142 research outputs found
Analysis of pivot sampling in dual-pivot Quicksort: A holistic analysis of Yaroslavskiy's partitioning scheme
The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-015-0041-7The new dual-pivot Quicksort by Vladimir Yaroslavskiy-used in Oracle's Java runtime library since version 7-features intriguing asymmetries. They make a basic variant of this algorithm use less comparisons than classic single-pivot Quicksort. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample. Surprisingly, dual-pivot Quicksort then needs more comparisons than a corresponding version of classic Quicksort, so it is clear that counting comparisons is not sufficient to explain the running time advantages observed for Yaroslavskiy's algorithm in practice. Consequently, we take a more holistic approach and give also the precise leading term of the average number of swaps, the number of executed Java Bytecode instructions and the number of scanned elements, a new simple cost measure that approximates I/O costs in the memory hierarchy. We determine optimal order statistics for each of the cost measures. It turns out that the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, we finally have a convincing explanation for the success of Yaroslavskiy's algorithm in practice: compared with corresponding versions of classic single-pivot Quicksort, dual-pivot Quicksort needs significantly less I/Os, both with and without pivot sampling.Peer ReviewedPostprint (author's final draft
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
Why Is Dual-Pivot Quicksort Fast?
I discuss the new dual-pivot Quicksort that is nowadays used to sort arrays
of primitive types in Java. I sketch theoretical analyses of this algorithm
that offer a possible, and in my opinion plausible, explanation why (a)
dual-pivot Quicksort is faster than the previously used (classic) Quicksort and
(b) why this improvement was not already found much earlier.Comment: extended abstract for Theorietage 2015
(https://www.uni-trier.de/index.php?id=55089) (v2 fixes a small bug in the
pseudocode
Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths
We present an average case analysis of a variant of dual-pivot quicksort. We
show that the used algorithmic partitioning strategy is optimal, i.e., it
minimizes the expected number of key comparisons. For the analysis, we
calculate the expected number of comparisons exactly as well as asymptotically,
in particular, we provide exact expressions for the linear, logarithmic, and
constant terms.
An essential step is the analysis of zeros of lattice paths in a certain
probability model. Along the way a combinatorial identity is proven.Comment: This article supersedes arXiv:1602.0403
How Good Is Multi-Pivot Quicksort?
Multi-Pivot Quicksort refers to variants of classical quicksort where in the
partitioning step pivots are used to split the input into segments.
For many years, multi-pivot quicksort was regarded as impractical, but in 2009
a 2-pivot approach by Yaroslavskiy, Bentley, and Bloch was chosen as the
standard sorting algorithm in Sun's Java 7. In 2014 at ALENEX, Kushagra et al.
introduced an even faster algorithm that uses three pivots. This paper studies
what possible advantages multi-pivot quicksort might offer in general. The
contributions are as follows: Natural comparison-optimal algorithms for
multi-pivot quicksort are devised and analyzed. The analysis shows that the
benefits of using multiple pivots with respect to the average comparison count
are marginal and these strategies are inferior to simpler strategies such as
the well known median-of- approach. A substantial part of the partitioning
cost is caused by rearranging elements. A rigorous analysis of an algorithm for
rearranging elements in the partitioning step is carried out, observing mainly
how often array cells are accessed during partitioning. The algorithm behaves
best if 3 to 5 pivots are used. Experiments show that this translates into good
cache behavior and is closest to predicting observed running times of
multi-pivot quicksort algorithms. Finally, it is studied how choosing pivots
from a sample affects sorting cost. The study is theoretical in the sense that
although the findings motivate design recommendations for multipivot quicksort
algorithms that lead to running time improvements over known algorithms in an
experimental setting, these improvements are small.Comment: Submitted to a journal, v2: Fixed statement of Gibb's inequality, v3:
Revised version, especially improving on the experiments in Section
Pivot Sampling in Dual-Pivot Quicksort
The new dual-pivot Quicksort by Vladimir Yaroslavskiy - used in Oracle's Java
runtime library since version 7 - features intriguing asymmetries in its
behavior. They were shown to cause a basic variant of this algorithm to use
less comparisons than classic single-pivot Quicksort implementations. In this
paper, we extend the analysis to the case where the two pivots are chosen as
fixed order statistics of a random sample and give the precise leading term of
the average number of comparisons, swaps and executed Java Bytecode
instructions. It turns out that - unlike for classic Quicksort, where it is
optimal to choose the pivot as median of the sample - the asymmetries in
Yaroslavskiy's algorithm render pivots with a systematic skew more efficient
than the symmetric choice. Moreover, the optimal skew heavily depends on the
employed cost measure; most strikingly, abstract costs like the number of swaps
and comparisons yield a very different result than counting Java Bytecode
instructions, which can be assumed most closely related to actual running time.Comment: presented at AofA 2014 (http://www.aofa14.upmc.fr/
Analysis of Branch Misses in Quicksort
The analysis of algorithms mostly relies on counting classic elementary
operations like additions, multiplications, comparisons, swaps etc. This
approach is often sufficient to quantify an algorithm's efficiency. In some
cases, however, features of modern processor architectures like pipelined
execution and memory hierarchies have significant impact on running time and
need to be taken into account to get a reliable picture. One such example is
Quicksort: It has been demonstrated experimentally that under certain
conditions on the hardware the classically optimal balanced choice of the pivot
as median of a sample gets harmful. The reason lies in mispredicted branches
whose rollback costs become dominating.
In this paper, we give the first precise analytical investigation of the
influence of pipelining and the resulting branch mispredictions on the
efficiency of (classic) Quicksort and Yaroslavskiy's dual-pivot Quicksort as
implemented in Oracle's Java 7 library. For the latter it is still not fully
understood why experiments prove it 10% faster than a highly engineered
implementation of a classic single-pivot version. For different branch
prediction strategies, we give precise asymptotics for the expected number of
branch misses caused by the aforementioned Quicksort variants when their pivots
are chosen from a sample of the input. We conclude that the difference in
branch misses is too small to explain the superiority of the dual-pivot
algorithm.Comment: to be presented at ANALCO 201
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