32 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 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