Computing approximate shortest paths on convex polytopes

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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eMap: A Web Application for Identifying and Visualizing Electron or Hole Hopping Pathways in Proteins

10.26434/chemrxiv.7889993 ◽

2019 ◽

Author(s):

Ruslan N. Tazhigulov ◽

James R. Gayvert ◽

Melissa Wei ◽

Ksenia B. Bravaya

Keyword(s):

Web Application ◽

Shortest Paths ◽

Coarse Grained ◽

Decay Parameter ◽

Electron Hole ◽

Web Based ◽

Aromatic Amino Acid Residue ◽

Pathways Model ◽

Visualization Tools ◽

Effective Decay

<p>eMap is a web-based platform for identifying and visualizing electron or hole transfer pathways in proteins based on their crystal structures. The underlying model can be viewed as a coarse-grained version of the Pathways model, where each tunneling step between hopping sites represented by electron transfer active (ETA) moieties is described with one eﬀective decay parameter that describes protein-mediated tunneling. ETA moieties include aromatic amino acid residue side chains and aromatic fragments of cofactors that are automatically detected, and, in addition, electron/hole residing sites that can be speciﬁed by the users. The software searches for the shortest paths connecting the user-speciﬁed electron/hole source to either all surface-exposed ETA residues or to the user-speciﬁed target. The identiﬁed pathways are ranked based on their length. The pathways are visualized in 2D as a graph, in which each node represents an ETA site, and in 3D using available protein visualization tools. Here, we present the capability and user interface of eMap 1.0, which is available at https://emap.bu.edu.</p>

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Bounds on the partition dimension of convex polytopes

Combinatorial Chemistry & High Throughput Screening ◽

10.2174/1386207323666201204144422 ◽

2020 ◽

Vol 23 ◽

Author(s):

Jia-Bao Liu ◽

Muhammad Faisal Nadeem ◽

Mohammad Azeem

Keyword(s):

Combinatorial Optimization ◽

Computer Science ◽

Facility Location ◽

Robot Navigation ◽

Convex Polytopes ◽

Pharmaceutical Chemistry ◽

Resolving Set ◽

Minimum Number ◽

Partition Dimension ◽

Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

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Computer algorithms

10.1093/oso/9780198805090.003.0008 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Data Structures ◽

Analysis Of Algorithms ◽

Shortest Paths ◽

Minimum Cut ◽

Network Data ◽

Computer Algorithms ◽

Breadth First Search ◽

Augmenting Path ◽

Basic Concepts ◽

Maximum Flows

This chapter introduces some of the fundamental concepts of numerical network calculations. The chapter starts with a discussion of basic concepts of computational complexity and data structures for storing network data, then progresses to the description and analysis of algorithms for a range of network calculations: breadth-first search and its use for calculating shortest paths, shortest distances, components, closeness, and betweenness; Dijkstra's algorithm for shortest paths and distances on weighted networks; and the augmenting path algorithm for calculating maximum flows, minimum cut sets, and independent paths in networks.

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Lower bounds for computing geometric spanners and approximate shortest paths

Discrete Applied Mathematics ◽

10.1016/s0166-218x(00)00280-8 ◽

2001 ◽

Vol 110 (2-3) ◽

pp. 151-167 ◽

Cited By ~ 12

Author(s):

Danny Z. Chen ◽

Gautam Das ◽

Michiel Smid

Keyword(s):

Lower Bounds ◽

Shortest Paths ◽

Geometric Spanners

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A short note on extreme points of certain polytopes

Special Matrices ◽

10.1515/spma-2020-0005 ◽

2020 ◽

Vol 8 (1) ◽

pp. 36-39

Author(s):

Lei Cao ◽

Ariana Hall ◽

Selcuk Koyuncu

Keyword(s):

Short Proof ◽

Convex Polytopes ◽

Short Note ◽

Convex Polytope ◽

Extreme Points ◽

Alternative Proof ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Birkhoff's Theorem ◽

Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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Shortest Paths on Cubes

College Mathematics Journal ◽

10.1080/07468342.2021.1866944 ◽

2021 ◽

Vol 52 (2) ◽

pp. 121-132

Author(s):

Richard Goldstone ◽

Rachel Roca ◽

Robert Suzzi Valli

Keyword(s):

Shortest Paths

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