68 research outputs found

    Resolving the Steiner Point Removal Problem in Planar Graphs via Shortcut Partitions

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    Recently the authors [CCLMST23] introduced the notion of shortcut partition of planar graphs and obtained several results from the partition, including a tree cover with O(1)O(1) trees for planar metrics and an additive embedding into small treewidth graphs. In this note, we apply the same partition to resolve the Steiner point removal (SPR) problem in planar graphs: Given any set KK of terminals in an arbitrary edge-weighted planar graph GG, we construct a minor MM of GG whose vertex set is KK, which preserves the shortest-path distances between all pairs of terminals in GG up to a constant factor. This resolves in the affirmative an open problem that has been asked repeatedly in literature.Comment: Manuscript not intended for publication. The results have been subsumed by arXiv:2308.00555 from the same author

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra

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    The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).Comment: 42 pages, 8 figures. Version 2: 47 pages, 8 figures; minor revisions and reformatting; to appear in Geom. Topo

    Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

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    Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion O(logn)min{logΦ,n}O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\} (or, O(d)min{logΦ,n}O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\} if the metric has doubling dimension dd), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio Φ\Phi from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion O(dlogΦ)O(d\cdot \log\Phi), generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of 2n2n metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio O(dlogΦ)O(d\cdot \log \Phi) and recourse O(logΦ)O(\log \Phi) against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using O(log2n)O(\log^2 n) recourse, that maintains a matching of weight at most O(logn)O(\log n) times the weight of the MST, i.e., a matching of lightness O(logn)O(\log n). We complement our upper bounds with a strategy for an oblivious adversary that, with recourse rr, establishes a lower bound of Ω(lognrlogr)\Omega(\frac{\log n}{r \log r}) for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA24

    Discrete Geometry

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    Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

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    We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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