An O(n^{2.75}) algorithm for online topological ordering

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n^{2.75}) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min(m^{3/2} log(n), m^{3/2} + n^2 log(n)) by Katriel and Bodlaender. We also provide an empirical comparison of our algorithm with other algorithms for online topological sorting. Our implementation outperforms them on certain hard instances while it is still competitive on random edge insertion sequences leading to complete DAGs.Comment: 20 pages, long version of SWAT'06 pape

QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average

In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithms that incur approximately n log n - 1.26n +o(n) comparisons on the average. A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n. Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs: when sorting integers it is slower by only 15% to STL-Introsort

RAM-Efficient External Memory Sorting

In recent years a large number of problems have been considered in external memory models of computation, where the complexity measure is the number of blocks of data that are moved between slow external memory and fast internal memory (also called I/Os). In practice, however, internal memory time often dominates the total running time once I/O-efficiency has been obtained. In this paper we study algorithms for fundamental problems that are simultaneously I/O-efficient and internal memory efficient in the RAM model of computation.Comment: To appear in Proceedings of ISAAC 2013, getting the Best Paper Awar

Computational Complexity Results for Genetic Programming and the Sorting Problem

Genetic Programming (GP) has found various applications. Understanding this type of algorithm from a theoretical point of view is a challenging task. The first results on the computational complexity of GP have been obtained for problems with isolated program semantics. With this paper, we push forward the computational complexity analysis of GP on a problem with dependent program semantics. We study the well-known sorting problem in this context and analyze rigorously how GP can deal with different measures of sortedness.Comment: 12 page

The Folklore of Sorting Algorithms

The objective of this paper is to review the folklore knowledge seen in research work devoted on synthesis, optimization, and effectiveness of various sorting algorithms. We will examine sorting algorithms in the folklore lines and try to discover the tradeoffs between folklore and theorems. Finally, the folklore knowledge on complexity values of the sorting algorithms will be considered, verified and subsequently converged in to theorems

Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

Engineering Parallel String Sorting

We discuss how string sorting algorithms can be parallelized on modern multi-core shared memory machines. As a synthesis of the best sequential string sorting algorithms and successful parallel sorting algorithms for atomic objects, we first propose string sample sort. The algorithm makes effective use of the memory hierarchy, uses additional word level parallelism, and largely avoids branch mispredictions. Then we focus on NUMA architectures, and develop parallel multiway LCP-merge and -mergesort to reduce the number of random memory accesses to remote nodes. Additionally, we parallelize variants of multikey quicksort and radix sort that are also useful in certain situations. Comprehensive experiments on five current multi-core platforms are then reported and discussed. The experiments show that our implementations scale very well on real-world inputs and modern machines.Comment: 46 pages, extension of "Parallel String Sample Sort" arXiv:1305.115