372 research outputs found
Light Euclidean Spanners with Steiner Points
The FOCS'19 paper of Le and Solomon, culminating a long line of research on
Euclidean spanners, proves that the lightness (normalized weight) of the greedy
-spanner in is for any
and any (where
hides polylogarithmic factors of ), and also shows the
existence of point sets in for which any -spanner
must have lightness . Given this tight bound on the
lightness, a natural arising question is whether a better lightness bound can
be achieved using Steiner points.
Our first result is a construction of Steiner spanners in with
lightness , where is the spread of the
point set. In the regime of , this provides an
improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime
of parameters is of practical interest, as point sets arising in real-life
applications (e.g., for various random distributions) have polynomially bounded
spread, while in spanner applications often controls the precision,
and it sometimes needs to be much smaller than . Moreover, for
spread polynomially bounded in , this upper bound provides a
quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019],
We then demonstrate that such a light spanner can be constructed in
time for polynomially bounded spread, where
hides a factor of . Finally, we extend the
construction to higher dimensions, proving a lightness upper bound of
for any and any .Comment: 23 pages, 2 figures, to appear in ESA 2
A Unified and Fine-Grained Approach for Light Spanners
Seminal works on light spanners from recent years provide near-optimal
tradeoffs between the stretch and lightness of spanners in general graphs,
minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a
"truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these
papers employ inherently different techniques than others. Moreover, the
runtime of these constructions is rather high.
In this work, we present a unified and fine-grained approach for light
spanners. Besides the obvious theoretical importance of unification, we
demonstrate the power of our approach in obtaining (1) stronger lightness
bounds, and (2) faster construction times. Our results include:
_ -minor-free graphs: A truly optimal spanner construction and a fast
construction.
_ General graphs: A truly optimal spanner -- almost and a linear-time
construction with near-optimal lightness.
_ Low dimensional Euclidean spaces: We demonstrate that Steiner points help
in reducing the lightness of Euclidean -spanners almost
quadratically for .Comment: We split this paper into two papers: arXiv:2106.15596 and
arXiv:2111.1374
Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the Lightness Barrier
An essential requirement of spanners in many applications is to be
fault-tolerant: a -spanner of a metric space is called (vertex)
-fault-tolerant (-FT) if it remains a -spanner (for the
non-faulty points) when up to faulty points are removed from the spanner.
Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been
extensively studied since the 90s.
For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03]
showed that the optimal guarantees , and on the size,
degree and lightness of -FT spanners can be achieved via a greedy algorithm,
which na\"{\i}vely runs in time. The question of
whether the optimal bounds of [CZ03] can be achieved via a fast construction
has remained elusive, with the lightness parameter being the bottleneck.
Moreover, in the wider family of doubling metrics, it is not even clear whether
there exists an -FT spanner with lightness that depends solely on (even
exponentially): all existing constructions have lightness
since they are built on the net-tree spanner, which is induced by a
hierarchical net-tree of lightness .
In this paper we settle in the affirmative these longstanding open questions.
Specifically, we design a construction of -FT spanners that is optimal with
respect to all the involved parameters (size, degree, lightness and running
time): For any -point doubling metric, any , and any integer
, our construction provides, within time ,
an -FT -spanner with size , degree and
lightness .Comment: Abstract is shortened to meet arxiv's requirement on the number of
character
Dynamic Matching Algorithms Under Vertex Updates
Dynamic graph matching algorithms have been extensively studied, but mostly under edge updates. This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges.
A basic setting arising in online algorithms and studied by Bosek et al. [FOCS\u2714] and Bernstein et al. [SODA\u2718] is that of dynamic approximate maximum cardinality matching (MCM) in bipartite graphs in which one side is fixed and vertices on the other side either arrive or depart via vertex updates. In the BASIC-incremental setting, vertices only arrive, while in the BASIC-decremental setting vertices only depart. When vertices can both arrive and depart, we have the BASIC-dynamic setting. In this paper we also consider the setting in which both sides of the bipartite graph are dynamic. We call this the MEDIUM-dynamic setting, and MEDIUM-decremental is the restriction when vertices can only depart. The GENERAL-dynamic setting is when the graph is not necessarily bipartite and the vertices can both depart and arrive.
