Guest Graph Research Articles

Extended Fault-Tolerant Cycle Embedding in Faulty Hypercubes

We consider fault-tolerant embedding, where an n-dimensional faulty hypercube, denoted by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , acts as the host graph, and the longest fault-free cycle represents the guest graph. Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> be a set of faulty nodes in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . Also, let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> be a set of faulty edges in which at least one end-node of each edge is faulty, and let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fe</i> be a set of faulty edges in which the end-nodes of each edge are both fault-free. An edge in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is said to be critical if it is either fault-free or in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> . In this paper, we prove that there exists a fault-free cycle of length at least 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -2| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> | in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ges 3) with | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-5, and | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> |+| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-4 , in which each node is incident to at least two critical edges. Our result improves on the previously best known results reported in the literature, where only faulty nodes or faulty edges are considered.

Approximating the maximum clique minor and some subgraph homeomorphism problems

We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to K h or a subgraph homeomorphic to K h , respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O ( n ) bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O ( n ) bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω ( n 1 / 2 − O ( 1 / ( log n ) γ ) ) lower bound (for some constant γ , unless NP ⊆ ZPTIME ( 2 ( log n ) O ( 1 ) ) ) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound. Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.

Mapping Cycles and Trees on Wrap-Around Butterfly Graphs

We give a new algebraic representation for the wrap-around butterfly interconnection network. This new representation is based on the direct product of groups and finite fields and allows an algebraic expression of the network connectivity. The abstract algebraic tools may then be employed to explore the structural properties of the butterfly. In this paper we exploit this model to map guest graphs on the butterfly. In particular, we provide designs of unit dilation mappings of all possible length cycles on butterflies. We also map the largest possible binary trees on butterfly networks with a dilation 2 if the network degree is less than 16, 3 if it is less than 32, and 4 if it is less than 64. This is a great improvement over previous results.

피라미드의 3-차원 메쉬로의 신장율 개선 임베딩

본 논문에서는 주어진 손님 그래프 모델의 정점들과 간선들을 신장율, 혼잡율 등의 성능 파라미터들을 보다 우수하게 유지하면서 주인 그래프의 대응되는 정점들 및 경로들오 매핑시키는 "그래프 임베딩 문제"라고 불리는 그래프이론 문제를 다른다. 먼저 높이가 N인 파라미트 모델을 높이가 <TEX>$(4^{(N+1)/3}+2)/3$</TEX> 이고 2-차원 정방현 메쉬의 한 변의 길이가 <TEX>$2^{(2N-1)/3}$</TEX>인 3-차원 메쉬 구조의 대규모 병렬처리시스템으로 임베딩 할 수 잇는 새로운 매핑함수를 제안하고, 해당 임베딩 하에서 인접된 두 정점들 상호간 통신에 필요한 단계의 수를 반영하는 신장율의 관점에서 성능을 분석한다. 본 임베딩의 신장율이 <TEX>$2{\cdot}4^{(N-2)/3}+4)/3$</TEX> 임을 증명한다. 이러한 결과는 동일한 조건 하에서 기존의 결과인 <TEX>$4^{N+183}+2)/3$</TEX> 보다 우수한 것이다.다 우수한 것이다. In this paper, we consider a graph-theoretic problem,, the so-called "graph embedding problem" that maps the vertices and edges of the given guest graph model into the corresponding vertices and paths of the host graph under the condition of maintaining better performance parameters such as dilation, congestion, and expansion. We firstly propose a new mapping function which can embed the pyramid model with height N into the 3-dimensional mesh massively parallel processor system with the height <TEX>$(4^{(N+1)/3}+2)/3$</TEX> and the regular 2-dimensional mesh of one side <TEX>$2^{(2N-1)/3}$</TEX>, and analyze the performance of the embedding in terms of the dilation parameter that reflects the number of communication steps between two adjacent vertices under the embedding. We prove that the dilation of the embedding is <TEX>$2{\cdot}4^{(N-2)/3}+4)/3$</TEX>. This is superior to the previous result of <TEX>$4^{N+183}+2)/3$</TEX> under the same condition.condition.

