Optimal routing in double loop networks
01 Aug 2007-Theoretical Computer Science (Elsevier Science Publishers Ltd.)-Vol. 381, Iss: 1, pp 68-85
TL;DR: This elementary and efficient shortest path algorithm has been derived from the Closest Vector Problem (CVP) of lattices in dimension two and with an @?"1 norm.
About: This article is published in Theoretical Computer Science.The article was published on 2007-08-01 and is currently open access. It has received 25 citations till now. The article focuses on the topics: Shortest path problem & Longest path problem.
Citations
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TL;DR: This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.
Abstract: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.
937 citations
TL;DR: The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks.
Abstract: Circulant graphs have been extensively investigated over the past 30 years because of their broad application to different fields of theory and practice. Two known surveys on circulant networks including a survey on undirected circulants have been published: by Bermond et al. [Distributed loop computer networks: A survey, J. Parallel Distributed Comput.24 (1995) 2–10] and by Hwang [A survey on multi-loop networks, Theoret. Comput. Sci.299 (2003) 107–121]. The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks. We focus on the survey connected with study of structural and communicative properties of circulant networks.
57 citations
TL;DR: In this paper , the authors optimize the diameters, mean path lengths, and bisection widths of circulant topologies to design low latency network topologies for high-performance computing clusters.
Abstract: Communication latency has become one of the determining factors for the performance of parallel clusters. To design low-latency network topologies for high-performance computing clusters, we optimize the diameters, mean path lengths, and bisection widths of circulant topologies. We obtain a series of optimal circulant topologies of size $$2^5$$ through $$2^{10}$$ and compare them with torus and hypercube of the same sizes and degrees. We further benchmark on a broad variety of applications including effective bandwidth, FFTE, Graph 500 and NAS parallel benchmarks to compare the optimal circulant topologies and Cartesian products of optimal circulant topologies and fully connected topologies with corresponding torus and hypercube. Simulation results demonstrate superior potentials of the optimal circulant topologies for communication-intensive applications. We also find the strengths of the Cartesian products in exploiting global communication with data traffic patterns of specific applications or internal algorithms.
17 citations
TL;DR: New versions of the algorithm improve the previously proposed shortest path search algorithm for optimal generalized Petersen graphs with an analytical description and is a promising solution for the use in networks-on-chip (NoCs).
Abstract: For a family of optimal two-dimensional circulant networks with an analytical description, two new improved versions of the shortest path search algorithm with a constant complexity estimate are obtained. A simple, based on the geometric model of circulant graphs, proof of the formulas used for the shortest path search algorithm is given. Pair exchange algorithms are presented, and their estimates are given for networks-on-chip (NoCs) with a topology in the form of the considered graphs. New versions of the algorithm improve the previously proposed shortest path search algorithm for optimal generalized Petersen graphs with an analytical description. The new proposed algorithm is a promising solution for the use in NoCs which was confirmed by an experimental study while synthesizing NoC communication subsystems and comparing the consumed hardware resources with those when other previously developed routing algorithms.
16 citations
Additional excerpts
...tables and adjacency matrices; it is adapted to node and link failures and load distribution of nodes; it has a constant complexityO(1) that does not depend on the size of the graph, in contrast to the following algorithms: 1) Dijkstra’s algorithm with quadratic complexity O(N 2) for any connected graph of order N ; 2) the algorithm from [8] with an estimate of O(N 1/2) of the time for calculating the shortest paths for circulants of type C(N ; s1, s2) and l = O(d) of the routing steps, where l is the distance between the vertices, and d is the diameter of the network; 3) the algorithm from [9] having the overall time complexity of one routing step O(logN ); 4) the algorithm from [10] with O(logN )−time preprocessing; 5) the algorithm from [11] with an estimate of O(logN ) arithmetic operations; 6) the algorithm from [12] having a constant estimate of the time complexity to calculate the shortest path for circulants of type C (N ; s1, s2) , but requiring preliminary calculation of the parameters for calculation with time complexityO (2 logN ) ; 7) the algorithm from [13] in circulants of typeC(Nd ; d, d+1) with an estimate ofO(d)....
