Minimum Tree Supports for Hypergraphs and Low-Concurrency Euler Diagrams Boris Klemz, Tamara Mchedlidze, and Martin Nollenburg¨ Institute of Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Germany Abstract. In this paper we present an O(n2(m + log n))-time algorithm for computing a minimum-weight tree support (if one exists) of a hypergraph H = (V; S) with n vertices and m hyperedges. This improves the previously best known algorithm with running time O(n4m2). A support of H is a graph G on V such that each hyperedge in S induces a connected subgraph in G. If G is a tree, it is called a tree support and it is a minimum tree support if its edge weight is minimum for a given edge weight function. Tree supports of hypergraphs have several applications, from social network analysis and network design problems to the visualization of hypergraphs and Euler diagrams. We show in particular how a minimum-weight tree support can be used to generate an area-proportional Euler diagram that satisfies typical well-formedness conditions and additionally minimizes the number of concurrent curves of the set boundaries in the Euler diagram. 1 Introduction A hypergraph H = (V; S) is a generalization of a graph that consists of a set of vertices V and a set of hyperedges S, which are arbitrary non-empty subsets of V (in contrast to graph edges, which are defined as pairs of vertices). Thus S is a subset of the power set (V ) = 2V . A graph G = (V; E) is called a support of a hypergraph H = (V; S) on theP same vertex set if every hyperedge s S induces a connected subgraph in G. Sparse support graphs are interesting from2 the perspective of network design as they represent realizations of hypergraphs as graphs, in which the vertices of each hyperedge induce a connected component. Korach and Stern [16] introduced the problem of finding a minimum(-weight) tree support (MTS) for a given hypergraph H = (V; S) and a V given edge-weight function w : R for the support graph. Here, an MTS is a 2 ! P support T = (V; E) that is a tree with minimum total edge weight e2E w(e). Not every hypergraph has a tree support, but the decision problem can be solved in linear time by testing whether its dual hypergraph is acyclic [14]. If a hypergraph has a tree support, it is called a tree-hypergraph. Korach and Stern [16] gave an algorithm to compute an MTS in O( V 4 S 2) time (if it exists). They later presented another polynomial-time algorithm forj j aj restrictedj variation, in which they ask for a tree support of minimum weight such that each subtree induced by a hyperedge is a star [17]. Hypergraphs and hypergraph supports are not as frequently used and studied as graphs themselves, but they still have many real-world applications. For example, in so- cial network analysis, minimum tree supports are used to compute maximum-likelihood social networks that serve as models to explain a collection of observed disease outbreaks 2 Boris Klemz, Tamara Mchedlidze, and Martin Nollenburg¨ that are modeled as hyperedges of the infected persons [1,2]. In topic-based peer-to-peer publish/subscribe systems [7, 13] the input is a set of users V and a set of topics S, where each topic t S is a subset of users (i.e., a hyperedge) interested in the topic. The task is to design an2 overlay network G on V (called minimum topic-connected overlay) with the minimum number of edges so that each topic forms a connected subgraph in G thus enabling private communication within each topic. If the underlying hypergraph H = (V; S) admits a tree support then this minimum overlay network will be a tree; if establishing edges in G is linked with a cost, the task is again to find an MTS. The unweighted problem is also known as subset interconnection design, which generally asks for a support graph with the minimum number of edges, i.e., not necessarily a tree. It has applications in the design of reconfigurable networks, e.g., vacuum sys- tems, in which valves correspond to edges in the support and their number needs to be minimized [6, 10, 11]. Of particular interest for hypergraph visualizations are planar supports. Johnson and Pollak [14] showed that a hypergraph is vertex-planar if and only if it has a planar support. A vertex-planar hypergraph has a representation of the vertices as faces in a planar subdivision such that for each hyperedge the union of the faces corresponding to the vertices incident to that hyperedge is a connected region. Simple and compact subdivision drawings [15] are an interesting restriction of vertex-planar hypergraph