833 research outputs found
Isometric Embeddings in Trees and Their Use in Distance Problems
International audienceWe present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows: (1) We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5âΔ}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs. (2) Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an O(n^{5/3})-time algorithm for computing all the eccentricities). (3) We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2. (4) Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, BergĂ© and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem. Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate
Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension
International audienceUnder the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes-where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| â 1. âą Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time Ă(k · mn 1âΔ_d), where Δ_d â (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. âą Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. âą Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublin-ear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of Δ-nets, region decomposition and other partition techniques
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively
Gradient metasurfaces: a review of fundamentals and applications
In the wake of intense research on metamaterials the two-dimensional
analogue, known as metasurfaces, has attracted progressively increasing
attention in recent years due to the ease of fabrication and smaller insertion
losses, while enabling an unprecedented control over spatial distributions of
transmitted and reflected optical fields. Metasurfaces represent optically thin
planar arrays of resonant subwavelength elements that can be arranged in a
strictly or quasi periodic fashion, or even in an aperiodic manner, depending
on targeted optical wavefronts to be molded with their help. This paper reviews
a broad subclass of metasurfaces, viz. gradient metasurfaces, which are devised
to exhibit spatially varying optical responses resulting in spatially varying
amplitudes, phases and polarizations of scattered fields. Starting with
introducing the concept of gradient metasurfaces, we present classification of
different metasurfaces from the viewpoint of their responses, differentiating
electrical-dipole, geometric, reflective and Huygens' metasurfaces. The
fundamental building blocks essential for the realization of metasurfaces are
then discussed in order to elucidate the underlying physics of various physical
realizations of both plasmonic and purely dielectric metasurfaces. We then
overview the main applications of gradient metasurfaces, including waveplates,
flat lenses, spiral phase plates, broadband absorbers, color printing,
holograms, polarimeters and surface wave couplers. The review is terminated
with a short section on recently developed nonlinear metasurfaces, followed by
the outlook presenting our view on possible future developments and
perspectives for future applications.Comment: Accepted for publication in Reports on Progress in Physic
Obstructions to Faster Diameter Computation: Asteroidal Sets
Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every -edge graph in can be computed in deterministic time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive -approximation of all vertex eccentricities in deterministic time. This is in sharp contrast with general -edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in time for any . As important special cases of our main result, we derive an -time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an -time algorithm for this problem on graphs of asteroidal number at most . We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions
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