Sergei L. Bezrukov

The congestion of n -cube layout on a rectangular grid

Abstract We consider the problem of embedding the n -dimensional cube into a rectangular grid with 2 n vertices in such a way as to minimize the congestion, the maximum number of edges along any point of the grid. After presenting a short solution for the cutwidth problem of the n -cube (in which the n -cube is embedded into a path), we show how to extend the results to give an exact solution for the congestion problem.

Embedding of Hypercubes into Grids

We consider one-to-one embeddings of the n-dimensional hypercube into grids with 2n vertices and present lower and upper bounds and asymptotic estimates for minimal dilation, edge-congestion, and their mean values. We also introduce and study two new cost-measures for these embeddings, namely the sum over i=1, ..., n of dilations and the sum of edge-congestions caused by the hypercube edges of the ith dimension. It is shown that, in the simulation via the embedding approach, such measures are much more suitable for evaluating the slowdown of uniaxial hypercube algorithms then the traditional cost measures.

Embedding ladders and caterpillars into the hypercube

We present an embedding of generalized ladders as subgraphs into the hypercube. Through an embedding of caterpillars into ladders, we obtain an embedding of caterpillars into the hypercube. In this way we get almost all known results concerning the embedding of caterpillars into the hypercube. In addition we construct an embedding for some new types of caterpillars. Our results support the conjecture of Havel (1984).

Embedding complete trees into the hypercube

Abstract We consider embeddings of the complete t -ary trees of depth k (denotation T k , t ) as subgraphs into the hypercube of minimum dimension n . This n , denoted by dim( T k , t ), is known if max{ k , t }⩽2. First, we study the next open cases t =3 and k =3. We improve the known upper bound dim( T k ,3 )⩽2 k +1 up to lim k →∞ dim( T k ,3 )/ k ⩽5/3 and show lim t →∞ dim( T 3, t )/ t =227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim( T k , t ) for arbitrary k and t .

Edge isoperimetric theorems for integer point arrays

Abstract We consider subsets of the n-dimensional grid with the Manhattan metrics, (i.e., the Cartesian product of chains of lengths k1,…,kn) and study those of them which have maximal number of induced edges of the grid, and those which are separable from their complement by the least number of edges. The first problem was considered for k1=…=kn by Bollobas and Leader [1]. Here we extend their result to arbitrary k1,…,kn, and give also a simpler proof based on a new approach. For the second problem, [1] offers only an inequality. We show that our approach to the first problem also gives a solution for the second problem, if all ki = ∞. If all kis are finite, we present an exact solution for n = 2.

New Spectral Lower Bounds on the Bisection Width of Graphs

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled by the graph bisection problem. The spectral lower bound of λ2|V|/4 for the bisection width of a graph is well-known. The bisection width is equal to λ2|V|/4 iff all vertices are incident to λ2/2 cut edges in every optimal bisection. We discuss the case for which this fact is not satisfied and present a new method to get tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. Under certain conditions we get a lower bound depending on λ2β|V| with 1/2 ≤ β < 1. We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3- and 4-regular graphs. We use them to establish tighter lower bounds for the bisection width of 3- and 4-regular Ramanujan graphs.

Edge-isoperimetric problems for cartesian powers of regular graphs

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best possible edge-isoperimetric inequalities for the cartesian powers of arbitrary regular, resp. regular bipartite, graphs with a high density.

New spectral lower bounds on the bisection width of graphs

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled using the graph bisection problem. The spectral lower bound of λ2|V|/4 for the bisection width of a graph is widely known. The bisection width is equal to λ2|V|/4 iff all vertices are incident to λ2/2 cut edges in every optimal bisection.We present a new method of obtaining tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. We define some global expansion properties and we show that the spectral lower bound increases with this global expansion. Under certain conditions we obtain a lower bound depending on λ2β|V with ½ < β ≤ 1. We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3- and 4-regular Ramanujan graphs.

A local-global principle for vertex-isoperimetric problems

We consider the vertex-isoperimetric problem (VIP) for cartesian powers of a graph G. A total order ≤ on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of ≤, and the ball around any initial segment is again an initial segment of ≤. We prove a local-global principle with respect to the so-called simplicial order on Gn (see Section 2 for the definition). Namely, we show that the simplicial order ≤n is isoperimetric for each n ≥ 1 iff it is so for n = 1,2. Some structural properties of graphs that admit simplicial isoperimetric orderings are presented. We also discuss new relations between the VIP and Macaulay posets.

The Spider Poset Is Macaulay

Let Q(k, l) be a poset whose Hasse diagram is a regular spider with k+1 legs having the same length l. We show that for any n?1 the nth cartesian power of the spider poset Q(k, l) is a Macaulay poset for any k?0 and l?1. In combination with our recent results (S. L. Bezrukov, 1998, J. Combin. Theory Ser. A84, 157?170) this provides a complete characterization of all Macaulay posets which are cartesian powers of upper semilattices, whose Hasse diagrams are trees.