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# New Challenges in Algorithms/Design/Process Optimization with Symmetry/Asymmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Engineering and Materials".

Deadline for manuscript submissions: 30 September 2024 | Viewed by 1020

## Special Issue Editors

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Guest Editor
College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou, China
Interests: machine vision; precision manufacturing; intelligent manufacturing

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Guest Editor
School of Mechanical and Electric Engineering, Soochow University, Suzhou 215123, China
Interests: precision manufacturing; bionic robot

## Special Issue Information

Dear Colleagues,

Symmetry and asymmetry are important concepts in algorithm optimization as they can help in improving the efficiency and effectiveness of algorithms. Here are some ways in which symmetry and asymmetry can be utilized for algorithm optimization:

1. Symmetry exploitation: Symmetry refers to a situation where certain elements or conditions in a problem have similar properties. By identifying and exploiting symmetry in a problem, algorithms can avoid redundant calculations or explore only a subset of the possible solutions. This can greatly improve the efficiency of the algorithm by reducing its time complexity;
2. Symmetry breaking: In some cases, symmetry can lead to a large number of equivalent solutions, making it hard for an algorithm to distinguish between them. Symmetry breaking techniques aim to introduce small variations to the problem representation or search strategies to break this symmetry and explore different regions of the problem space. This can help in finding better solutions or improving the convergence speed of an algorithm;
3. Asymmetry detection: Asymmetry refers to a lack of symmetry or regularity in a problem. By detecting and utilizing the asymmetry in a problem, algorithms can focus on areas where significant differences exist, leading to better solutions or faster convergence. For example, in graph algorithms, identifying and exploiting the asymmetry in the structure of a graph can help in reducing the search space and improving the efficiency of traversal or optimization algorithms;
4. Asymmetric data structures: Using asymmetric data structures can also contribute to algorithm optimization. By designing data structures that take advantage of the specific properties or patterns in a problem, algorithms can achieve better time and space complexity. For example, using a hash table instead of a linear search can result in faster lookup times, especially if the data exhibit certain patterns or distributions;
5. Exploiting symmetry in parallel computing: Symmetry can also be exploited in parallel computing to improve performance. When executing algorithms on multiple processors or cores, considering symmetry can help in distributing the workload more evenly, reducing communication and synchronization overhead, and achieving better load balancing. This can result in faster execution and the better utilization of resources.

In conclusion, symmetry and asymmetry play vital roles in algorithm optimization. By appropriately leveraging symmetry or breaking it when necessary, algorithms can improve efficiency, convergence speed, and the quality of solutions. Similarly, utilizing asymmetric properties or structures can lead to optimized algorithms in terms of time and space complexity.

Dr. Jun Zhao
Prof. Dr. Cheng Fan
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

## Keywords

• symmetry exploitation
• symmetry breaking
• parallel computing
• asymmetric data structures
• algorithms with symmetry and asymmetry
• optimization with symmetry and asymmetry
• design with symmetry and asymmetry
• process with symmetry and asymmetry

## Published Papers (1 paper)

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## Research

20 pages, 1277 KiB
Article
A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation
by Meijuan Wang and Shugong Zhang
Symmetry 2023, 15(12), 2144; https://doi.org/10.3390/sym15122144 - 2 Dec 2023
Viewed by 702
Abstract
As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion [...] Read more.
As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method. Full article
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