Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality

Delocalizing Entanglement of Anisotropic Black Branes

We study the mutual information between pairs of regions on the two asymptotic boundaries of maximally-extended anisotropic black-brane solutions. This quantity characterizes the local pattern of entanglement of thermofield double states which are dual to these geometries. We analyse the disruption of the mutual information in anisotropic shock wave geometries and show that the entanglement velocity plays an important role in this phenomenon. Besides that we compute several chaos-related properties of this system, like the entanglement velocity, the butterfly velocity and the scrambling time. We find that the butterfly velocity and the entanglement velocity violate the upper bounds proposed in 1311.1200 and 1612.00082, but remain bounded by their corresponding values in the infrared effective theory.Comment: 34 pages, 10 figures. V2: typos corrected and references added. Analysis extended to higher anisotropies. Figures 3, 6(a) and 8(b) replaced to include higher anisotropies. Figures 6(b), 7(a) and 7(b) replaced to improve visualization. Minor changes in the end of the abstract and introduction. Two figures added in App. C. Discussion and App.C expanded. V3: Matches published versio

Duality between Feature Selection and Data Clustering

The feature-selection problem is formulated from an information-theoretic perspective. We show that the problem can be efficiently solved by an extension of the recently proposed info-clustering paradigm. This reveals the fundamental duality between feature selection and data clustering,which is a consequence of the more general duality between the principal partition and the principal lattice of partitions in combinatorial optimization

Algorithms for Game Metrics

Simulation and bisimulation metrics for stochastic systems provide a quantitative generalization of the classical simulation and bisimulation relations. These metrics capture the similarity of states with respect to quantitative specifications written in the quantitative {\mu}-calculus and related probabilistic logics. We first show that the metrics provide a bound for the difference in long-run average and discounted average behavior across states, indicating that the metrics can be used both in system verification, and in performance evaluation. For turn-based games and MDPs, we provide a polynomial-time algorithm for the computation of the one-step metric distance between states. The algorithm is based on linear programming; it improves on the previous known exponential-time algorithm based on a reduction to the theory of reals. We then present PSPACE algorithms for both the decision problem and the problem of approximating the metric distance between two states, matching the best known algorithms for Markov chains. For the bisimulation kernel of the metric our algorithm works in time O(n^4) for both turn-based games and MDPs; improving the previously best known O(n^9\cdot log(n)) time algorithm for MDPs. For a concurrent game G, we show that computing the exact distance between states is at least as hard as computing the value of concurrent reachability games and the square-root-sum problem in computational geometry. We show that checking whether the metric distance is bounded by a rational r, can be done via a reduction to the theory of real closed fields, involving a formula with three quantifier alternations, yielding O(|G|^O(|G|^5)) time complexity, improving the previously known reduction, which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated to approximate the metrics using binary search.Comment: 27 pages. Full version of the paper accepted at FSTTCS 200