representations that puts additional constraint on the geometry of hyperedge representations. Johnson and Pollak [14] proved, however, that deciding the existence of a planar support is NP-complete; Buchin et al. [4, 5] extended the NP-completeness to testing the existence of a 2-outerplanar support. On the other hand, it can be decided in polynomial time, whether a hypergraph has a path-, cycle-, tree-, or cactus-support [3, 14, 16]. Contributions. In this paper we study minimum tree supports from a perspective that was initially motivated by generating (area-proportional) Euler diagrams with low con- currency of set contours. Euler diagrams are set visualizations and thus closely related to hypergraph visualizations. Section 2 describes the background of Euler diagrams, defines our algorithmic problem in that context, and sketches a solution approach based on minimum tree supports. In Section 3 we introduce some necessary definitions and notations, before we present our main technical contribution in Section 4. Our result is initially tailored for the problem to generate low-concurrency Euler diagrams. Hence we first transform an abstract Euler diagram description D into a so-called labeled hypergraph H(D) and then present an algorithm that computes an MTS for H(D) in O(n2m) time, where n is the number of vertices and m is the number of hyperedges. The algorithm itself is a simple modification of Kruskal’s algorithm for incrementally constructing minimum spanning trees, but its correctness proof relies on several crucial properties of tree supports and the special order in which the algorithm adds edges to the growing tree support. Finally, in Section 5 we generalize our result and show that every hypergraph can easily be translated into an equivalent labeled hypergraph. Then we can apply our algorithm for labeled hypergraphs to compute minimum tree supports for arbitrary hypergraphs that admit a tree support. This improves the result of Korach and Stern [16], who gave an algorithm with running time O(n4m2), to an algorithm with time complexity O(n2(m + log n)). Minimum Tree Supports for Hypergraphs 3 a c b abc b b a abc a abc bc bc bc abcd a abcd b bd abcd ad ad bd ad bd d d d d (a) (b) (c) Fig. 1: (a) Euler diagram realizing AEDD D = (fa; b; c; dg; fa; b; d; ad; bd; bc; abc; abcdg), (b) tree support T for H(D), (c) area-proportional Euler diagram based on T 2 Euler Diagrams An Euler diagram is a visualization of a set system as a collection of simple closed curves, whose interiorsD represent the sets, see Fig. 1(a) for an example. The arrangement of curves forms a subdivision of the plane and each face is called a (concrete) zone of the Euler diagram. We define an abstract Euler diagram description (AEDD) as a pair D = (L; Z), where L is a set of labels (each representing one set in a set system) and Z (L) is a set of label subsets that we call zones (each representing a non-empty intersection⊆ P of a particular set of labels). We say that an Euler diagram realizes an AEDD D if there is a bijection ' between L and the set of curves in D, as well as between Z and the set of concrete zones of such that for each zone z ZDthe concrete zone '(z) is in the interior of curve '(l) forD each l z and exterior to2 '(l0) for each l0 L z. AEDDs are closely related to hypergraphs2 since we can interpret each zone as2 a vertexn and each label l as a hyperedge containing all zones that carry the label l. Similarly, an Euler diagram that realizes a given AEDD is a subdivision drawing of its corresponding hypergraph. Euler diagrams must adhere to certain well-formedness conditions [8,12] that control the visual appearance of the diagrams, e.g., a zone with a certain set of labels L0 may exist if and only if there exists an element that is contained in all sets corresponding to L0 and that is not contained in any other set. Moreover, every zone must be uniquely labeled, i.e., there cannot be two distinct zones that lie in the interior of exactly the same set of curves. Other common well-formedness criteria require convex zones, disallow triple points, i.e., points that lie in the intersection of three or more curves, or disallow concurrent curves that run partially in parallel, i.e., connected intersections of two distinct curves c and c0 that contain more than just one point. There is a number of algorithms and complexity results for generating Euler diagrams with certain well-formedness properties [8, 12, 18]. Since one can easily come up with AEDDs that require concurrencies (see the example in Fig.