A complete set of filters $F_n$ for the optimal-depth $n$-input sorting network problem is such that if there exists an $n$-input sorting network of depth $d$ then there exists one of the form $C \oplus C'$ for some $C \in F_n$. Previous work on the topic presents a method for finding complete set of filters $R_{n, 1}$ and $R_{n, 2}$ that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. Our main contribution is a practical approach for finding a complete set of filters $R_{n, 3}$ containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For $n < 17$, we empirically compute the complete set of filters $R_{n, 2}$, $R_{n, 3}$, $R_{n, 2} \upharpoonright w $ and $R_{n, 3}^w$ of the representative minimal up to permutation and reflection $n$-input networks, where all but $R_{n, 2}$ are novel to this work.
Sorting networks are oblivious sorting algorithms with many practical applications and rich theoretical properties. Propositional encodings of sorting networks are a key tool for proving concrete bounds on the minimum number of comparators or depth (number of parallel steps) of sorting networks. In this paper, we present new SAT encodings that reduce the number of variables and clauses of the sorting constraint of optimality problems. Moreover, the proposed SAT encodings can be applied to a broader class of problems, such as the search of optimal single-exception sorting networks and $ε-$halve…
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.
In this paper a new method for checking the subsumption relation for the optimal-size sorting network problem is described. The new approach is based on creating a bipartite graph and modelling the subsumption test as the problem of enumerating all perfect matchings in this graph. Experiments showed significant improvements over the previous approaches when considering the number of subsumption checks and the time needed to find optimal-size sorting networks. We were able to generate all the complete sets of filters for comparator networks with 9 channels, confirming that the 25-comparators so…
Sorting networks are oblivious sorting algorithms with many interesting theoretical properties and practical applications. One of the related classical challenges is the search of optimal networks respect to size (number of comparators) of depth (number of layers). However, up to our knowledge, the joint size-depth optimality of small sorting networks has not been addressed before. This paper presents size-depth optimality results for networks up to $12$ channels. Our results show that there are sorting networks for $n\leq9$ inputs that are optimal in both size and depth, but this is not the c…
A new algorithm, Guidesort, for sorting in the uniprocessor variant of the parallel disk model (PDM) of Vitter and Shriver is presented. The algorithm is deterministic and executes a number of (parallel) I/O operations that comes within a constant factor $C$ of the optimum. The algorithm and its analysis are simpler than those proposed in previous work, and the achievable constant factor $C$ of essentially 3 appears to be smaller than for all other known deterministic algorithms, at least for plausible parameter values.
This paper studies properties of the back end of a sorting network and illustrates the utility of these in the search for networks of optimal size or depth. All previous works focus on properties of the front end of networks and on how to apply these to break symmetries in the search. The new properties help shed understanding on how sorting networks sort and speed-up solvers for both optimal size and depth by an order of magnitude.
One of the fundamental problem in the theory of sorting is to find the pessimistic number of comparisons sufficient to sort a given number of elements. Currently 16 is the lowest number of elements for which we do not know the exact value. We know that 46 comparisons suffices and that 44 do not. There is an open question if 45 comparisons are sufficient. We present an attempt to resolve that problem by performing an exhaustive computer search. We also present an algorithm for counting linear extensions which substantially speeds up computations.
We present new parallel sorting networks for $17$ to $20$ inputs. For $17, 19,$ and $20$ inputs these new networks are faster (i.e., they require less computation steps) than the previously known best networks. Therefore, we improve upon the known upper bounds for minimal depth sorting networks on $17, 19,$ and $20$ channels. Furthermore, we show that our sorting network for $17$ inputs is optimal in the sense that no sorting network using less layers exists. This solves the main open problem of [D. Bundala & J. Zavodný. Optimal sorting networks, Proc. LATA 2014].
