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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…

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 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…

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…

Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be…

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…

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 $n=16$, $17$,…

Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science. Recent studies suggest that networks often exhibit hierarchical organization, where vertices divide into…

The structure entropy is an important index to illuminate the structure property of the complex network. Most of the existing structure entropies are based on the degree distribution of the complex network. But the structure entropy based…

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…

Common experience suggests that many networks might possess community structure - division of vertices into groups, with a higher density of edges within groups than between them. Here we describe a new computer algorithm that detects…

Hierarchy seems to pervade complexity in both living and artificial systems. Despite its relevance, no general theory that captures all features of hierarchy and its origins has been proposed yet. Here we present a formal approach resulting…

The sorting problem is one of the most relevant problems in computer science. Within the scope of modern computer science it has been studied for more than 70 years. In spite of these facts, new sorting algorithms have been developed in…

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 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…

Networks in nature possess a remarkable amount of structure. Via a series of data-driven discoveries, the cutting edge of network science has recently progressed from positing that the random graphs of mathematical graph theory might…

To understand the structure of a large-scale biological, social, or technological network, it can be helpful to decompose the network into smaller subunits or modules. In this article, we develop an information-theoretic foundation for the…

What is a complex network? How do we characterize complex networks? Which systems can be studied from a network approach? In this text, we motivate the use of complex networks to study and understand a broad panoply of systems, ranging from…

Modular structure is ubiquitous among complex networks. We note that most such systems are subject to multiple structural and functional constraints, e.g., minimizing the average path length and the total number of links, while maximizing…