Edge-isoperimetric Problem Research Articles

Minimum Congestion and Wirelength of Embedding the nth Cartesian Product of K4 into Various Kinds of Grid Networks

Embedding an interconnection network into another network is one of the main problems in parallel processing and computing systems. Interconnection networks in multiprocessing systems facilitate effective communication among various system components. Obtaining the minimum cutwidth, congestion, and wirelength in graph embedding problems is of great significance for network design and architecture, where the minimum is taken over all embedding of guest graphs into host graphs, and addressing these issues can reduce time and cost in embedded design. This work aims to accurately calculate the minimum congestion and wirelength for the [Formula: see text] (the [Formula: see text]th Cartesian product of [Formula: see text]) layout into [Formula: see text]-dimensional, [Formula: see text]-dimensional, and [Formula: see text]-dimensional grids by the optimal solution of edge isoperimetric problem on [Formula: see text].

Reflect-Push Methods Part I: Two Dimensional Techniques

We determine all maximum weight downsets in the product of two chains, where the weight function is a strictly increasing function of the rank. Many discrete isoperimetric problems can be reduced to the maximum weight downset problem. Our results generalize Lindsay's edge-isoperimetric theorem in two dimensions in several directions. They also imply and strengthen (in several directions) a result of Ahlswede and Katona concerning graphs with maximal number of adjacent pairs of edges. We find all optimal shifted graphs in the Ahlswede-Katona problem. Furthermore, the results of Ahlswede-Katona are extended to posets with a rank increasing and rank constant weight function. Our results also strengthen a special case of a recent result by Keough and Radcliffe concerning graphs with the fewest matchings. All of these results are achieved by applications of a key lemma that we call the reflect-push method. This method is geometric and combinatorial. Most of the literature on edge-isoperimetric inequalities focuses on finding a solution, and there are no general methods for finding all possible solutions. Our results give a general approach for finding all compressed solutions for the above edge-isoperimetric problems.
 By using the Ahlswede-Cai local-global principle, one can conclude that lexicographic solutions are optimal for many cases of higher dimensional isoperimetric problems. With this and our two dimensional results we can prove Lindsay's edge-isoperimetric inequality in any dimension. Furthermore, our results show that lexicographic solutions are the unique solutions for which compression techniques can be applied in this general setting.

Optimization of layout for embedding half hypercube into conventional tree architectures

SummaryEmbedding a graph into another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. This has the potential to enhance the physical layout of network‐on‐chip (NoC) devices as well as to investigate their virtualization possibilities. Layout is one of the many indicators of graph embedding. An optimal layout in NoC design can result in a decreased wiring area and cost, as well as the reduction in communication delay between parallel processing components. In this work, the guest graph is the half hypercube, which possess efficient routing, fault tolerance, and Hamiltonian features. The edge isoperimetric problems is solved for the half hypercube using a novel technique called the isochronal ordering. The host graphs considered in our work are complete binary trees, ‐rooted complete binary trees, and other few predominantly investigated tree based architectures.

Pull-push method: A new approach to edge-isoperimetric problems

We prove a generalization of the Ahlswede-Cai local-global principle for the edge-isoperimetric problems. A new technique to handle edge-isoperimetric problems is introduced which we call the pull-push method. Our main result includes all previously published results in this area as special cases with the only exception of the edge-isoperimetric problem for grids. With this we partially answer a question of Harper on local-global principles. We also describe a strategy for further generalization of our results so that the case of grids would be covered, which would completely settle Harper's question.

Optimal embedding of hypercube into cylinder

In the paper of Manuel et al. [10] the minimum wirelength of embedding hypercube into cylinder was given as a conjecture. We show that Gray code embedding is an optimal strategy for the conjecture. In the paper, we prove the lower bound of the wirelength as two parts respectively, one of which is studied by the edge isoperimetric problem (EIP) related to the type sequence. The technique is totally different from the previous work in Refs. [10,11,1,6].