We empirically study sorting in the evolving data model. In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an adversary randomly swaps the order of adjacent items in the true sorted order. Previous work studies only two versions of quicksort, and has a gap between the lower bound of Omega(n) and the best upper bound of O(n log log n). The experiments we perform in this paper provide empirical evidence that some quadratic-time algorithms such as insertion sort and bubble sor…
We describe an algorithm computing an optimal prefix free code for $n$ unsorted positive weights in time within $O(n(1+\lg α))\subseteq O(n\lg n)$, where the alternation $α\in[1..n-1]$ measures the amount of sorting required by the computation. This asymptotical complexity is within a constant factor of the optimal in the algebraic decision tree computational model, in the worst case over all instances of size $n$ and alternation $α$. Such results refine the state of the art complexity of $Θ(n\lg n)$ in the worst case over instances of size $n$ in the same computational model, a landmark in co…
We show that 11-channel sorting networks have at least 35 comparators and that 12-channel sorting networks have at least 39 comparators. This positively settles the optimality of the corresponding sorting networks given in The Art of Computer Programming vol. 3 and closes the two smallest open instances of the Bose-Nelson sorting problem. We obtain these bounds by generalizing a result of Van Voorhis from sorting networks to a more general class of comparator networks. From this we derive a dynamic programming algorithm that computes the optimal size for a sorting network with a given number o…
This paper settles the optimality of sorting networks given in The Art of Computer Programming vol. 3 more than 40 years ago. The book lists efficient sorting networks with n <= 16 inputs. In this paper we give general combinatorial arguments showing that if a sorting network with a given depth exists then there exists one with a special form. We then construct propositional formulas whose satisfiability is necessary for the existence of such a network. Using a SAT solver we conclude that the listed networks have optimal depth. For n <= 10 inputs where optimality was known previously, our algo…
This paper describes a computer-assisted non-existence proof of nine-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29-comparator network found by Waksman in 1969 is optimal when sorting ten inputs. This closes the two smallest open instances of the optimal size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to eight inputs. The proof involves a combination of two methodologie…
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…
We solve a 40-year-old open problem on the depth optimality of sorting networks. In 1973, Donald E. Knuth detailed, in Volume 3 of "The Art of Computer Programming", sorting networks of the smallest depth known at the time for n =< 16 inputs, quoting optimality for n =< 8. In 1989, Parberry proved the optimality of the networks with 9 =< n =< 10 inputs. In this article, we present a general technique for obtaining such optimality results, and use it to prove the optimality of the remaining open cases of 11 =< n =< 16 inputs. We show how to exploit symmetry to construct a small set of two-layer…
This paper studies new properties of the front and back ends of a sorting network, and illustrates the utility of these in the search for new bounds on optimal sorting networks. Search focuses first on the "outsides" of the network and then on the inner part. All previous works focus only on properties of the front end of networks and on how to apply these to break symmetries in the search. The new, out-side-in, properties help shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimal sorting networks. We present new parallel sorting networks for …
The suffix array is a fundamental data structure for many applications that involve string searching and data compression. Designing time/space-efficient suffix array construction algorithms has attracted significant attention and considerable advances have been made for the past 20 years. We obtain the \emph{first} in-place suffix array construction algorithms that are optimal both in time and space for (read-only) integer alphabets. Concretely, we make the following contributions: 1. For integer alphabets, we obtain the first suffix sorting algorithm which takes linear time and uses only $O(…
Previous work identifying depth-optimal $n$-channel sorting networks for $9\leq n \leq 16$ is based on exploiting symmetries of the first two layers. However, the naive generate-and-test approach typically applied does not scale. This paper revisits the problem of generating two-layer prefixes modulo symmetries. An improved notion of symmetry is provided and a novel technique based on regular languages and graph isomorphism is shown to generate the set of non-symmetric representations. An empirical evaluation demonstrates that the new method outperforms the generate-and-test approach by orders…
We consider the problem of sorting $n$ elements in the case of \emph{persistent} comparison errors. In this model (Braverman and Mossel, SODA'08), each comparison between two elements can be wrong with some fixed (small) probability $p$, and \emph{comparisons cannot be repeated}. Sorting perfectly in this model is impossible, and the objective is to minimize the \emph{dislocation} of each element in the output sequence, that is, the difference between its true rank and its position. Existing lower bounds for this problem show that no algorithm can guarantee, with high probability, \emph{maximu…
A long-standing open question in the algorithms and complexity literature is whether there exist sorting circuits of size $o(n \log n)$. A recent work by Asharov, Lin, and Shi (SODA'21) showed that if the elements to be sorted have short keys whose length $k = o(\log n)$, then one can indeed overcome the $n\log n$ barrier for sorting circuits, by leveraging non-comparison-based techniques. More specifically, Asharov et al.~showed that there exist $O(n) \cdot \min(k, \log n)$-sized sorting circuits for $k$-bit keys, ignoring $poly\log^*$ factors. Interestingly, the recent works by Farhadi et al…
In this paper we extend the knowledge on the problem of empirically searching for sorting networks of minimal depth. We present new search space pruning techniques for the last four levels of a candidate sorting network by considering only the output set representation of a network. We present an algorithm for checking whether an $n$-input sorting network of depth $d$ exists by considering the minimal up to permutation and reflection itemsets at each level and using the pruning at the last four levels. We experimentally evaluated this algorithm to find the optimal depth sorting networks for al…
Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy Sorting problem, where each comparison result is flipped independently with probability $p$ for some fixed $p\in (0, \frac 12)$. As our main result, we show that $$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$ noisy comparisons are both necessary and sufficient to sort $n$ elements with error probabili…
We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM}, which studies the case $k=1$. Here we show that, for $k=2$, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each $k \ge 3$. Next we describe an optimal sorting algorithm, again for the case $k=2$. We then analyze two types of le…
Sorted data is usually easier to compress than unsorted permutations of the same data. This motivates a simple compression scheme: specify the sorted permutation of the data along with a representation of the sorted data compressed recursively. The sorted permutation can be specified by recording the decisions made by quicksort. If the size of the data is known, then the quicksort decisions describe the data at a rate that is nearly as efficient as the minimal prefix-free code for the distribution, which is bounded by the entropy of the distribution. This is possible even though the distributi…
In the 60's, Knuth introduced stack-sorting and serial compositions of stacks. In particular, one significant question arise out of the work of Knuth: how to decide efficiently if a given permutation is sortable with 2 stacks in series? Whether this problem is polynomial or NP-complete is still unanswered yet. In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and d…
Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maxim…
