78 research outputs found
Fast Deterministic Selection
The Median of Medians (also known as BFPRT) algorithm, although a landmark
theoretical achievement, is seldom used in practice because it and its variants
are slower than simple approaches based on sampling. The main contribution of
this paper is a fast linear-time deterministic selection algorithm
QuickselectAdaptive based on a refined definition of MedianOfMedians. The
algorithm's performance brings deterministic selection---along with its
desirable properties of reproducible runs, predictable run times, and immunity
to pathological inputs---in the range of practicality. We demonstrate results
on independent and identically distributed random inputs and on
normally-distributed inputs. Measurements show that QuickselectAdaptive is
faster than state-of-the-art baselines.Comment: Pre-publication draf
Worst-Case Efficient Sorting with QuickMergesort
The two most prominent solutions for the sorting problem are Quicksort and
Mergesort. While Quicksort is very fast on average, Mergesort additionally
gives worst-case guarantees, but needs extra space for a linear number of
elements. Worst-case efficient in-place sorting, however, remains a challenge:
the standard solution, Heapsort, suffers from a bad cache behavior and is also
not overly fast for in-cache instances.
In this work we present median-of-medians QuickMergesort (MoMQuickMergesort),
a new variant of QuickMergesort, which combines Quicksort with Mergesort
allowing the latter to be implemented in place. Our new variant applies the
median-of-medians algorithm for selecting pivots in order to circumvent the
quadratic worst case. Indeed, we show that it uses at most
comparisons for large enough.
We experimentally confirm the theoretical estimates and show that the new
algorithm outperforms Heapsort by far and is only around 10% slower than
Introsort (std::sort implementation of stdlibc++), which has a rather poor
guarantee for the worst case. We also simulate the worst case, which is only
around 10% slower than the average case. In particular, the new algorithm is a
natural candidate to replace Heapsort as a worst-case stopper in Introsort
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
Sesquickselect: One and a half pivots for cache-efficient selection
Because of unmatched improvements in CPU performance, memory transfers have become a bottleneck of program execution. As discovered in recent years, this also affects sorting in internal memory. Since partitioning around several pivots reduces overall memory transfers, we have seen renewed interest in multiway Quicksort. Here, we analyze in how far multiway partitioning helps in Quickselect. We compute the expected number of comparisons and scanned elements (approximating memory transfers) for a generic class of (non-adaptive) multiway Quickselect and show that three or more pivots are not helpful, but two pivots are. Moreover, we consider "adaptive" variants which choose partitioning and pivot-selection methods in each recursive step from a finite set of alternatives depending on the current (relative) sought rank. We show that "Sesquickselect", a new Quickselect variant that uses either one or two pivots, makes better use of small samples w.r.t. memory transfers than other Quickselect variants
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
Max-Plus Algebraic Statistical Leverage Scores
The statistical leverage scores of a matrix A ∈ Rn×d record the degree of alignment between col(A) and the coordinate axes in Rn. These scores are used in random sampling algorithms for solving certain numerical linear algebra problems. In this paper we present a max-plus algebraic analogue of statistical leverage scores. We show that max-plus statistical leverage scores can be used to calculate the exact asymptotic behavior of the conventional statistical leverage scores of a generic radial basis function network (RBFN) matrix. We also show how max-plus statistical leverage scores can provide a novel way to approximate the conventional statistical leverage scores of a fixed, nonparametrized matrix
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/
- …