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 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. The proofs use
methods originally developed for random graphs.Comment: 31 PAGE
