Abstract
We consider the problem of approximate sorting in I/O model. The goal of approximate sorting in I/O model is to find out a permutation that is as close as possible to the true ordering of elements in t I/O operations. However, the quality of approximate sorting in I/O model can not be well measured by the existing metrics on permutation space. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocation that happened in each I/O block. We consider the External Spearman’s footrule metric (short for ESP) (Spearman’s footrule metric in RAM model) and a new metric external errors (short for EE) (errors in RAM model). ESP shows the block dislocation distance of each element and EE states the number of the dislocation elements in the approximate result. According to the rate-distortion relationship endowed by these two metrics, we find the lower bound of these two metrics of the permutation generated by any external approximate sorting algorithm with t I/O operations. Finally, we propose a k-pass external approximate sorting algorithm that is asymptotically optimal in I/O model.
This work was supported by the National Natural Science Foundation of China under grants 61832003, 61972110 and U1811461.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Commun. ACM 31(9), 1116–1127 (1988). https://doi.org/10.1145/48529.48535
Disser, Y., Kratsch, S.: Robust and adaptive search (2017). http://arxiv.org/abs/1702.05932
Estivill-Castro, V., Mannila, H., Wood, D.: Right invariant metrics and measures of presortedness. Discret. Appl. Math. 42, 1–16 (1993). https://doi.org/10.1016/0166-218X(93)90175-N
Farnoud, F., Schwartz, M., Bruck, J.: Rate-distortion for ranking with incomplete information (2014). http://arxiv.org/abs/1401.3093
Farnoud, F., Yaakobi, E., Bruck, J.: Approximate sorting of data streams with limited storage. J. Comb. Optim. 32(4), 1133–1164 (2015). https://doi.org/10.1007/s10878-015-9930-6
Gao, T., Li, J.: Analysis of approximate sorting in I/O model (2022). https://doi.org/10.48550/ARXIV.2208.10298
Giesen, J., Schuberth, E., Stojaković, M.: Approximate sorting. Fundam. Inf. 90(1–2), 67–72 (2009)
Klove, T., Lin, T.T., Tsai, S.C., Tzeng, W.G.: Permutation arrays under the Chebyshev distance. IEEE Trans. Inf. Theory 56(6), 2611–2617 (2010). https://doi.org/10.1109/TIT.2010.2046212
Knuth, D.E.: The Art of Computer Programming, vol. 3, 2nd edn. Sorting and Searching. Addison Wesley Longman Publishing Co., Inc. (1998)
Wang, D., Mazumdar, A., Wornell, G.W.: A rate-distortion theory for permutation spaces. In: 2013 IEEE International Symposium on Information Theory, pp. 2562–2566 (2013). https://doi.org/10.1109/ISIT.2013.6620689
Gao, X., Li, J., Miao, D., Liu, X.: Recognizing the tractability in big data computing. Theor. Comput. Sci. 838, 195–207 (2020). https://doi.org/10.1016/j.tcs.2020.07.026
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Gao, T., Li, J. (2022). Analysis of Approximate Sorting in I/O Model. In: Zhang, Y., Miao, D., Möhring, R. (eds) Computing and Combinatorics. COCOON 2022. Lecture Notes in Computer Science, vol 13595. Springer, Cham. https://doi.org/10.1007/978-3-031-22105-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-22105-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-22104-0
Online ISBN: 978-3-031-22105-7
eBook Packages: Computer ScienceComputer Science (R0)