We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a “pebble ” which has a unique destination π(v) in G (so that π is a permutation of the vertices of G). It is required to route all the pebbles to their respective destinations by performing a sequence of moves of the following type: A disjoint set of edges is selected and the pebbles at each edge’s endpoints are interchanged. The problem of interest is to minimize the number of steps required for any possible permutation π. In this paper we investigate this routing problem for a variety of graphs G, including trees, complete graphs, hypercubes, Cartesian products of graphs, expander graphs and Cayley graphs. In addition, we relate this routing problem to certain network flow problems, and to several graph invariants including diameter, eigenvalues and expansion coefficients. 2 1

In this paper we present efficient algorithms for sorting, selection and packet routing on the AROB (Array with Reconfigurable Optical Buses) model.

We survey routing problems on fixed-connection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, k-relation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.

This paper first presents some general properties of product networks pertinent to parallel architectures and then focuses on three case studies. These are products of complete binary trees, shuffle-exchange, and de Bruijn networks. It is shown that all of these are powerful architectures for parallel computation, as evidenced by their ability to efficiently emulate numerous other architectures. In particular, r-dimensional grids, and r-dimensional meshes of trees can be embedded efficiently in products of these graphs, i.e. either as a subgraph or with small constant dilation and congestion. In addition, the shuffle-exchange network can be embedded in r-dimensional product of shuffle exchange networks with dilation cost 2r and congestion cost 2. Similarly, the de Bruijn network can be embedded in r-dimensional product of de Bruijn networks with dilation cost r and congestion cost 4. Moreover, it is well known that shuffle-exchange and de Bruijn graphs can emulate the hypercu...

The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (so-called) caterpillar tree partition, we prove that any permutation on an n-node tree (and thus graph) can be routed in 3 2 n + O(log n) steps. This answers an open problem of Alon, Chung and Graham. Key words. matching routing, off-line algorithms, trees AMS subject classifications. 05C, 68M, 68R 1 Introduction Routing problems on networks arise in different fields such as communications, parallel architectures and VLSI theory, and have been extensively studied in recent years (see [9, 10] for a comprehensive survey). In this paper, we study permutation routing under the matching model, which was proposed by Alon, Chung and Graham[2]. The routing of this type is described as follows. Given a graph G = (V; E) with vertex set V and edge set E. Initially, each vertex v of G is occupied by a "packet" p. To each packet p is associated a destination ß(v) 2 V , so that di...

In this paper we develop generalized methods to layout homogeneous product networks with any number of dimensions, and analyze their VLSI complexity by deriving upper and lower bounds on the area and maximum wire length. In the literature, lower bounds are generally obtained by computing lower bounds on the bisection width or the crossing number of the network being laid out. In this paper we define a new measure that we call "maximal congestion," that can be used to obtain both the bisection width and the crossing number, thereby unifying the two approaches. Upper bounds are traditionally obtained by constructing layouts based on separators or bifurcators. Both methods have the basic limitation that they are applicable only for graphs with bounded vertex degree. The separators approach generally yields good layouts when good separators can be found, but it is difficult to find a good separator for an arbitrary graph. The bifurcators approach is easier to apply, but it gener...

The grid and the mesh of trees (or MOT) are among the best-known parallel architectures in the literature. Both of them enjoy efficient VLSI layouts, simplicity of topology, and a large number of parallel algorithms that can efficiently execute on them. One drawback of these architectures is that algorithms that perform best on one of them do not perform very well on the other. Thus there is a gap between the algorithmic capabilities of these two architectures. We propose a new class of parallel architectures, called the mesh-connected trees (or MCT) that can execute grid algorithms as efficiently as the grid, and MOT algorithms as efficiently as the MOT, up to a constant amount of slowdown. In particular, the MCT topology contains the MOT as a subgraph and emulates the grid via embedding with dilation 3 and congestion 2. This significant amount of computational versatility offered by the MCT comes at no additional VLSI area cost over these earlier networks. Many topological,...

In this paper we present an extensive study of many-to-many routing on trees under the matching routing model. Our study includes on-line and off-line algorithms. We present an asymptotically optimal on-line algorithm which routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps, where d is the degree of tree T on which the routing takes place and dist is the maximum distance any packet has to travel. We also present an off-line algorithm that solves the same problem within 2(k \Gamma 1)+dist steps. The analysis of our algorithms is based on the establishment of a close relationship between the matching and the hot-potato routing models that allows us to apply tools which were previously used exclusively in the analysis of hot-potato routing.

A sparse-mesh, which has PUs on the diagonal of a two-dimensional grid only, is a cost effective distributed memory machine. Variants of this machine have been considered before, but none of them is so simple and pure as a sparse-mesh. Various fundamental problems (routing, sorting, list ranking) are analyzed, proving that sparse-meshes have a great potential. The results are extended for higher dimensional sparse-meshes. 1 Introduction On ordinary two-dimensional meshes we must accept that, due to their small bisection width, for most problems the maximum achievable speed-up with n 2 processing units (PUs) is only \Theta(n). On the other hand, networks such as hypercubes impose increasing conditions on the interconnection modules with increasing network sizes. Cube-connected-cycles do not have this problem, but are harder to program due to their irregularity. Anyway, because of a basic theorem from VLSI lay-out [18], all planar architectures have an area that is quadratic in their...

In this paper we study the packet routing problem under the matching model proposed by Alon, Chung and Graham [1]. We extend the model to allow more than one packet per origin and destination node. We give tight bounds for the many-to-one routing number for complete graphs, complete bipartite graphs and linear arrays. We also present an efficient algorithm for many-to-one routing on an trees (and therefore any graph). Finally, we give bounds for routing arbitrary relations in this model. 1 Introduction Routing packets arises naturally in the design of large-scale parallel computers and the study of data flow in parallel computing. Packet routing consists of moving packets of data from each node of a network to the other nodes in the network. The goal is to move all of the packets to their desired locations as quickly as possible. Various routing problems have been extensively studied under different models. We refer the reader to [4] for a survey of the topic. In this paper, we study ...