scholarly journals Towards tight approximation bounds for graph diameter and eccentricities

2018 ◽  
Author(s):  
Arturs Backurs ◽  
Liam Roditty ◽  
Gilad Segal ◽  
Virginia Vassilevska Williams ◽  
Nicole Wein
Keyword(s):  
Graph Diameter

scholarly journals Modelling Selected Parameters of Power Grid Network in the South-Eastern Part of Poland: The Case Study

Energies ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 239
Author(s):  
Piotr Hadaj ◽  
Dominik Strzałka
Keyword(s):  
Distribution System ◽  
Fault Tolerant ◽  
Real Data ◽  
Electrical Energy ◽  
Vertex Degree ◽  
Clustering Coefficient ◽  
Network Efficiency ◽  
Graph Diameter ◽  
Graph Parameters

In this paper, a case study is conducted based on the real data obtained from the local Distribution System Operator (DSO) of electrical energy. The analyzed network represents connections and high-voltage switchgears of 110 kV. Selected graph parameters—vertex degree distribution, the average vertex degree, the graph density, network efficiency, the clustering coefficient, the average path length, and the graph diameter were examined, taking into account that in the analysis, some nodes were removed due to the different failures. For each failure, the possible effects on network parameters were tested. As a final result, it was shown that in the analyzed case, the removal of only five nodes could cause a significant (almost four times) fall of graph efficiency. In turn, this means that the whole analyzed network cannot be considered as a fault-tolerant.

scholarly journals Random Geometric Graph Diameter in the Unit Ball

Algorithmica ◽  
2007 ◽  
Vol 47 (4) ◽  
pp. 421-438 ◽  
Author(s):  
Robert B. Ellis ◽  
Jeremy L. Martin ◽  
Catherine Yan
Keyword(s):  
Unit Ball ◽  
Geometric Graph ◽  
Random Geometric Graph ◽  
Graph Diameter

scholarly journals Analysis and limitations of modified circuit-to-Hamiltonian constructions

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 94 ◽  
Author(s):  
Johannes Bausch ◽  
Elizabeth Crosson
Keyword(s):  
Upper Bound ◽  
Ground State ◽  
Spectral Gap ◽  
General Setting ◽  
Quantum Circuit ◽  
Many Body ◽  
Graph Diameter ◽  
Uniform Distributions ◽  
Labeled Graphs ◽  
Constant Output

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining theΩ(T−2)scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits.Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tightO(T−2)upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond theΩ(T−3)scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend theO(T−2)bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.

scholarly journals Graph diameter in long-range percolation

2010 ◽  
Vol 39 (2) ◽  
pp. 210-227 ◽  
Author(s):  
Marek Biskup
Keyword(s):  
Long Range ◽  
Graph Diameter

Better Approximation Algorithms for the Graph Diameter

2013 ◽  
Author(s):  
Shiri Chechik ◽  
Daniel H. Larkin ◽  
Liam Roditty ◽  
Grant Schoenebeck ◽  
Robert E. Tarjan ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Graph Diameter

Edges versus circuits: a hierarchy of diameters in polyhedra

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
S. Borgwardt ◽  
J. A. De Loera ◽  
E. Finhold
Keyword(s):  
Open Problem ◽  
Linear Optimization ◽  
Graph Diameter ◽  
Polyhedral Geometry

AbstractThe study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [

scholarly journals Distributed Graph Diameter Approximation

Algorithms ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 216
Author(s):  
Matteo Ceccarello ◽  
Andrea Pietracaprina ◽  
Geppino Pucci ◽  
Eli Upfal
Keyword(s):  
Approximation Algorithm ◽  
Linear Space ◽  
State Of The Art ◽  
Large Family ◽  
Proof Of Concept ◽  
Undirected Graphs ◽  
Graph Diameter ◽  
Similar Approximation ◽  
The One ◽  
Approximation Guarantee

We present an algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. In order to be efficient in terms of both time and space, our algorithm is based on a decomposition strategy which partitions the graph into disjoint clusters of bounded radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; most importantly, for a large family of graphs, it features a round complexity asymptotically smaller than the one exhibited by a natural approximation algorithm based on the state-of-the-art Δ-stepping SSSP algorithm, which is its only practical, linear-space competitor in the distributed setting. We complement our theoretical findings with a proof-of-concept experimental analysis on large benchmark graphs, which suggests that our algorithm may attain substantial improvements in terms of running time compared to the aforementioned competitor, while featuring, in practice, a similar approximation ratio.

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