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

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

Applying Length-Dependent Stochastic Context-Free Grammars to RNA Secondary Structure Prediction

Weinberg F, Nebel M. Applying Length-Dependent Stochastic Context-Free Grammars to RNA Secondary Structure Prediction. Algorithms. 2011;4(4):223--238

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/

Random generation of RNA secondary structures according to native distributions

Nebel M, Scheid A, Weinberg F. Random generation of RNA secondary structures according to native distributions. Algorithms for Molecular Biology. 2011;6(1): 24

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

GeFaST: An improved method for OTU assignment by generalising Swarm’s fastidious clustering approach

Müller R, Nebel M. GeFaST: An improved method for OTU assignment by generalising Swarm’s fastidious clustering approach. BMC Bioinformatics. 2018;19(1): 321.**Background**: Massive genomic data sets from high-throughput sequencing allow for new insights into complex biological systems such as microbial communities. Analyses of their diversity and structure are typically preceded by clustering millions of 16S rRNA gene sequences into OTUs. Swarm introduced a new clustering strategy which addresses important conceptual and performance issues of the popular de novo clustering approach. However, some parts of the new strategy, e.g. the fastidious option for increased clustering quality, come with their own restrictions. **Results:** In this paper, we present the new exact, alignment-based de novo clustering tool GeFaST, which implements a generalisation of Swarm’s fastidious clustering. Our tool extends the fastidious option to arbitrary clustering thresholds and allows to adjust its greediness. GeFaST was evaluated on mock-community and natural data and achieved higher clustering quality and performance for small to medium clustering thresholds compared to Swarm and other de novo tools. Clustering with GeFaST was between 6 and 197 times as fast as with Swarm, while the latter required up to 38% less memory for non-fastidious clustering but at least three times as much memory for fastidious clustering. **Conclusions:** GeFaST extends the scope of Swarm’s clustering strategy by generalising its fastidious option, thereby allowing for gains in clustering quality, and by increasing its performance (especially in the fastidious case). Our evaluations showed that GeFaST has the potential to leverage the use of the (fastidious) clustering strategy for higher thresholds and on larger data sets

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