Abstract
We settle a long-standing open question, namely whether it is possible to sort a sequence of n elements stably (i.e. preserving the original relative order of the equal elements), using O(1) auxiliary space and performing O(n log n) comparisons and O(n) data moves. Munro and Raman stated this problem in [J. Algorithms, 13, 1992] and gave an in-place but unstable sorting algorithm that performs O(n) data moves and O(n 1 + ε) comparisons. Subsequently [Algorithmica, 16, 1996] they presented a stable algorithm with these same bounds. Recently, Franceschini and Geffert [FOCS 2003] presented an unstable sorting algorithm that matches the asymptotic lower bounds on all computational resources.
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References
Franceschini, G., Geffert, V.: An In-Place Sorting with O(nlogn) Comparisons and O(n) Moves. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS (2003)
Franceschini, G., Grossi, R.: Optimal cache-oblivious implicit dictionaries. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719. Springer, Heidelberg (2003)
Hoare, C.A.R.: Quicksort. The Computer Journal 5(1), 10–16 (1962)
Katajainen, J., Pasanen, T.: Sorting multisets stably in minimum space. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 410–421. Springer, Heidelberg (1992)
Katajainen, J., Pasanen, T.: Stable minimum space partitioning in linear time. BIT 32(4), 580–585 (1992)
Katajainen, J., Pasanen, T.: In-place sorting with fewer moves. Inform. Process. Lett. 70, 31–37 (1999)
Knuth, D.E.: The Art of Computer Programming. Sorting and Searching, vol. 3. Addison-Wesley, Reading (1973)
Kronrod, M.A.: Optimal ordering algorithm without operational field. Soviet Math. Dokl. 10, 744–746 (1969)
Lai, T.W., Wood, D.: Implicit selection. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 14–23. Springer, Heidelberg (1988)
Munro, J.I.: An implicit data structure supporting insertion, deletion, and search in O(log2 n) time. J. Comput. System Sci. 33, 66–74 (1986)
Munro, J.I., Raman, V.: Sorting with minimum data movement. J. Algorithms 13, 374–393 (1992)
Munro, J.I., Raman, V.: Fast stable in-place sorting with O(n) data moves. Algorithmica 16, 151–160 (1996)
Munro, J.I., Raman, V.: Selection from read-only memory and sorting with minimum data movement. Theoret. Comput. Sci. 165, 311–323 (1996)
Pardo, L.T.: Stable sorting and merging with optimal space and time bounds. SIAM Journal on Computing 6(2), 351–372 (1977)
Salowe, J., Steiger, W.: Simplified stable merging tasks. Journal of Algorithms 8(4), 557–571 (1987)
Williams, J.W.J.: Heapsort (Algorithm 232). Comm. Assoc. Comput. Mach. 7, 347–348 (1964)
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Franceschini, G. (2005). Sorting Stably, In-Place, with O(n log n) Comparisons and O(n) Moves. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_52
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DOI: https://doi.org/10.1007/978-3-540-31856-9_52
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