11 research outputs found
A Spanner for the Day After
We show how to construct -spanner over a set of
points in that is resilient to a catastrophic failure of nodes.
Specifically, for prescribed parameters , the
computed spanner has edges, where . Furthermore, for any , and
any deleted set of points, the residual graph is -spanner for all the points of except for
of them. No previous constructions, beyond the trivial clique
with edges, were known such that only a tiny additional fraction
(i.e., ) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension,
and then showing a surprisingly simple and elegant construction in higher
dimensions, that uses the one-dimensional construction in a black box fashion
A Spanner for the Day After
We show how to construct (1+epsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters theta, epsilon in (0,1), the computed spanner G has O(epsilon^{-7d} log^7 epsilon^{-1} * theta^{-6} n log n (log log n)^6) edges. Furthermore, for any k, and any deleted set B subseteq P of k points, the residual graph G B is (1+epsilon)-spanner for all the points of P except for (1+theta)k of them. No previous constructions, beyond the trivial clique with O(n^2) edges, were known such that only a tiny additional fraction (i.e., theta) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one dimensional construction in a black box fashion
A spanner for the day after
We show how to construct (1 + ε)-spanner over a set P of n points in ℝd that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ, ε ∈ (0, 1), the computed spanner G has O(ε−7d log7 ε−1 · ϑ−6n log n(log log n)6) edges. Furthermore, for any k, and any deleted set B ⊆ P of k points, the residual graph G \ B is (1 + ε)-spanner for all the points of P except for (1 + ϑ)k of them. No previous constructions, beyond the trivial clique with O(n2) edges, were known such that only a tiny additional fraction (i.e., ϑ) lose their distance preserving connectivity. Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one dimensional construction in a black box fashion.</p
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
Research on the integrated management and mapping method of BOM multi-view for complex products
The bill of materials (BOM) runs through all stages of the life cycle of manufacturing products, which is the core of manufacturing enterprises. With increasing complexity of modern manufacturing engineering and widespread using of intelligent manufacturing technology, the BOM data keeps rising and transformation process is increasingly frequent and complicated. In order to improve efficiency of BOM management and ensure the diversity, accuracy and consistency of BOM in the product development, the BOM multi-view integrated management and mapping method for complex products were researched. First, a complex product BOM integrated management framework and the evolution model based on multiple views were established which described the BOM integrated management mechanism and transformation relationship among different BOMs. Subsequently, process of BOM transformation was analyzed, and a BOM transformation model was proposed. Moreover, a rule-based BOM multi-view mapping algorithm was proposed. With the rule definition and mathematical modelling for key components, the complex mapping principle was elaborated. Finally, the BOM multi-view transformation cases and the prototype system were illustrated and discussed, which verified the feasibility and versatility of model and method
A spanner for the day after
We show how to construct (1+ε)-spanner over a set P of n points in Rd that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ,ε∈(0,1), the computed spanner G has O(ε−cϑ−6nlogn(loglogn)6) edges, where c=O(d). Furthermore, for any k, and any deleted set B⊆P of k points, the residual graph G∖B is (1+ε)-spanner for all the points of P except for (1+ϑ)k of them. No previous constructions, beyond the trivial clique with O(n2) edges, were known such that only a tiny additional fraction (i.e., ϑ) lose their distance preserving connectivity.Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion
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Greedy Spanners in Euclidean Spaces Admit Sublinear Separators
The greedy spanner in low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness