1,301 research outputs found

    On the random greedy F-free hypergraph process

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    Let FF be a strictly kk-balanced kk-uniform hypergraph with e(F)≥∣F∣−k+1e(F)\geq |F|-k+1 and maximum co-degree at least two. The random greedy FF-free process constructs a maximal FF-free hypergraph as follows. Consider a random ordering of the hyperedges of the complete kk-uniform hypergraph KnkK_n^k on nn vertices. Start with the empty hypergraph on nn vertices. Successively consider the hyperedges ee of KnkK_n^k in the given ordering, and add ee to the existing hypergraph provided that ee does not create a copy of FF. We show that asymptotically almost surely this process terminates at a hypergraph with O~(nk−(∣F∣−k)/(e(F)−1))\tilde{O}(n^{k-(|F|-k)/(e(F)-1)}) hyperedges. This is best possible up to logarithmic factors

    Block Crossings in Storyline Visualizations

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    Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016
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