Optimally Sorting Evolving Data

Authors Juan Jose Besa , William E. Devanny, David Eppstein, Michael T. Goodrich, Timothy Johnson

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Author Details

Juan Jose Besa
  • Dept. of Computer Science, Univ. of California, Irvine, Irvine, CA 92697 USA
William E. Devanny
  • Dept. of Computer Science, Univ. of California, Irvine, Irvine, CA 92697 USA
David Eppstein
  • Dept. of Computer Science, Univ. of California, Irvine, Irvine, CA 92697 USA
Michael T. Goodrich
  • Dept. of Computer Science, Univ. of California, Irvine, Irvine, CA 92697 USA
Timothy Johnson
  • Dept. of Computer Science, Univ. of California, Irvine, Irvine, CA 92697 USA

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Juan Jose Besa, William E. Devanny, David Eppstein, Michael T. Goodrich, and Timothy Johnson. Optimally Sorting Evolving Data. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We give optimal sorting algorithms in the evolving data framework, where an algorithm's input data is changing while the algorithm is executing. In this framework, instead of producing a final output, an algorithm attempts to maintain an output close to the correct output for the current state of the data, repeatedly updating its best estimate of a correct output over time. We show that a simple repeated insertion-sort algorithm can maintain an O(n) Kendall tau distance, with high probability, between a maintained list and an underlying total order of n items in an evolving data model where each comparison is followed by a swap between a random consecutive pair of items in the underlying total order. This result is asymptotically optimal, since there is an Omega(n) lower bound for Kendall tau distance for this problem. Our result closes the gap between this lower bound and the previous best algorithm for this problem, which maintains a Kendall tau distance of O(n log log n) with high probability. It also confirms previous experimental results that suggested that insertion sort tends to perform better than quicksort in practice.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
  • Sorting
  • Evolving data
  • Insertion sort


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  1. Aris Anagnostopoulos, Ravi Kumar, Mohammad Mahdian, and Eli Upfal. Sorting and selection on dynamic data. Theoretical Computer Science, 412(24):2564-2576, 2011. Special issue on selected papers from 36th International Colloquium on Automata, Languages and Programming (ICALP 2009). URL: http://dx.doi.org/10.1016/j.tcs.2010.10.003.
  2. Aris Anagnostopoulos, Ravi Kumar, Mohammad Mahdian, Eli Upfal, and Fabio Vandin. Algorithms on evolving graphs. In 3rd ACM Innovations in Theoretical Computer Science Conference (ITCS), pages 149-160, 2012. URL: http://dx.doi.org/10.1145/2090236.2090249.
  3. Bahman Bahmani, Ravi Kumar, Mohammad Mahdian, and Eli Upfal. Pagerank on an evolving graph. In 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 24-32, 2012. URL: http://dx.doi.org/10.1145/2339530.2339539.
  4. Juan Jose Besa Vial, William E. Devanny, David Eppstein, Michael T. Goodrich, and Timothy Johnson. Quadratic time algorithms appear to be optimal for sorting evolving data. In Proc. Algorithm Engineering &Experiments (ALENEX 2018), pages 87-96, 2018. URL: http://dx.doi.org/10.1137/1.9781611975055.8.
  5. Mark Braverman and Elchanan Mossel. Noisy sorting without resampling. In 19th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 268-276, 2008. Google Scholar
  6. Thomas H. Cormen, Clifford Stein, Ronald L. Rivest, and Charles E. Leiserson. Introduction to Algorithms. McGraw-Hill Higher Education, 2nd edition, 2001. Google Scholar
  7. Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM Journal on Computing, 23(5):1001-1018, 1994. URL: http://dx.doi.org/10.1137/S0097539791195877.
  8. Michael T. Goodrich and Roberto Tamassia. Algorithm Design and Applications. Wiley Publishing, 1st edition, 2014. Google Scholar
  9. Benoit Groz and Tova Milo. Skyline queries with noisy comparisons. In 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS), pages 185-198, 2015. URL: http://dx.doi.org/10.1145/2745754.2745775.
  10. Dorit S. Hochbaum. Ranking sports teams and the inverse equal paths problem. In Paul Spirakis, Marios Mavronicolas, and Spyros Kontogiannis, editors, 2nd Int. Workshop on Internet and Network Economics (WINE), volume 4286 of Lecture Notes in Computer Science, pages 307-318, Berlin, Heidelberg, 2006. Springer. URL: http://dx.doi.org/10.1007/11944874_28.
  11. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13-30, 1963. URL: http://dx.doi.org/10.1080/01621459.1963.10500830.
  12. Qin Huang, Xingwu Liu, Xiaoming Sun, and Jialin Zhang. Partial sorting problem on evolving data. Algorithmica, 79(3):1-24, 2017. URL: http://dx.doi.org/10.1007/s00453-017-0295-3.
  13. Varun Kanade, Nikos Leonardos, and Frédéric Magniez. Stable Matching with Evolving Preferences. In Klaus Jansen, Claire Mathieu, José D. P. Rolim, and Chris Umans, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), volume 60 of LIPIcs, pages 36:1-36:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.36.
  14. Donald Ervin Knuth. The Art of Computer Programming: Sorting and Searching, volume 3. Pearson Education, 2nd edition, 1998. Google Scholar
  15. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Sorting noisy data with partial information. In 4th ACM Conference on Innovations in Theoretical Computer Science (ITCS), pages 515-528, 2013. URL: http://dx.doi.org/10.1145/2422436.2422492.
  16. Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York, NY, USA, 2005. Google Scholar
  17. Jean Vuillemin. A unifying look at data structures. Commun. ACM, 23(4):229-239, 1980. URL: http://dx.doi.org/10.1145/358841.358852.
  18. Jialin Zhang and Qiang Li. Shortest paths on evolving graphs. In H. Nguyen and V. Snasel, editors, 5th Int. Conf. on Computational Social Networks (CSoNet), volume 9795 of Lecture Notes in Computer Science, pages 1-13, Berlin, Heidelberg, 2016. Springer. URL: http://dx.doi.org/10.1007/978-3-319-42345-6_1.
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