743 research outputs found
The limiting distribution for the number of symbol comparisons used by QuickSort is nondegenerate (extended abstract)
In a continuous-time setting, Fill (2010) proved, for a large class of
probabilistic sources, that the number of symbol comparisons used by QuickSort,
when centered by subtracting the mean and scaled by dividing by time, has a
limiting distribution, but proved little about that limiting random variable Y
-- not even that it is nondegenerate. We establish the nondegeneracy of Y. The
proof is perhaps surprisingly difficult
Parallel String Sample Sort
We discuss how string sorting algorithms can be parallelized on modern
multi-core shared memory machines. As a synthesis of the best sequential string
sorting algorithms and successful parallel sorting algorithms for atomic
objects, we propose string sample sort. The algorithm makes effective use of
the memory hierarchy, uses additional word level parallelism, and largely
avoids branch mispredictions. Additionally, we parallelize variants of multikey
quicksort and radix sort that are also useful in certain situations.Comment: 34 pages, 7 figures and 12 table
Distributional convergence for the number of symbol comparisons used by QuickSort
Most previous studies of the sorting algorithm QuickSort have used the number
of key comparisons as a measure of the cost of executing the algorithm. Here we
suppose that the n independent and identically distributed (i.i.d.) keys are
each represented as a sequence of symbols from a probabilistic source and that
QuickSort operates on individual symbols, and we measure the execution cost as
the number of symbol comparisons. Assuming only a mild "tameness" condition on
the source, we show that there is a limiting distribution for the number of
symbol comparisons after normalization: first centering by the mean and then
dividing by n. Additionally, under a condition that grows more restrictive as p
increases, we have convergence of moments of orders p and smaller. In
particular, we have convergence in distribution and convergence of moments of
every order whenever the source is memoryless, that is, whenever each key is
generated as an infinite string of i.i.d. symbols. This is somewhat surprising;
even for the classical model that each key is an i.i.d. string of unbiased
("fair") bits, the mean exhibits periodic fluctuations of order n.Comment: Published in at http://dx.doi.org/10.1214/12-AAP866 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm
There is excitement within the algorithms community about a new partitioning
method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly
faster than the case when it runs under classic partitioning methods. We show
that this improved performance in Quicksort is not sustained in Quickselect; a
variant of Quicksort for finding order statistics. We investigate the number of
comparisons made by Quickselect to find a key with a randomly selected rank
under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator
over all individual distributions for specific fixed order statistics. We give
the exact grand average. The grand distribution of the number of comparison
(when suitably scaled) is given as the fixed-point solution of a distributional
equation of a contraction in the Zolotarev metric space. Our investigation
shows that Quickselect under older partitioning methods slightly outperforms
Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random
rank. Similar results are obtained for extremal order statistics, where again
we find the exact average, and the distribution for the number of comparisons
(when suitably scaled). Both limiting distributions are of perpetuities (a sum
of products of independent mixed continuous random variables).Comment: full version with appendices; otherwise identical to Algorithmica
versio
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