Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed.The Jaccard index [8] is a classical similarity measure on sets with a lot of practical applications in information retrieval, data mining, machine learning, and many more (cf., e.g., [7]). Measuring the relative size of the overlap of two finite sets A and B, the Jaccard index J and the associated Jaccard distance J δ are formally defined as:

Automatic and interactive data analysis is instrumental in making use of increasing amounts of complex data. Owing to novel sensor modalities, analysis of data generated in professional team sport leagues such as soccer, baseball, and basketball has recently become of concern, with potentially high commercial and research interest. The analysis of team ball games can serve many goals, e.g., in coaching to understand effects of strategies and tactics, or to derive insights improving performance. Also, it is often decisive to trainers and analysts to understand why a certain movement of a player or groups of players happened, and what the respective influencing factors are. We consider team sport as group movement including collaboration and competition of individuals following specific rule sets. Analyzing team sports is a challenging problem as it involves joint understanding of heterogeneous data perspectives, including high-dimensional, video, and movement data, as well as considering team behavior and rules (constraints) given in the particular team sport. We identify important components of team sport data, exemplified by the soccer case, and explain how to analyze team sport data in general. We identify challenges arising when facing these data sets and we propose a multi-facet view and analysis including pattern detection, context-aware analysis, and visual explanation. We also present applicable methods and technologies covering the heterogeneous aspects in team sport data.

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In this paper, we study the problem of finding a periodic attractor of a Boolean network (BN), which arises in computational systems biology and is known to be NP-hard. Since a general case is quite hard to solve, we consider special but biologically important subclasses of BNs. For finding an attractor of period 2 of a BN consisting of n OR functions of positive literals, we present a polynomial time algorithm. For finding an attractor of period 2 of a BN consisting of n AND/OR functions of literals, we present an O(1:985(n)) time algorithm. For finding an attractor of a fixed period of a BN consisting of n nested canalyzing functions and having constant treewidth w, we present an O(n(2p(w+1))poly(n)) time algorithm.

Abstract. We study the complexity of counting the number of elements in intervals of feasible partial orders. Depending on the properties that partial orders may have, such counting functions have different complexities. If we consider total, polynomial-time decidable orders then we obtain exactly the #P functions. We show that the interval size functions for polynomial-time adjacency checkable orders are tightly related to the class FPSPACE(poly): Every FPSPACE(poly) function equals a polynomial-time function subtracted from such an interval size function. We study the function #DIV that counts the nontrivial divisors of natural numbers, and we show that #DIV is the interval size function of a polynomial-time decidable partial order with polynomial-time adjacency checks if and only if primality is in polynomial time.