68 research outputs found
Resolving the Steiner Point Removal Problem in Planar Graphs via Shortcut Partitions
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 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 of
terminals in an arbitrary edge-weighted planar graph , we construct a minor
of whose vertex set is , which preserves the shortest-path distances
between all pairs of terminals in 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
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra
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
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 (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , 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 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 and recourse 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 recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
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
LIPIcs, Volume 244, ESA 2022, Complete Volum
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