Wirelength of hypercubes into certain trees (original) (raw)

Embedding of hypercubes into sibling trees

Discrete Applied Mathematics, 2014

The aim of this paper is to generalize the Congestion Lemma, which has been considered an efficient tool to compute the minimum wirelength and thereby obtain the minimum wirelength of embedding hypercubes into sibling trees.

Optimal Dynamic Embeddings of Complete Binary Trees into Hypercubes

Journal of Parallel and Distributed Computing, 2001

It is folklore that the double-rooted complete binary tree is a spanning tree of the hypercube of the same size. Unfortunately, the usual construction of an embedding of a double-rooted complete binary tree into a hypercube does not provide any hint on how this embedding can be extended if each leaf spawns two new leaves. In this paper, we present simple dynamic embeddings of double-rooted complete binary trees into hypercubes which do not suffer from this disadvantage. We also present edge-disjoint embeddings with optimal load and unit dilation. Furthermore, all these embeddings can be efficiently implemented on the hypercube itself such that the embedding of each new level of leaves can be computed in constant time. Because of the similarity, our results can be immediately transfered to complete binary trees.

Embedding of hypercubes into complete binary trees

Proc. 16th Australasian Workshop on …, 2005

Let G and H be finite graphs with n vertices. V (G) and V (H) denote the vertex sets of G and H respectively. E(G) and E(H) denote the edge sets of G and H respectively. A 1 − 1 mapping f : V (G) → V (H) is called an embedding of G into H. H is normally called a host graph. The ...

A New Efficient Algorithm for Embedding an Arbitrary Binary Tree into Its Optimal Hypercube

Journal of Algorithms, 1996

The d-dimensional binary hypercube is a very popular model of parallel computation. On the other hand, the execution of many algorithms can be represented by binary trees, making desirable fast simulations of binary trees on hypercubes. In this paper, we present a simple one-to-one embedding of arbitrary binary trees into their optimal hypercubes with dilation 8 and constant congestion. The novelty of our method is based on the use of an intermediate quadtree data structure, which also permits the embedding to be efficiently computed on the hypercube itself. 2 Topics: Algorithms and Data Structures, Theory of Parallel and Distributed Computation.

Two new classes of trees embeddable into hypercubes

RAIRO - Operations Research, 2004

The problem of embedding graphs into other graphs is much studied in the graph theory. In fact, much effort has been devoted to determining the conditions under which a graph G is a subgraph of a graph H, having a particular structure. An important class to study is the set of graphs which are embeddable into a hypercube. This importance results from the remarkable properties of the hypercube and its use in several domains, such as: the coding theory, transfer of information, multicriteria rule, interconnection networks ... In this paper we are interested in defining two new classes of embedding trees into the hypercube for which the dimension is given.

Embeddings Between Hypercubes and Hypertrees

Journal of Graph Algorithms and Applications, 2015

Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we embed the rooted hypertree RHT (r) into r-dimensional hypercube Q r with dilation 2, r ≥ 2. Also, we compute the exact wirelength of the embedding of the r-dimensional hypercube Q r into the rooted hypertree RHT (r), r ≥ 2.

Embedding perfectly balanced 2-caterpillar into its optimal hypercube

2021

A long-standing conjecture on spanning trees of a hypercube states that a balanced tree on 2n vertices with maximum degree at most 3 spans the hypercube of dimension n [4]. In this paper, we settle the conjecture for a special family of binary trees. A 0-caterpillar is a path. For k ≥ 1, a k-caterpillar is a binary tree consisting of a path with j-caterpillars (0 ≤ j ≤ k−1) emanating from some of the vertices on the path. A k-caterpillar that contains a perfect matching is said to be perfectly balanced. In this paper, we show that a perfectly balanced 2-caterpillar on 2n vertices spans the hypercube of dimension n.

Embedding complete trees into the hypercube

Discrete Applied Mathematics, 2001

We consider embeddings of the complete t-ary trees of depth k (denotation T k;t ) as subgraphs into the hypercube of minimal dimension n. This n, denoted by dim(T k;t ), is known if one of the parameters k; t equals 2. Here we study the next open case when one of k; t equals 3. We improve the known upper bound dim(T k;3 ) 2k + 1 up to lim k!1 dim(T k;3 )=k 5=3 and present the asymptotic lim t!1 dim(T 3;t )=t = 227=120. As a byresult, we found an exact formula for the dimension of arbitrary trees of depth 2, as a function of its vertex degrees. These results and new techniques have lead to the improvement of the known upper bound for dim(T k;t ) for arbitrary k and t.