Abstract. We propose a generic framework and methods for simplification of large networks. The methods can be used to improve the understandability of a given network, to complement user-centric analysis methods, or as a pre-processing step for computationally more complex methods. The approach is path-oriented: edges are pruned while keeping the original quality of best paths between all pairs of nodes (but not necessarily all best paths). The framework is applicable to different kinds of graphs (for instance flow networks and random graphs) and connections can be measured in different ways (for instance by the shortest path, maximum flow, or maximum probability). It has relative neighborhood graphs, spanning trees, and certain Pathfinder graphs as its special cases. We give four algorithmic variants and report on experiments with 60 real biological networks. The simplification methods are part of ongoing projects for intelligent analysis of networked information. 1

Abstract. The disjoint-set data structure is used to maintain a collection of non-overlapping sets of elements from a finite universe. Algorithms that operate on this data structure are often referred to as Union-Find algorithms. They are used in numerous practical applications and are also available in several software libraries. This paper presents an extensive experimental study comparing the time required to execute 55 variations of Union-Find algorithms. The study includes all the classical algorithms, several recently suggested enhancements, and also different combinations and optimizations of these. Our results clearly show that a somewhat forgotten simple algorithm developed by Rem in 1976 is the fastest, in spite of the fact that its worst-case time complexity is inferior to that of the commonly accepted “best ” algorithms. Keywords: Union-Find, Disjoint Set, Experimental Algorithms. 1

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Abstract. Let P be a set of points in R d. We propose GEOFILTERKRUSKAL,an algorithm that computes the minimum spanning tree of P using well separated pair decomposition in combination with a simple modification of Kruskal’s algorithm. When P is sampled from uniform random distribution, we show that our algorithm takes one parallel sort plus a linear number of additional steps, with high probability, to compute the minimum spanning tree. Experiments show that our algorithm works better in practice for most data distributions compared to the current state of the art [31]. Our algorithm is easy to parallelize and to our knowledge, is currently the best practical algorithm on multi-core machines for d>2.