One-to-Many Embeddings of Hypercubes into Cayley Graphs Generated by Reversals

Among Cayley graphs on the symmetric group, some that have a noticeably low diameter relative to the degree of regularity are examples such as the ``star'' network and the ``pancake'' network, the latter a representative of a variety of Cayley graphs generated by reversals. These diameter and degree conditions make these graphs potential candidates for parallel computation networks. Thus it is natural to investigate how well they can simulate other standard parallel networks, in particular hypercubes. For this purpose, constructions have previously been given for low dilation embeddings of hypercubes into star networks. Developing this theme further, in this paper we construct especially low dilation maps (e.g., with dilation 1, 2, 3, or 4) of hypercubes into pancake networks and related Cayley graphs generated by reversals. Whereas obtaining such results by the use of ``traditional'' graph embeddings (i.e., one-to-one or many-to-one embeddings) is sometimes difficult or impossible, we achieve many of these results by using a nontraditional simulation technique known as a ``one-to-many'' graph embedding. That is, in such embeddings we allow each vertex in the guest (i.e., domain) graph to be associated with some nonempty subset of the vertex set of the host (i.e., range) graph, these subsets satisfying certain distance and connection requirements which make the simulation possible.

Routing of 2-D switching networks by their embedding into cubes

The proposed all-optical 2-D switching networks are (i) M× N-gon prism switches ( M⩾2, N⩾3 ) and (ii) 3-D grids of any geometry N⩾3. For the routing we assume (1) the projection of the spatial architectures onto plane graphs (2) the embedding of the latter guest graphs into (in)complete host hypercubes ( N=4) and generally, into N-cube networks ( N⩾3) and (3) routing by means of the cube algorithms of the host. By the embedding mainly faulty cubes (synonyms: injured cubes, incomplete cubes) arise which complicate the routing and analysis. The application of N-cube networks (i) extend the hypercube principles to any N⩾3 (ii) increase the number of plane host graphs and (iii) reduce the incompleteness of the host cubes. Several different embeddings of the intersection graphs (IGs) of 2-D switching networks and several different routings are explained for N=4 and 6 by various examples. By the expansion of the grids (enlargement) internal waveguides (WGs) and internal switches are introduced which interact with the switches of the original 3-D grid without increasing the number of stages (NS). The embeddings by expansion apply to interconnection networks whereas dilation-2 embeddings (dilation ≡ distance of the nearest-neighbour nodes of the guest graph at the host) are rather suitable for the emulation of algorithms. Concepts for fault-tolerant routing and algorithm mapping are briefly explained.

Embedding complete binary trees in product graphs

This paper shows how to embed complete binary trees in products of complete binary trees, products of shuffle?exchange graphs, and products of de Bruijn graphs with small dilation and congestion. In the embedding results presented here the size of the host graph can be fixed to an arbitrary size, while we define no bound on the size of the guest graph. This is motivated by the fact that the host architecture has a fixed number of processors due to its physical design, while the guest graph can grow arbitrarily large depending on the application. The results of this paper widen the class of computations that can be performed on these product graphs which are often cited as being low?cost alternatives for hypercubes.

Edge-Packing in Planar Graphs

Maximum G Edge-Packing (EPack-sub G) is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H. This paper investigates the computational complexity of edge-packing for planar guests and planar hosts. Edge-packing is solvable in polynomial time when both G and H are either a 3-cycle or a k-star (graphs isomorphic to K(sub 1,k). Edge-packing is NP-complete when H is planar and G is either a cycle or a tree with greater than or equal to 3 edges. A strategy for developing polynomial-time approximation algorithms for planar hosts is exemplified by a linear-time approximation algorithm that finds a k-star edge-packing of size at least 1/2 optimal.

Edge-packing planar graphs by cyclic graphs

Maximum G edge-packing is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H. This paper considers the cases where G and H are planar and G is cyclic. Recent work on the general problem is surveyed, inadequacies and limitations in these results are identified, and NP-completeness proofs for key cases are presented.

Graph embeddings and simplicial maps

An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhatt et al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of the k-dimensional torus are determined. As corollaries to the main lower bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genus g graphs, and k-dimensional meshes in a butterfly host graph.

