Hypercube Embedding Research Articles

Topological properties and algorithms for two-level hypernet networks

Although many networks have been proposed as the topology of a large-scale parallel and distributed system, most of them are neither expansible nor of equal degree. A network with these two properties will gain the advantages of easy implementation and low cost when it is manufactured. The hypernet, which was proposed by Hwang and Ghosh, represents a family of recursively scalable networks that are both expansible and of equal degree. In addition to the two merits, the hypernet has proven efficient for communication and computation. But, unfortunately, most topological properties and the problem of shortest-path routing for the hypernet are still unsolved. The reason is that the structure of the hypernet is complex and asymmetric, and, especially, no mathematical description was given before. In this paper, considering current hardware restrictions, we concentrate our effort on the hypernet of moderate size. We first give a concise mathematical definition for the hypernet and then solve the following problems for the hypernet of two levels: (1) shortest-path routing, (2) diameter, (3) connectivity, (4) minimum-height spanning trees, and (5) embedding of rings, tori, and hypercubes. © 1998 John Wiley & Sons, Inc. Networks 31:105–118, 1998

The Hyperstar Interconnection Network

In this paper, a multiprocessor interconnection topology, the hyperstar, based on the Cartesian product of star graphs is studied. The basic properties of the hyperstar are discussed and proved. This includes reduced degree and diameter, hierarchical structure, vertex symmetry, optimal routing, and shortest path characterization. The hyperstar is shown to be a member of the Cayley class of symmetric graphs. Embeddings of hypercubes, star graphs, and meshes are discussed. An optimal one-to-all broadcasting algorithm is obtained and analyzed. Some results on fault tolerance, parallel paths, Hamiltonian cycles, and VLSI layouts are obtained. Furthermore, a comparative study between the hyperstar and seven related networks is conducted. The comparison is based on scalability, broadcasting cost, link requirements, cost/performance ratio, and other static parameters such as degree, diameter, and average diameter.

Cyclic-cubes: a new family of interconnection networks of even fixed-degrees

We introduce a new family of interconnection networks that are Cayley graphs with fixed degrees of any even number greater than or equal to four. We call the proposed graphs cyclic-cubes because contracting some cycles in such a graph results in a generalized hypercube. These Cayley graphs have optimal fault tolerance and logarithmic diameters. For comparable number of nodes, a cyclic-cube can have a diameter smaller than previously known fixed-degree networks. The proposed graphs can adopt an optimum routing algorithm known for one of its subfamilies of Cayley graphs. We also show that a graph in the new family has a Hamiltonian cycle and, hence, there is an embedding of a ring. Embedding of meshes and hypercubes are also discussed.

Index-Shuffle Graphs

Index-shuffle graphs are introduced as candidate interconnection networks for parallel computers. The comparative advantages of index-shuffle graphs over the standard bounded-degree "approximations" of the hypercube, namely butterfly-like and shuffle-like graphs, are demonstrated in the theoretical framework of graph embedding and network emulations. An N-node index-shuffle graph emulates: • an N-node shuffle-exchange graph with no slowdown, which the currently best emulations of shuffle-like graphs by hypercubes and butterflies incur a slowdown of Ω( log N). • its like-sized butterfly graph with a slowdown O( log log log N), while the currently best emulations of butterfly-like graphs by shuffle-like graphs incur a slowdown of Ω( log log N). • an N-node hypercube that executes an on-line leveled algorithm with a slowdown O( log log N), while the slowdown of currently best such emulations of the hypercube by its bounded-degree shuffle-like and butterfly-like derivatives remains Ω( log N). Our emulation is based on an embedding of an N-node hypercube into an N-node index-shuffle graph with dilation O( log log N), while the currently best embeddings of the hypercube into its bounded-degree shuffle-like and butterfly-like derivatives incur a dilation of Ω( log N).

