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 matching upper and lower bounds for $α$-merge sort, when $φ< α< 2$, where $φ$ is the golden ratio.
Our bounds are in terms of merge cost; this upper bounds the number of comparisons and accurately models runtime. The merge strategies can be used for any stable merge sort, not just natural merge sorts. The new $2$-merge and $α$-merge sorts have better worst-case merge cost upper bounds and are slightly simpler to implement than the widely-used Timsort; they also perform better in experiments. We report also experimental comparisons with algorithms developed by Munro-Wild and Jugé subsequently to the results of the present paper.
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.
Sorting is one of the oldest computing problems and is still very important in the age of big data. Various algorithms and implementation techniques have been proposed. In this study, we focus on comparison based, internal sorting algorithms. We created 12 data types of various sizes for experiments and tested extensively various implementations in a single setting. Using some effective techniques, we discovered that quicksort is adaptive to nearly sorted inputs and is still the best overall sorting algorithm. We also identified which techniques are effective in timsort, one of the most popula…
The two most prominent solutions for the sorting problem are Quicksort and Mergesort. While Quicksort is very fast on average, Mergesort additionally gives worst-case guarantees, but needs extra space for a linear number of elements. Worst-case efficient in-place sorting, however, remains a challenge: the standard solution, Heapsort, suffers from a bad cache behavior and is also not overly fast for in-cache instances. In this work we present median-of-medians QuickMergesort (MoMQuickMergesort), a new variant of QuickMergesort, which combines Quicksort with Mergesort allowing the latter to be i…
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.
Sorting is a primitive operation that is a building block for countless algorithms. As such, it is important to design sorting algorithms that approach peak performance on a range of hardware architectures. Graphics Processing Units (GPUs) are particularly attractive architectures as they provides massive parallelism and computing power. However, the intricacies of their compute and memory hierarchies make designing GPU-efficient algorithms challenging. In this work we present GPU Multiway Mergesort (MMS), a new GPU-efficient multiway mergesort algorithm. MMS employs a new partitioning techniq…
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…
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.
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…
The objective behind the Twin Sort technique is to sort the list of unordered data elements efficiently and to allow efficient and simple arrangement of data elements within the data structure with optimization of comparisons and iterations in the sorting method. This sorting technique effectively terminates the iterations when there is no need of comparison if the elements are all sorted in between the iterations. Unlike Quick sort, Merge sorting technique, this new sorting technique is based on the iterative method of sorting elements within the data structure. So it will be advantageous for…
In this paper we present a framework for designing algorithms in shared memory of GPUs without incurring memory bank conflicts. Using our framework we develop the first comparison-based shared memory sorting algorithm that incurs no bank conflicts. It can be used as a subroutine for GPU sorting algorithms to replace current use of sorting networks in shared memory. Using our bank conflict free shared memory sorting subroutine as a black box, we design BCFMergesort, an algorithm for merging sorted streams of data that are larger than shared memory. Our algorithm performs all accesses to global …
This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is $n \lg n - 1.4427n + O(\log n)$. For many efficient algorithms, the first $n\lg n$ term is easy to achieve and our focus is on the (negative) constant factor of the linear term. The current best value is $-1.3999$ for the MergeInsertion sort. Our new value is $-1.4106$, narrowing the gap by some $25\%$. An important building block of our algorithm is "two-element insertion," which inserts two numbers $A$ and $B$, $A<B$, into a sorted sequence $T$. This inserti…
We consider the fundamental problem of internally sorting a sequence of $n$ elements. In its best theoretical setting QuickMergesort, a combination Quicksort with Mergesort with a Median-of-$\sqrt{n}$ pivot selection, requires at most $n \log n - 1.3999n + o(n)$ element comparisons on the average. The questions addressed in this paper is how to make this algorithm practical. As refined pivot selection usually adds much overhead, we show that the Median-of-3 pivot selection of QuickMergesort leads to at most $n \log n - 0{.}75n + o(n)$ element comparisons on average, while running fast on eleme…