Optimal emulations by butterfly-like networks

The power of butterfly-like networks as multicomputer interconnection networks is studied, by considering how efficiently the butterfly can emulate other networks. Emulations are studied formally via graph embeddings, so the topic here becomes : How efficiently can one embed the graph underlying a given interconnection network in the graph underlying the butterfly network ? Within this framework, the slowdown incurred by an emulation is measured by the sum of the dilation and the congestion of the corresponding embedding (respectively, the maximum amount that the embedding stretches an edge of the guest graph, and the maximum traffic across any edge of the host graph) ; the efficiency of resource utilization in an emulation is measured by the expansion of the corresponding embedding (the ratio of the sizes of the host to guest graph). Three main results expose a number of optimal emulations by butterfly networks. Call a family of graphs balanced if complete binary trees can be embedded in the family with simultaneous dilation, congestion, and expansion 0(1). (1) The family of butterfly graphs is balanced. (2) (a) Any graph < from a family of maxdegree-d graphs having a recursive separator of size S(x) can be embedded in any balanced graph family with simultaneous dilation O(log(d Σ i S(2 -i |G|))) and expansion O(1). (b) Any dilation-D embedding of a maxdegree-d graph in a butterfly graph can be converted to an embedding having simultaneous dilation O(D) and congestion O(dD). (3) Any embedding of a planar graph G in a butterfly graph must have dilation Ω(log Σ (G)/Φ(G), where : Σ(G) is the size of the smallest (1/3, 2/3)-node-separator of G, and Φ(G) is the size of G's largest interior face. Applications of these results include : (1) The n-node X-tree network can be emulated by the butterfly network with slowdown O(log log n) and expansion 0(1) ; no embedding has dilation smaller than Ω(log log n), independent of expansion. (2) Every embedding of the n x n mesh in the butterfly graph has dilation Ω(log n) ; any expansion-O(1) embedding in the butterfly graph achieves dilation O(log n). These applications provide the first examples of networks that can be embedded more efficiently in hypercubes than in butterflies. We also show that analogues of these results hold for networks that are structurally related to the butterfly network. The upper bounds hold for the hypercube and the de Bruijn networks, possibly with altered constants. The lower bounds hold-at least in weakened form-for the de Bruijn network.

Efficient approach to embed binary trees in 3-D rectangular arrays

The complete binary tree has long been known to support important applications. This paper presents an efficient embedding of a complete binary tree in a 3-D rectangular array. The array hosting the binary tree is called the host, and the binary tree is referred to as the guest. The scheme is modular and high level trees are made of low level trees inductively. It is shown that all PEs (except one) of the host array may be utilised in embedding a k-level complete binary tree. The dilation (maximum length of an edge in the guest graph in terms of the number of edges in the host graph) is kept as low as possible by using building blocks, which allow tree embedding, with dilation of two at most occurring between leaf nodes and their parents where the message traffic is minimum. Upperbounds on propagation delay are reduced from O(2k/2 to O(2k/3) as a result of the use of the third dimension. Results are compared with those dealing with 2-D layouts. As discussed, these bounds are the best known to date for structures with practically implementable dimensionality.

Storage representations for tree-like data structures

Adata encoding is a formal counterpart of the “translation of a logical data structure into a physical storage structure". In this paper, both kinds of structures are represented by graphs; and an encoding is a type of embedding of the guest (logical structure) graph in the host (physical structure) graph. Thecost of an encoding is the average amount that the edges of the guest graph are stretched as they are replaced by paths in the host graph via the encoding; this “average” is weighted by an assignment of probabilities to the edges of the guest graph (called ausage pattern) that weights the edges according to their anticipated frequencies of traversals. This paper continues the study begun in [13] of encodings of data structures in the leaves of complete trees. The major thrust of the paper is to show that tree-hosts do not accommodate tree-like guests as congenially as they do array-like guests. Specifically, we study two encodings of tree-guests in tree-hosts, and three usage patterns for the guests. One of these usage patterns can be accommodated with cost independent of the size of the guest by one of the encodings but not by the other; the second usage pattern engenders the complementary situation; and the third usage pattern resists size-independent cost not only on the part of the two encodings studied here but also on the part ofany encoding of trees into trees. We do, however, find a third encoding of trees in trees whose cost relative to this intransigent usage pattern is exponentially smaller than the costs of our two encodings and is, in fact, optimal in rate of growth. Motivated by the demonstrated weaknesses of trees as hosts in data encodings, we introduce a new tree-like storage structure called adree; and we show that drees are quite congenial hosts for array-like and tree-like (and dree-like) guests. Throughout the paper, demonstrations of size-independence in the costs of encodings are accompanied by proofs of the (asympotic) near-optimality of the attained costs.

Data encodings and their costs

This paper is devoted to developing and studying a precise notion of the "encoding" of a "logical data structure" in a "physical storage structure," that is motivated by considerations of computational efficiency. The development builds upon the notion of an encoding of one graph in another. The cost of such an encoding is then defined so as to reflect the structural compatibility of the two graphs, the (externally specified) costs of "implementing" the host graph, and the (externally specified) set of intended "usage patterns" of the guest graph. The stability of the constructed framework is demonstrated in terms of a number of results; the faithfulness of the formalism is argued in terms of a number of examples from the literature; and the tractability of the model is hinted at by several results and by further references to the literature.