Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1605.00863

On the Mathematics of Data Centre Network Topologies

Iain A. Stewart
View original source ↗Read source document ↗Export metadata ↓

Abstract

The theory of combinatorial designs has recently been used in order to build switch-centric data centre networks incorporating a large number of servers, in comparison with the popular Fat-Tree data centre network. The construction employed, called the 3-step method, revolves around an appropriately chosen (but relatively small) bipartite graph and a transversal design. In this paper, we clarify and extend these recent results. In particular, we prove the following path diversity results: in a one-to-one context, we prove that in these data centre networks there are pairwise link-disjoint paths joining all the servers adjacent to some switch with all the servers adjacent to any other switch so that we retain control of the path lengths (these results are optimal in terms of the numbers of paths constructed and we prove that we have a wide choice of bipartite graph and transversal design to which we can apply the 3-step method); and in a one-to-many context, we prove that there are pairwise link-disjoint paths from all the servers adjacent to some switch to any identically-sized collection of target servers where these target servers need not be adjacent to the same switch (again, we keep control of the path lengths). Our constructions and analysis are undertaken on bipartite graphs with the applications to data centre networks being easily derived. Our results strengthen the overall competitiveness of data centre networks constructed using the 3-step method, in comparison with Fat-Tree data centre networks, and, more generally, show the potential of results and methodologies from combinatorics to data centre network design.

Explore related subjects

cs.DM

Keep this discovery

BibTeXRIS

Iain A. Stewart. 2016-05-03. On the Mathematics of Data Centre Network Topologies. https://arxiv.org/abs/1605.00863

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An FPTAS for Two-Machine Open-Shop Scheduling with a Single Unavailability Interval

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

cs.DM↗
arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗
arXiv2026

The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.

cs.DM↗