We study the impact of merging routines in merge-based sorting algorithms. More precisely, we focus on the galloping routine that TimSort uses to merge monotonic sub-arrays, hereafter called runs, and on the impact on the number of element comparisons performed if one uses this routine instead of a naïve merging routine.
This routine was introduced in order to make TimSort more efficient on arrays with few distinct values. Alas, we prove that, although it makes TimSort sort array with two values in linear time, it does not prevent TimSort from requiring up to $Θ(n \log(n))$ element comparisons to sort arrays of length~$n$ with three distinct values. However, we also prove that slightly modifying TimSort's galloping routine results in requiring only $\mathcal{O}(n + n \log(σ))$ element comparisons in the worst case, when sorting arrays of length $n$ with $σ$ distinct values.
We do so by focusing on the notion of dual runs, which was introduced in the 1990s, and on the associated dual run-length entropy. This notion is both related to the number of distinct values and to the number of runs in an array, which came with its own run-length entropy that was used to explain TimSort's otherwise "supernatural" efficiency. We also introduce new notions of fast- and middle-growth for natural merge sorts (i.e., algorithms based on merging runs), which are found in several merge sorting algorithms similar to TimSort.
We prove that algorithms with the fast- or middle-growth property, provided that they use our variant of TimSort's galloping routine for merging runs, are as efficient as possible at sorting arrays with low run-induced or dual-run-induced complexities.
We introduce new stable natural merge sort algorithms, called $2$-merge sort and $α$-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shivers' sort, $α$-stack sorts, and our new $2$-merge and $α$-merge sorts. The upper and lower bounds have the forms $c \cdot n \log m$ and $c \cdot n \log n$ for inputs of length~$n$ comprising $m$~monotone runs. For Timsort, we prove a lower bound of $(1.5 - o(1)) n \log n$. For $2$-merge sort, we prove optimal upper and lower bounds of approximately $(1.089 \pm o(1))n \log m$. We prove similar asymptotically ma…
We present two stable mergesort variants, "peeksort" and "powersort", that exploit existing runs and find nearly-optimal merging orders with practically negligible overhead. Previous methods either require substantial effort for determining the merging order (Takaoka 2009; Barbay & Navarro 2013) or do not have a constant-factor optimal worst-case guarantee (Peters 2001; Auger, Nicaud & Pivoteau 2015; Buss & Knop 2018). We demonstrate that our methods are competitive in terms of running time with state-of-the-art implementations of stable sorting methods.
We present one stable mergesort algorithm, called \Adaptive Shivers Sort, that exploits the existence of monotonic runs for sorting efficiently partially sorted data. We also prove that, although this algorithm is simple to implement, its computational cost, in number of comparisons performed, is optimal up to a small additive linear term.
The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can manipulate our graph such that the resulting topological sort is the sorted array. By taking advantage of the many relations between Hamiltonian paths and topological sorts in comparison graphs, we design a Divide-and-Conquer algorithm that runs in the optimal $O(n \log n)$ time. In the process, we construct a new merge process for graphs with relevant invariant pr…
FAST problem is finding minimum feedback arc set problem in tournaments. In this paper we present some algorithms that are similar to sorting algorithms for FAST problem and we analyze them. We present Pseudo_InsertionSort algorithm for FAST problem and we show that average number of all backward edges in output of that is equal to ((n^2-5n+8)/4)-2^(1-n). We introduce Pseudo_MergeSort algorithm and we find the probability of being backward for an edge. Finally we introduce other algorithms for this problem.
Sorting is an essential operation which is widely used and is fundamental to some very basic day to day utilities like searches, databases, social networks and much more. Optimizing this basic operation in terms of complexity as well as efficiency is cardinal. Optimization is achieved with respect to space and time complexities of the algorithm. In this paper, a novel left-field N-dimensional cartesian spaced sorting method is proposed by combining the best characteristics of bucket sort, counting sort and radix sort, in addition to employing hashing and dynamic programming for making the meth…
Sorting is the one of the fundamental tasks of modern data management systems. With Disk I/O being the most-accused performance bottleneck and more computation-intensive workloads, it has come to our attention that in heterogeneous environment, performance bottleneck may vary among different infrastructure. As a result, sort kernels need to be adaptive to changing hardware conditions. In this paper, we propose Leyenda, a hybrid, parallel and efficient Radix Most-Significant-Bit (MSB) MergeSort algorithm, with utilization of local thread-level CPU cache and efficient disk/memory I/O. Leyenda is…
QuickXsort is a highly efficient in-place sequential sorting scheme that mixes Hoare's Quicksort algorithm with X, where X can be chosen from a wider range of other known sorting algorithms, like Heapsort, Insertionsort and Mergesort. Its major advantage is that QuickXsort can be in-place even if X is not. In this work we provide general transfer theorems expressing the number of comparisons of QuickXsort in terms of the number of comparisons of X. More specifically, if pivots are chosen as medians of (not too fast) growing size samples, the average number of comparisons of QuickXsort and X di…
In the age of big data, sorting is an indispensable operation for DBMSes and similar systems. Having data sorted can help produce query plans with significantly lower run times. It also can provide other benefits like having non-blocking operators which will produce data steadily (without bursts), or operators with reduced memory footprint. Sorting may be required on any step of query processing, i.e., be it source data or intermediate results. At the same time, the data to be sorted may not fit into main memory. In this case, an external sort operator, which writes intermediate results to dis…
In this paper, we present several improvements in the parallelization of the in-place merge algorithm, which merges two contiguous sorted arrays into one with an O(T) space complexity (where T is the number of threads). The approach divides the two arrays into as many pairs of partitions as there are threads available; such that each thread can later merge a pair of partitions independently of the others. We extend the existing method by proposing a new algorithm to find the median of two partitions. Additionally, we provide a new strategy to divide the input arrays where we minimize the data …
