20 research outputs found
Average-Case Complexity of Shellsort
We prove a general lower bound on the average-case complexity of Shellsort:
the average number of data-movements (and comparisons) made by a -pass
Shellsort for any incremental sequence is \Omega (pn^{1 + 1/p) for all . Using similar arguments, we analyze the average-case complexity
of several other sorting algorithms.Comment: 11 pages. Submitted to ICALP'9
The Average-Case Area of Heilbronn-Type Triangles
From among triangles with vertices chosen from points in
the unit square, let be the one with the smallest area, and let be the
area of . Heilbronn's triangle problem asks for the maximum value assumed by
over all choices of points. We consider the average-case: If the
points are chosen independently and at random (with a uniform distribution),
then there exist positive constants and such that for all large enough values of , where is the expectation of
. Moreover, , with probability close to one. Our proof
uses the incompressibility method based on Kolmogorov complexity; it actually
determines the area of the smallest triangle for an arrangement in ``general
position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a
roll, {\em New Scientist}, November 6, 1999, 44--4
Spin-the-bottle Sort and Annealing Sort: Oblivious Sorting via Round-robin Random Comparisons
We study sorting algorithms based on randomized round-robin comparisons.
Specifically, we study Spin-the-bottle sort, where comparisons are
unrestricted, and Annealing sort, where comparisons are restricted to a
distance bounded by a \emph{temperature} parameter. Both algorithms are simple,
randomized, data-oblivious sorting algorithms, which are useful in
privacy-preserving computations, but, as we show, Annealing sort is much more
efficient. We show that there is an input permutation that causes
Spin-the-bottle sort to require expected time in order to
succeed, and that in time this algorithm succeeds with high
probability for any input. We also show there is an implementation of Annealing
sort that runs in time and succeeds with very high probability.Comment: Full version of a paper appearing in ANALCO 2011, in conjunction with
SODA 201