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TL;DR: An algorithm for computing shortest paths and the exact value of their diameters is given for the family of circulant graphs of degree 4 and lower and upper bounds for their forwarding and optical indices are obtained.
Abstract: Cayley graphs are highly attractive structures for communication networks because of their many desirable properties, including vertex-transitivity and efficient routing algorithms [4]. The families of circulants and cube-connected graphs are among the most popular Cayley graphs for efficient communication networks [11, 12]. The diameter, forwarding and optical indices, bisection width and Wiener index of a network are among the most important parameters to measure the efficiency of the network [2, 5–7]. Circulant graphs and, in particular, circulant graphs with small degrees are interesting models for communication networks [1]. However, our knowledge of many of their parameters, including the arc-forwarding index, edge-forwarding index, directed and undirected optical indices, are very limited, except for very few special cases. We study the family of circulant graphs of degree 4 and obtain lower and upper bounds for their forwarding and optical indices. We give approximation algorithms for the corresponding problems of the forwarding indices and optical indices with a small constant performance ratio. Our results on the family of circulant graphs of degree 4 are published in [3]. The family of recursive cubes of rings has received a lot of attention for communication networks [10], but many aspects of them have remained unknown. We study this family of graphs by redefining each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and obtain the exact value of their diameters. We obtain
14 citations
Cites background or methods from "Optimal routing in double loop netw..."
...asserted that the shortest path problem on circulant graphs with fixed k is solvable in polynomial time [56]....
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...The idea of an integer lattice with l1 norm is rooted from the well-known Minokowski theorem [24, 56]....
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...However, they divulged that they could not derive a similar algorithm for k ≥ 3 [56]....
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References
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Book•
01 Dec 1986TL;DR: Introduction and Preliminaries.
Abstract: Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.
7,005 citations
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01 Jan 1988
TL;DR: In this article, the Fulkerson Prize was won by the Mathematical Programming Society and the American Mathematical Society for proving polynomial time solvability of problems in convexity theory, geometry, and combinatorial optimization.
Abstract: This book develops geometric techniques for proving the polynomial time solvability of problems in convexity theory, geometry, and - in particular - combinatorial optimization. It offers a unifying approach based on two fundamental geometric algorithms: - the ellipsoid method for finding a point in a convex set and - the basis reduction method for point lattices. The ellipsoid method was used by Khachiyan to show the polynomial time solvability of linear programming. The basis reduction method yields a polynomial time procedure for certain diophantine approximation problems. A combination of these techniques makes it possible to show the polynomial time solvability of many questions concerning poyhedra - for instance, of linear programming problems having possibly exponentially many inequalities. Utilizing results from polyhedral combinatorics, it provides short proofs of the poynomial time solvability of many combinatiorial optimization problems. For a number of these problems, the geometric algorithms discussed in this book are the only techniques known to derive polynomial time solvability. This book is a continuation and extension of previous research of the authors for which they received the Fulkerson Prize, awarded by the Mathematical Programming Society and the American Mathematical Society.
3,676 citations
"Optimal routing in double loop netw..." refers background in this paper
...Lattices are geometric objects that have been used to solve many problems in mathematics and computer science, see for instance [2,18,15,26,27,29]....
[...]
TL;DR: This paper presents a polynomial-time algorithm to solve the following problem: given a non-zeroPolynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q (X).
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).
3,513 citations
"Optimal routing in double loop netw..." refers background in this paper
...Lattices are geometric objects that have been used to solve many problems in mathematics and computer science, see for instance [2,18,15,26,27,29]....
[...]
01 Jan 1982
TL;DR: In this paper, a polynomial-time algorithm was proposed to decompose a primitive polynomials into irreducible factors in Z(X) if the greatest common divisor of its coefficients is 1.
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).
3,248 citations
Book•
28 May 1999
TL;DR: This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.
Abstract: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.
1,917 citations