Emerging memory technologies have a significant gap between the cost, both in time and in energy, of writing to memory versus reading from memory. In this paper we present models and algorithms that account for this difference, with a focus on write-efficient sorting algorithms. First, we consider the PRAM model with asymmetric write cost, and show that sorting can be performed in $O\left(n\right)$ writes, $O\left(n \log n\right)$ reads, and logarithmic depth (parallel time). Next, we consider a variant of the External Memory (EM) model that charges $ω> 1$ for writing a block of size $B$ to th…
We present a stable mergesort variant, Multiway Powersort, that exploits existing runs and finds nearly-optimal merging orders for k-way merges with negligible overhead. This builds on Powersort (Munro & Wild, ESA2018), which has recently replaced Timsort's suboptimal merge policy in the CPython reference implementation of Python, as well as in PyPy and further libraries. Multiway Powersort reduces the number of memory transfers, which increasingly determine the cost of internal sorting (as observed with Multiway Quicksort (Kushagra et al., ALENEX 2014; Aumüller & Dietzfelbinger, TALG 2016; Wi…
Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of $O(N (\log N)^{M - 1}…
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…
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…
Research Institute for Symbolic Computation, Johannes Kepler University（约翰内斯·开普勒大学符号计算研究所）
Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as a set of rewrite rules based on the witnesses found in the proof of the synthesis conjecture produced from the specification of the desired function (input and output conditions). The proofs are in natural style, using standard rules, but most importantly domain specific inference rules and strategies. In particular the use of multisets allows us to develop …
MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of $n \log n - 1.4005n + o(n)$ comparisons. We also give an exact description of the probability distribution of the length of the chain a g…
We present a deterministic sorting algorithm, SPMS (Sample, Partition, and Merge Sort), that interleaves the partitioning of a sample sort with merging. Sequentially, it sorts $n$ elements in $O(n \log n)$ time cache-obliviously with an optimal number of cache misses. The parallel complexity (or critical path length) of the algorithm is $O(\log n \cdot \log\log n)$, which improves on previous bounds for optimal cache oblivious sorting. The algorithm also has low false sharing costs. When scheduled by a work-stealing scheduler in a multicore computing environment with a global shared memory and…
Sorting is at the core of many database operations, such as index creation, sort-merge joins, and user-requested output sorting. As GPUs are emerging as a promising platform to accelerate various operations, sorting on GPUs becomes a viable endeavour. Over the past few years, several improvements have been proposed for sorting on GPUs, leading to the first radix sort implementations that achieve a sorting rate of over one billion 32-bit keys per second. Yet, state-of-the-art approaches are heavily memory bandwidth-bound, as they require substantially more memory transfers than their CPU-based …
Multi-Pivot Quicksort refers to variants of classical quicksort where in the partitioning step $k$ pivots are used to split the input into $k + 1$ segments. For many years, multi-pivot quicksort was regarded as impractical, but in 2009 a 2-pivot approach by Yaroslavskiy, Bentley, and Bloch was chosen as the standard sorting algorithm in Sun's Java 7. In 2014 at ALENEX, Kushagra et al. introduced an even faster algorithm that uses three pivots. This paper studies what possible advantages multi-pivot quicksort might offer in general. The contributions are as follows: Natural comparison-optimal a…
We propose a conceptually simple oblivious sort and oblivious random permutation algorithms called bucket oblivious sort and bucket oblivious random permutation. Bucket oblivious sort uses $6n\log n$ time (measured by the number of memory accesses) and $2Z$ client storage with an error probability exponentially small in $Z$. The above runtime is only $3\times$ slower than a non-oblivious merge sort baseline; for $2^{30}$ elements, it is $5\times$ faster than bitonic sort, the de facto oblivious sorting algorithm in practical implementations.
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…
