2007/01/10 by Svante Janson, Janson, Svante
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60C05 #68W40 #Algorithms and Data Compression #FOS: Mathematics #Genome Rearrangement Algorithms #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60C05 #msc:68W40
paper · pdf · doi:10.48550/arxiv.math/0701288
31 PAGES
arxiv created 2007/01/10 · openalex publication_date 2007/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 then 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 n1/2, 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 n1/3. We also treat some variations, including priority queues. The proofs use methods originally developed for random graphs.