Denote by K the total number of edges inserted and deleted to and from the graph throughout the entire update sequence. A well-studied measure, the recourse of a dynamic matching algorithm is the number of changes made to the matching per step. We largely focus on Maximal Matching (MM) which is a 2-approximation to the MCM. Our main results are as follows.
- In the BASIC-dynamic setting, there is a straightforward algorithm for maintaining a MM, with a total runtime of O(K) and constant worst-case recourse. In fact, this algorithm never removes an edge from the matching; we refer to such an algorithm as irrevocable.
- For the MEDIUM-dynamic setting we give a strong conditional lower bound that even holds in the MEDIUM-decremental setting: if for any fixed ? > 0, there is an irrevocable decremental MM algorithm with a total runtime of O(K ? n^{1-?}), this would refute the OMv conjecture; a similar (but weaker) hardness result can be achieved via a reduction from the Triangle Detection conjecture.
- Next, we consider the GENERAL-dynamic setting, and design an MM algorithm with a total runtime of O(K) and constant worst-case recourse. We achieve this result via a 1-revocable algorithm, which may remove just one edge per update step. As argued above, an irrevocable algorithm with such a runtime is not likely to exist.
- Finally, back to the BASIC-dynamic setting, we present an algorithm with a total runtime of O(K), which provides an (e/(e-1))-approximation to the MCM.
To this end, we build on the classic "ranking" online algorithm by Karp et al. [STOC\u2790]. Beyond the results, our work draws connections between the areas of dynamic graph algorithms and online algorithms, and it proposes several open questions that seem to be overlooked thus far
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
New Cross-Bridged Cyclam Ligands and Their Transition Metal Complexes as CXCR4 Antagonists
CXCR4 is a co-receptor on the surface of immune cells that has been proven to facilitate the entry of HIV into the cells. Within the last 15 years the CXCR4 and CCR5 coreceptors have influenced new therapeutic approaches to the treatment of HIV via fusion inhibitor drugs that target these receptors. Our aim is to develop new antagonists for the CXCR4 coreceptor. Specifically, the goal was the synthesis of Propyl Cross-Bridged, linked analogues of the known CXCR4 antagonist AMD-3100
Turbulence-driven magnetic reconnection and the magnetic correlation length: observations from magnetospheric multiscale in Earth's magnetosheath
Turbulent plasmas generate a multitude of thin current structures that can be sites for magnetic reconnection. The Magnetospheric Multiscale (MMS) mission has recently enabled the detailed examination of such turbulent current structures in Earth's magnetosheath and revealed that a novel type of reconnection, known as electron-only reconnection, can occur. In electron-only reconnection, ions do not have enough space to couple to the newly reconnected magnetic fields, suppressing ion jet formation and resulting in thinner sub-proton-scale current structures with faster super-AlfvĂ©nic electron jets. In this study, MMS observations are used to examine how the magnetic correlation length (λC) of the turbulence, which characterizes the size of the large-scale magnetic structures and constrains the length of the current sheets formed, influences the nature of turbulence-driven reconnection. We systematically identify 256 reconnection events across 60 intervals of magnetosheath turbulence. Most events do not appear to have ion jets; however, 18 events are identified with ion jets that are at least partially coupled to the reconnected magnetic field. The current sheet thickness and electron jet speed have a weak anti-correlation, with faster electron jets at thinner current sheets. When âČ20 ion inertial lengths, as is typical near the sub-solar magnetosheath, a tendency for thinner current sheets and potentially faster electron jets is present. The results are consistent with electron-only reconnection being more prevalent for turbulent plasmas with relatively short λC and may be relevant to the nonlinear dynamics and energy dissipation in turbulent plasmas
What is Life Worth? A Rough Guide to Valuation
In this speculative article, the aim is to elaborate a definition of life that is not biological, and a valuation of it that is not commodified. This is undertaken by the
development of an understanding of death as a process which is embedded in the life of a community. The idea is that we can best understand what life is worth by first
understanding what death means
- âŠ