A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology--to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimization problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was also studied by Fernau et al. A similar reduction to a graph crossing problem also helps to solve an open problem they posed, showing a fixed-parameter tractable algorithm for minimizing the number of crossings over all d-ary trees. For the case where one tree is fixed, we show an O(n log n) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman's footrule distance and give an O(n²) algorithm. We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber to minimize crossings.

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Popular approximate membership query structures such as Bloom filters and cuckoo filters are widely used in databases, security, and networking. These structures represent sets approximately, and support at least two operations-insert and lookup; lookup always returns true on elements in the structure; it also returns true with some probability 𝜀 on elements not in the structure. These latter elements are called false positives. Compensatory for these false positives, filters can be much smaller than hash tables that represent the same set. However, unlike hash tables, cuckoo filters, and Bloom filters must be initialized with the intended number of maximum entries, and cannot grow larger-inserts beyond this number fail or significantly increase the false positive probability. This article presents designs and implementations of filters than can grow without inserts failing and without meaningfully increasing the false positive probability, even if the filters are created with a small initial size. The resulting code is available on GitHub under a permissive open source license.

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