Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence: Extended abstract.

Svante Janson (1)

1 Department of Mathematics [Uppsala]

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.

Mots clés: sorting algorithm; runs; priority queues; sock-sorting; evolution of random strings; Brownian motion

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Géométrie algorithmique [cs.CG]

Svante Janson. Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence: Extended abstract.. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. pp.259-270, ⟨10.46298/dmtcs.3548⟩. ⟨hal-01184796⟩