scholarly journals Tight Lower Bounds for Greedy Routing in Higher-Dimensional Small-World Grids

2013 ◽  
Author(s):  
Martin Dietzfelbinger ◽  
Philipp Woelfel
Keyword(s):  
Lower Bounds ◽  
Small World ◽  
Greedy Routing ◽  
Higher Dimensional

TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

scholarly journals Greedy routing in small-world networks with power-law degrees

2014 ◽  
Vol 27 (4) ◽  
pp. 231-253 ◽  
Author(s):  
Pierre Fraigniaud ◽  
George Giakkoupis
Keyword(s):  
Power Law ◽  
Small World ◽  
Greedy Routing ◽  
Small World Networks

scholarly journals LS-category of moment-angle manifolds and higher order Massey products

2021 ◽  
Vol 33 (5) ◽  
pp. 1179-1205
Author(s):  
Piotr Beben ◽  
Jelena Grbić
Keyword(s):  
Simplicial Complex ◽  
Structural Properties ◽  
Lower Bounds ◽  
Higher Order ◽  
Hyperbolic Manifolds ◽  
Upper And Lower Bounds ◽  
Connected Sum ◽  
Massey Products ◽  
Higher Dimensional ◽  
Lusternik Schnirelmann

Abstract Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

Smooth Mappings with Higher Dimensional Critical Sets

2010 ◽  
Vol 53 (3) ◽  
pp. 542-549 ◽  
Author(s):  
Cornel Pintea
Keyword(s):  
Lower Bounds ◽  
High Dimensional ◽  
Higher Dimensional ◽  
Critical Sets

AbstractIn this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

scholarly journals Grid Minors in Damaged Grids

10.37236/3872 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
David Eppstein
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Grid Graph ◽  
Square Grid ◽  
Higher Dimensional

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

Greedy Routing and the Algorithmic Small-World Phenomenon

2017 ◽  
Author(s):  
Karl Bringmann ◽  
Ralph Keusch ◽  
Johannes Lengler ◽  
Yannic Maus ◽  
Anisur Rahaman Molla
Keyword(s):  
Small World ◽  
Greedy Routing

scholarly journals Optimisation of the coalescent hyperbolic embedding of complex networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bianka Kovács ◽  
Gergely Palla
Keyword(s):  
Small World ◽  
Hyperbolic Distance ◽  
Greedy Routing ◽  
Scale Free ◽  
Recent Approach ◽  
Strong Motivation ◽  
Angular Coordinates ◽  
Small World Property ◽  
Logarithmic Loss ◽  
Hyperbolic Embedding

AbstractSeveral observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small-world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high-quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.

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