Edge-isoperimetric inequalities and ball-noise stability: Linear programming and probabilistic approaches

Let Qnr be the graph with vertex set {−1,1}n in which two vertices are joined if their Hamming distance is at most r. The edge-isoperimetric problem for Qnr is that: For every (n,r,M) such that 1≤r≤n and 1≤M≤2n, determine the minimum edge-boundary size of a subset of vertices of Qnr with a given size M. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is probabilistic. Our bound derived by the first approach generalizes the tight bound for M=2n−1 derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also tight for M=2n−2 and r≤n2−1. Our bounds derived by the second approach are expressed in terms of the noise stability, and they are shown to be asymptotically tight as n→∞ when r=2⌊βn2⌋+1 and M=⌊α2n⌋ for fixed α,β∈(0,1), and is tight up to a factor 2 when r=2⌊βn2⌋ and M=⌊α2n⌋. In fact, the edge-isoperimetric problem is equivalent to a ball-noise stability problem which is a variant of the traditional (i.i.d.-) noise stability problem. Our results can be interpreted as bounds for the ball-noise stability problem.

Relating the bisection width of dual-port, server-centric datacenter networks and the solution of edge isoperimetric problems in graphs

Stellar datacenter networks are a recent generic construction designed to transform a base-graph into a dual-port, server-centric datacenter network. We prove that the S-bisection width of any stellar datacenter network can be obtained from the solution of isoperimetric problems on the base-graph, provided that the base-graph is regular. We extend previous research on the stellar datacenter networks GQ⁎, instantiated with generalized hypercubes, and show that with respect to S-bisection width, GQ⁎ performs well in comparison with the dual-port datacenter network FiConn. Our work develops a strong combinatorial link between graph bisection width and throughput metrics for stellar datacenter networks.

N^{3/4} Law in the Cubic Lattice

We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M_n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most mathrm{O}(n^{3/4}) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations

We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph G for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of G. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on G, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibility of computing the lower bounds on some of the smaller energy eigenvalues of Heisenberg models. This amounts to estimating the isoperimetric inequalities of the symmetric product of graphs. By using connections with discrete Sobolev inequalities, we show that this can be performed by considering just the vertex-induced subgraphs of G. If our conjecture for a polynomial time approximation algorithm to solve the edge-isoperimetric problem holds, then our proposed method of estimating the energy eigenvalues via approximating the edge-isoperimetric properties of vertex-induced subgraphs will yield a polynomial time algorithm for estimating the smaller energy eigenvalues of the Heisenberg ferromagnet.

Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs.

Vertex decomposition method for wirelength problem and its applications to enhanced hypercube networks

In this study, the authors discuss the vertex congestion of any embedding from the guest graph into the host graph and outline a rigorous mathematical method to compute the wirelength of that embedding. Further, they show that the computation of the optimal wirelength depends on finding optimal solutions for another graph partition problem such as edge isoperimetric problem in that guest graph. On the other side, they consider an important variant of the popular hypercube network, the enhanced hypercube, and obtain the nested optimal solutions for the edge isoperimetric problem. As a combined output, they illustrate the authors' technique by embedding enhanced hypercube into a caterpillar and from that reducing the linear layout of the enhanced hypercube. As another application of their technique, they embed the hypercube as well as the enhanced hypercube on the two rows extended grid structure with optimal wirelength for the first time and showing that the existing edge congestion technique cannot be used to solve this problem.

Isoperimetry in integer lattices

The edge isoperimetric problem for a graph $G$ is to determine, for each $n$, the minimum number of edges leaving any set of $n$ vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example when $G$ is the usual integer lattice. We solve the edge isoperimetric problem asymptotically for every Cayley graph on $\mathbb Z^d$. The near-optimal shapes that we exhibit are zonotopes generated by line segments corresponding to the generators of the Cayley graph.

New infinite family of regular edge-isoperimetric graphs

We introduce a new infinite family of regular graphs admitting nested solutions in the edge-isoperimetric problem for all their Cartesian powers. The obtained results include as special cases most of previously known results in this area.

