Aalto computer scientists in SoCG 2024
Accepted papers
In alphabetical order. Click the title to see the authors and the abstract.
Authors
Karl Bringmann, Frank Staals, Karol Wegrzycki and Geert van Wordragen
Abstract
The Earth Mover's Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover's Distance under Translation (EMDuT) is a translation-invariant version thereof. It minimizes the Earth Mover's Distance over all translations of one point set. For EMDuT in ℝ1, we present an ˜(n2)-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For EMDuT in ℝd, we present an ˜(n2d+2)-time algorithm for the L1 and L∞ metric. We show that this dependence on d is asymptotically tight, as an no(d)-time algorithm for L1 or L∞ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.
Authors
Sándor Kisfaludi-Bak, Jana Masaříková, Erik Jan van Leeuwen, Bartosz Walczak and Karol Węgrzycki
Abstract
The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ≥0, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G−Zinduces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is poly(δ)⋅logn. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running in npolylog(n)⋅2(δ2)⋅ε−(δ) time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no no(δ)-time algorithm on planar δ-hyperbolic graphs, unless ETH fails.
Authors
Sándor Kisfaludi-Bak and Geert van Wordragen
Abstract
We propose a data structure in d-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. Using these quadtrees and the L-order we build geometric spanners. Near-linear size (1+ϵ)-spanners do not exist in hyperbolic spaces, but we are able to create a Steiner spanner that achieves a spanning ratio of 1+ϵ with d,ϵ(n) edges, using a simple construction that can be maintained dynamically. As a corollary we also get a (2+ϵ)-spanner (in the classical sense) of the same size, where the spanning ratio 2+ϵ is almost optimal among spanners of subquadratic size. Finally, we show that our Steiner spanner directly provides a solution to the approximate nearest neighbour problem: given a point set P in d-dimensional hyperbolic space we build the data structure in d,ϵ(nlogn) time, using d,ϵ(n) space. Then for any query point q we can find a point p∈P that is at most 1+ϵ times farther from q than its nearest neighbour in P in d,ϵ(logn) time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time d,ϵ(logn).
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