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13 pages, 499 KiB

Open AccessArticle

Routing and Spectrum Allocation in Spectrum-Sliced Elastic Optical Path Networks: A Primal-Dual Framework

by Yang Wang, Chaoyang Li, Qian Hu, Jabree Flor and Maryam Jalalitabar

Electronics 2021, 10(22), 2809; https://doi.org/10.3390/electronics10222809 - 16 Nov 2021

Cited by 7 | Viewed by 2598

The recent decade has witnessed a tremendous growth of Internet traffic, which is expected to continue climbing for the foreseeable future. As a new paradigm, Spectrum-sliced Elastic Optical Path (SLICE) networks promise abundant (elastic) bandwidth to address the traffic explosion, while bearing other inherent advantages including enhanced signal quality and extended reachability. The fundamental problem in SLICE networks is to route each traffic demand along a lightpath with continuously and consecutively available sub-carriers, which is known as the Routing and Spectrum Allocation (RSA) problem. Given its NP-Hardness, the solutions to the RSA problem can be classified into two categories: optimal solutions using link-based, path-based, and channel-based Integer Linear Programming (ILP) models, which require extensive computational time; and sub-optimal heuristic and meta-heuristic algorithms, which have no guarantee on the solution quality. In this work, inspired by a channel-based ILP model, we propose a novel primal-dual framework to address the RSA problem, which can obtain a near-optimal solution with guaranteed per-instance closeness to the optimal solution. Full article

(This article belongs to the Special Issue Advanced Photonic Technologies for High-Speed Communications)

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12 pages, 251 KiB

Open AccessArticle

On the Number of Witnesses in the Miller–Rabin Primality Test

by Shamil Talgatovich Ishmukhametov, Bulat Gazinurovich Mubarakov and Ramilya Gakilevna Rubtsova

Symmetry 2020, 12(6), 890; https://doi.org/10.3390/sym12060890 - 1 Jun 2020

Cited by 9 | Viewed by 4621

Abstract

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let [...] Read more.

W (n)

denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer

a < n

is a primality witness for n. For composite n, the power of

W (n)

is less than or equal to

φ (n) / 4

where

φ (n)

is Euler’s Totient function. We derive new exact formulas for the power of

W (n)

depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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