A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on mixed volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the edges of the original graph. We demonstrate the efficacy of this method by revisiting some previously solved classical edge-isoperimetric problems. We then apply our method to some discrete isoperimetric problems which had not previously been solved. The complexity of those solutions suggest that it would be quite difficult to find them using discrete methods only.

Sharp N^{3/4} Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice

We investigate the edge-isoperimetric problem (EIP) for sets of n points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of n. In view of this isoperimetric characterizations, EIP minimizers M_n are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers M_n are estimated to deviate from such hexagonal configurations by at most K_t, n^{3/4}+mathrm{o}(n^{3/4}) points. The constant K_t is explicitly determined and shown to be sharp.

Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi Functions

Let $G$ be a finite abelian group of exponent $m\ge 2$. For subsets $A,S\subseteq G$, denote by $\partial_S(A)$ the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$. We show that if $S$ generates $G$, and $A$ is nonempty, then $\partial_S(A) \ge \frac{e}m\,|A|\ln\frac{|G|}{|A|}\, . $ Here the coefficient $e=2.718\ldots$ is best possible and cannot be replaced with a number larger than $e$. For homocyclic groups $G$ of exponent $m$, we find an explicit closed-form expression for $\partial_S(A)$ in the case where $S$ is the “standard" generating subset of $G$, and $A$ is an initial segment of $G$ with respect to the lexicographic order induced by $S$. Namely, we show that in this situation $ \partial_S(A) = |G|\,\omega_m(|A|/|G|), $ where $\omega_2$ is the Takagi function, and $\omega_m$ for $m\ge3$ is an appropriate generalization thereof. This particular case is of special interest, since for $m\in\{2,3,4\}$ it is known to yield the smallest possible value of $\partial_S(A)$, over all sets $A\subseteq G$ of given size. We give this classical result a new proof, somewhat different from the standard one. We also give a new, short proof of the Boros--Páles inequality $\omega_2(\frac{x+y}2) \le \frac{\omega_2(x)+\omega_2(y)}2 + \frac12\,|y-x|,$ establish an extremal characterization of the Takagi function as the (pointwise) maximal function, satisfying this inequality and the boundary condition $\max\{\omega_2(0),$ $\omega_2(1)\}\le 0$, and obtain similar results for the $3$-adic analogue $\omega_3$ of the Takagi function.

Embeddings of circulant networks

In this paper we solve the edge isoperimetric problem for circulant networks and consider the problem of embedding circulant networks into various graphs such as arbitrary trees, cycles, certain multicyclic graphs and ladders to yield the minimum wirelength.

Embedding hypercubes into cylinders, snakes and caterpillars for minimizing wirelength

We consider the problem of embedding hypercubes into cylinders to minimize the wirelength. Further, we show that the edge isoperimetric problem solves the wirelength problem of regular graphs and, in particular, hypercubes into triangular snakes and caterpillars.

Linear Wirelength of Folded Hypercubes

Manuel et al. (Discret. Appl. Math. 157(7):1486–1495, 2009) obtained the exact wirelength of an r-dimensional hypercube into a path as well as a $${2^{\lfloor r/2\rfloor}\times 2^{\lceil r/2\rceil}}$$ grid and conjectured the same for a folded hypercube. In this paper we solve the edge isoperimetric problem for folded hypercubes and thereby obtain the exact wirelength of folded hypercubes into paths. Further we compute the exact wirelength of the r-dimensional folded hypercubes into 2 k × 2 r−k grids.

On partitioning of hypergraphs

We study edge-isoperimetric problems (EIP) for hypergraphs and extend some technique in this area from graphs to hypergraphs. In particular, we establish some new results on a relationship between the EIP and some extremal poset problems, and apply them to obtain an exact solution of the EIP for certain hypergraph families. We also show how to solve the EIP on hypergraphs in some cases when the link to posets does not work. Another outcome of our results is a new series of hypergraphs admitting nested solutions in the EIP.