STATE ENCODING OF FINITE STATE MACHINES FOR LOW POWER DESIGN

We address the problem of state encoding for synchronous finite state machines (FSMs), targeted for low power design. Most previous work in FSM state encoding has been focused on minimizing chip area and does not consider switching activity of the circuit. As a result, this does not always lead to a power efficient implementation. Especially in CMOS circuits, the switching activity is a very important factor to power dissipation. In this work, we define a function λ for automatic tradeoff between switching activity and area that contribute to power dissipation. λ is used in determining the encoding affinity between states and is observed to be related to the number of states of an FSM in our experiments. A state encoding algorithm, based on hypercube embedding, is proposed to find encodings of states such that the sum of bit toggles between each pair of states times the encoding affinity between them is minimized. The proposed approach does not require any change in the functional specification of the state machine and can be easily incorporated in present design flow. Results over a wide range of MCNC benchmark examples which show the efficacy of our technique are presented. A simple function for λ is provided, and it is shown to be robust in finding low-power state encodings.

A new F‐H embedding algorithm for input encoding

AbstractThis paper presents a new encoding algorithm based on the face hypercube embedding for satisfying given input constraints. the algorithm works for a constraint relation matrix derived from the sets of the input symbols grouped for each output symbol independently and finds the encoding through row‐by‐row (group‐by‐group) cover assignment. A cover is found by the extension of a core cover generated from the magnitude estimations and adjacent relations of the currently known covers. Each cover is formed to be a single cube; and, if it is unsuccessful for specified code length, then the encoding length is extended. the experimental results indicate that this encoding technique has advantages over well‐known existing methods in the quality of encoding.

Transposition networks as a class of fault-tolerant robust networks

The paper proposes designs of interconnection networks (graphs) which can tolerate link failures. The networks under study belong to a subclass of Cayley graphs whose generators are subsets of all possible transpositions. We specifically focus on star and bubble sort networks. Our approach is to augment existing dimensions (or generators) with one or more dimensions. If the added dimension is capable of replacing any arbitrary failed dimension, it is called a wildcard dimension. It is shown that, up to isomorphism among digits used in labeling the vertices, the generators of the star graph are unique. The minimum number of extra dimensions needed to acquire i wildcard dimensions is derived for the star and bubble sort networks. Interestingly, the optimally augmented star network coincides with the Transposition network, T/sub n/. Transposition networks are studied rigorously. These networks are shown to be optimally fault tolerant. T/sub n/ is also shown to possess wide containers with short length. Fault diameter of T/sub n/ is shown to be n. While the T can efficiently embed star and bubble sort graphs, it can also lend itself to an efficient embedding of meshes and hypercubes.

Benchmarking the computation and communication performance of the CM-5

Thinking Machines' CM-5 machine is a distributed-memory, message-passing computer. In the paper we devise a performance benchmark for the base and vector units and the data communication networks of the CM-5 machine. We model the communication characteristics such as communication latency and bandwidths of point-to-point and global communication primitives. We show, on a simple Gaussian elimination code, that an accurate static performance estimation of parallel algorithms is possible by using those basic machine properties connected with computation, vectorization, communication and synchronization. Futhermore, we describe the embedding of meshes or hypercubes on the CM-5 fat-tree topology and illustrate the performance results of their basic communication primitives.

Executing algorithms with hypercube topology on torus multicomputers

Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost-per-node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, an embedding of hypercubes onto toruses of any given dimension is proposed. This novel embedding is called xor embedding. The paper presents a set of performance figures for both the standard and the xor embeddings and shows that the latter outperforms the former for any torus. In addition, it is proven that for a one-dimensional torus (a ring) the xor embedding is optimal in the sense that it minimizes the execution time of a class of parallel algorithms with hypercube topology. This class of algorithms is frequently found in real applications, such as FFT and some class of sorting algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Embeddings of Hypercubes and Grids into de Bruijn Graphs

An embedding of a graph G into a graph H is an injective mapping f from the vertices of G into the vertices of H together with a mapping P f of edges of G into paths in H. The dilation of the embedding is tile maximum taken over all the lengths of the paths P f ( xy) associated with the edges xy of G. We show that it is possible to embed a D-dimensional hypercube into the binary de Bruijn graph of the same order and diameter with dilation at most [ D/2]. Similarly a majority of planar grids can be embedded into a binary de Bruijn graph of the same or nearly the same order with dilation at most [ D/2] where D is the diameter of the de Bruijn graph.

