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Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 12, no 4, 417-447 p.Article in journal (Refereed) Published
Abstract [en]

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent   to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n (1/2), and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n (1/3). We also treat some variations, including priority queues and sock-sorting. The proofs use methods originally   developed for random graphs.

Place, publisher, year, edition, pages
2009. Vol. 12, no 4, 417-447 p.
Keyword [en]
sorting algorithm, runs, priority queues, sock-sorting, evolution of random strings, Brownian motion
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URN: urn:nbn:se:uu:diva-114320DOI: 10.1007/s00026-009-0007-zISI: 000264809400005OAI: oai:DiVA.org:uu-114320DiVA: diva2:293731
Available from: 2010-02-12 Created: 2010-02-12 Last updated: 2010-08-03Bibliographically approved

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Janson, Svante
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