10.0001, 10.0005\), then the minimum distance is 0.0004 not 0.04.
- 4.
Note: 130–120 means exponents interval [-120,...,-111].
- 5.
the binary exponent is shifted, i.2. a value \(2^{-128}\) has the shifted exponent 0.
- 6.
The array list, i.e. extensible arrays were used in the actual implementation.
- 7.
The value \(k \doteq 33\) is obtained by solving \(10^{10}*2^k=1\), which is \(k \doteq -33\) and the shifted exponent \(Ex=128-33=95\).
- 8.
In the case of \(10^6\) values, over \(10^3\) bucket extensions were called, but with the \(20\%\) additional memory allocation, the extension was called only 7 times.
- 9.
Sort-CPP - standard Shell sort in C++.
- 10.
Note, that p depends on the FP precision used.
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Acknowledgments
The author would like to thank colleagues at the University of West Bohemia, Plzen for their comments and suggestions, comments and hints provided, especially to Martin Cervenka and Lukas Rypl for some additional additional counter tests. Thanks also belong to anonymous reviewers for their critical view and recommendations that helped to improve this manuscript.
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Skala, V.: theoretical part, algorithm design, algorithm implementation and verification, manuscript preparation; Esteban Martinez, A., Esteban Martinez, D., Hernandez Moreno, F.: algorithm implementation and experimental verification.
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Skala, V., Martinez, A.E., Martinez, D.E., Moreno, F.H. (2023). A New Algorithm for the Closest Pair of Points for Very Large Data Sets Using Exponent Bucketing and Windowing. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds) Computational Science – ICCS 2023. ICCS 2023. Lecture Notes in Computer Science, vol 14074. Springer, Cham. https://doi.org/10.1007/978-3-031-36021-3_40
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