Near embeddings of hypercubes into Cayley graphs on the symmetric group

Simulations of hypercube networks by certain Cayley graphs on the symmetric group are investigated. Let Q(k) be the familiar k-dimensional hypercube, and let S(n) be the star network of dimension n defined as follows. The vertices of S(n) are the elements of the symmetric group of degree n, two vertices x and y being adjacent if xo(1,i)=y for some i. That is, xy is an edge if x and y are related by a transposition involving some fixed symbol (which we take to be /spl I.bold/1). This network has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages of S(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube. The first step in such a simulation is the construction of a one-to-one map f:Q(k)/spl rarr/S(n) of dilation d, for d small. That is, one wants a map f such that images of adjacent points are at most distance d apart in S(n). An alternative approach, best applicable when one-to-one maps are difficult or impossible to find, is the construction of a one-to-many map g of dilation d, defined as follows. For each point x/spl isin/Q(k), there is an associated subset g(x)/spl sube/V(S(n)) such that for each edge xy in Q(k), every x'/spl isin/g(x) is at most distance d in S(n) from some y'/spl isin/g(y). Such one-to-many maps allow one to achieve the low interprocessor communication time desired in the usual one-to-one embedding underlying a simulation. This is done by capturing the local structure of Q(k) inside of S(n) (via the one-to-many embedding) when the global structure cannot be so captured. Our results are the following. 1) There exist the following one-to-many embeddings: a) f:Q(k)/spl rarr/S(3k+1) with dilation (f)=1; b) f:Q(11k+2)/spl rarr/S(13k+2) with dilation (f)=2. 2) There exists a one-to-one embedding f:Q(n2/sup n/spl minus/1/)/spl rarr/S(2/sup n/) with dilation (f)=3.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Hypersphere Mapper: A Nonlinear Programming Approach to the Hypercube Embedding Problem

A nonlinear programming approach is introduced for solving the hypercube embedding problem. The basic idea of the proposed approach is to approximate the discrete space of an n-dimensional hypercube, i.e., { z: z ∈ {0, 1} n }, with the continuous space of an n-dimensional hypersphere, i.e., { x: x ∈ R n & || x|| 2 = 1}. The mapping problem is initially solved in the continuous domain by employing the gradient projection technique to a continuously differentiable objective function. The optimal process "locations" from the solution of the continuous hypersphere mapping problem are then discretized onto the n-dimensional hypercube. The proposed approach can solve, directly, the problem of mapping P processes onto N nodes for the general case where P > N. In contrast, competing embedding heuristics from the literature can produce only one-to-one mappings and cannot, therefore, be directly applied when P > N.

Hypercube embedding heuristics: An evaluation

The hypercube embedding problem, a restricted version of the general mapping problem, is the problem of mapping a set of communicating processes to a hypercube multiprocessor. The goal is to find a mapping that minimizes the length of the paths between communicating processes. Unfortunately the hypercube embedding problem has been shown to be NP-hard. Thus many heuristics have been proposed for hypercube embedding. This paper evaluates several hypercube embedding heuristics, including simulated annealing, local search, greedy, and recursive mincut bipartitioning. In addition to known heuristics, we propose a new greedy heuristic, a new Kernighan-Lin style heuristic, and some new features to enhance local search. We then assess variations of these strategies (e.g., different neighborhood structures) and combinations of them (e.g., greedy as a front end of iterative improvement heuristics). The asymptotic running times of the heuristics are given, based on efficient implementations using a priority-queue data structure.