Introspective Sorting and Selection Algorithms

Abstract

SUMMARYQuicksort is the preferred in-place sorting algorithm in many contexts, since its average computing time on uniformly distributed inputs is Θ (N log N), and it is in fact faster than most other sorting algorithms on most inputs. Its drawback is that its worst-case time bound is Θ(N 2 . Previous attempts to protect against the worst case by improving the way quicksort chooses pivot elements for partitioning have increased the average computing time too much -one might as well use heapsort, which has a Θ(N log N) worst-case time bound, but is on the average 2-5 times slower than quicksort. A similar dilemma exists with selection algorithms (for finding the i-th largest element) based on partitioning. This paper describes a simple solution to this dilemma: limit the depth of partitioning, and for subproblems that exceed the limit switch to another algorithm with a better worst-case bound. Using heapsort as the 'stopper' yields a sorting algorithm that is just as fast as quicksort in the average case, but also has an Θ(N log N) worst case time bound. For selection, a hybrid of Hoare's FIND algorithm, which is linear on average but quadratic in the worst case, and the Blum-Floyd-Pratt-Rivest-Tarjan algorithm is as fast as Hoare's algorithm in practice, yet has a linear worst-case time bound. Also discussed are issues of implementing the new algorithms as generic algorithms, and accurately measuring their performance in the framework of the C++ Standard Template Library.

Faster set intersection with SIMD instructions by reducing branch mispredictions

“…An adaptive fallback using a runtime check is a standard heuristic technique to avoid worst case performance in many algorithms. For example, introsort [19] used in the STL's std::sort library method uses quicksort with adaptive fallback to heapsort to avoid the O(N 2 ) worst-case performance of quicksort. From the results shown in Figure 10, for two input arrays with comparable sizes, we start execution with our SIMD algorithm using the block size setting of 4x4 (or 8x8 if we use STTNI on Xeon).…”

Massively parallel sort-merge joins in main memory multi-core database systems

“…For the disk-enabled Vectorwise this is achieved by executing the query several times and reporting only the execution times of runs after the data was fully resident in RAM. In order to cover the most important scenarios, we report benchmark results using datasets Core 2 (8,40) Core 4 (16,48) Core 6 (24,56) Core 1 (4,36) Core 3 (12,44) Core 5 (20,52) Core 7…”

“…Section 5, Figure 11). We therefore instantiated 32 threads to work on one relation with a total of 1600M (throughout the paper we use M = 2 20 ) tuples, each consisting of a 64-bit sort key and a 64-bit payload, in parallel. (1) We first chunked the relation and sorted the chunks of 50M tuples each as runs in parallel.…”

A Practical Quicksort Algorithm for Graphics Processors

“…Introsort is based on Quicksort, but switches to heap-sort when the recursion depth gets too large. Since it is highly dependent on the computer system and compiler used, we only included it to give a hint as to what could be gained by sorting on the GPU instead of on the CPU [19].…”