Related papers: Joint Size and Depth Optimization of Sorting Netwo…
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…
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…
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…
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…
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…
We establish new depth upper bounds for sorting networks on 27 and 28 channels, improving the previous best bound of 14 to 13. Our 28-channel network is constructed with reflectional symmetry by combining high-quality prefixes of 16- and…
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…
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…
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…
In this paper, we address sorting networks that are constructed from comparators of arity $k > 2$. That is, in our setting the arity of the comparators -- or, in other words, the number of inputs that can be sorted at the unit cost -- is a…
The sorting number of a graph with $n$ vertices is the minimum depth of a sorting network with $n$ inputs and outputs that uses only the edges of the graph to perform comparisons. Many known results on sorting networks can be stated in…
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…
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…
Sorting a set of items is a task that can be useful by itself or as a building block for more complex operations. The more sophisticated and fast sorting algorithms become asymptotically, the less efficient they are for small sets of items…
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…
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…
Sorting and ranking supervision is a method for training neural networks end-to-end based on ordering constraints. That is, the ground truth order of sets of samples is known, while their absolute values remain unsupervised. For that, we…
In recent work, we formalized the theory of optimal-size sorting networks with the goal of extracting a verified checker for the large-scale computer-generated proof that 25 comparisons are optimal when sorting 9 inputs, which required more…
It is a long-standing open question to determine the minimum number of comparisons $S(n)$ that suffice to sort an array of $n$ elements. Indeed, before this work $S(n)$ has been known only for $n\leq 22$ with the exception for $n=16$, $17$,…
Many real-world networks are large, complex and thus hard to understand, analyze or visualize. The data about networks is not always complete, their structure may be hidden or they change quickly over time. Therefore, understanding how…