Many popular interconnection network topologies, such as hypercubes and toroidal meshes, are based on Cayley graphs of Abelian groups. The symmetry and algebraic structure of these graphs result in many nice physical properties of the network concerning layout, routing algorithms, and load balancing. There has been interest in low-diameter Abelian--Cayley graphs because of their smaller communication delay and reduced congestion. For any fixed number of nodes n, and any fixed out-degree k, we are interested in how small the diameter of directed Cayley graphs of Abelian groups can be and what these low-diameter graphs look like. We give an upper bound of $n \leq \frac{3(d + 3)^3}{25}$ for the size of directed Abelian--Cayley graphs with k = 3 and diameter d, correcting a previously published result by Hsu and Jia [SIAM J. Discrete Math., 7 (1994), pp. 57--71]. Our method is based on translational tiling techniques and is a generalization of Wong and Coppersmith's method for k = 2 [J. Assoc. Comput. Mach., 21 (1974), pp. 392--402]. Moreover, our method works for all Abelian groups, not just the cyclic case. For k = 3 we give computational results for the largest Abelian--Cayley graph as a function of diameter. When n = 84 m3, for integer m, there is a network with $n = \frac{(d + 3)^3}{11.95}$ whose diameter is approximately three-fourths of that of a three-dimensional toroidal cube.

Let k be a positive integer. We call a graph G = (V, E) a k-dot product graph if there is a function f:V --> R-k SO that for all vertices v not equal w we have vw is an element of E if and only if f(v).f(w) greater than or equal to 1. The least k for which G is a k-dot product graph is called the dot product dimension of G and is denoted rho(G).We discuss the significance of dot product dimension and obtain various results about the dot product dimension of various sorts of graphs.

The problem of choosing a static shortest-path system that minimizes maximum edge congestion in a network is studied. Bounds based on parameters, such as diameter, bisection width, and average distance, are derived and conditions for producing uniform congestion on all edges are explored. Trees are shown to have maximum congestion on edges that are incident to a centroid node. Cartesian product graphs, which generalize multidimensional meshes, are shown to satisfy several closure properties and a generic factor-routing scheme is defined and shown to be optimal in many cases.

Pin-efficient single-instruction multiple-data networks, with p/spl ap//spl radic/(2m) pins per cell that can/spl minus/in one clock tick/spl minus/shift data by any amount k in an interval /spl lsqb//spl minus/m,m/spl rsqb/ are considered. Perfect barrel shifters, which perform the group of permutations c/spl rarr/c+k(mod n), 0/spl les/k/spl les/n/spl minus/1, using p=q+1 pins per cell, are known to exist for all n=q/sup 2/+q+1, where q is any prime power. In sharp contrast, it is shown that for any permutation /spl pi/ of order greater than 3m, one-tick perfect shifters for the set of permutations /spl pi//sup /spl lsqb//spl minus/m,m/spl rsqb//=/spl lcub//spl pi//sup k//spl verbar//spl minus/m/spl les/k/spl les/m/spl rcub/ exist only for the three cases (m=1, p=2), (m=3, p=3), and (m=6, p=4). In particular, only three perfect linear arrays, c/spl rarr/c+k, exist. The proof is based on a relationship between the difference covers and zero-one solutions to certain quadratic equations. >

Pin-efficient bussed network families are discussed that can-in one clock tick-simultaneously shift all data in a k-dimensional grid to neighboring processors in any one of the 3/sup k/-1 'compass directions' x/spl I.oarr//spl rarr/x/spl I.oarr/+/spl delta//spl I.oarr/, for every nonzero vector /spl delta//spl I.oarr/ /spl isin/ {-1,0,1}/sup k/. The networks have the advantages of being simple to describe (using a single 5-state automaton), extendible (the k-dimensional network is obtained by extending the busses of the (k-1)-dimensional network), and provably optimal for k/spl les/3. The networks use only [3/2(/spl radic/3)/sup k/] pins per processor, which is within 3/2 of the theoretical minimum number of pins required. The best previously known family uses 2/sup k/ pins.< >

Pin limitations are a fundamental obstacle in the construction of massively parallel computers. The paper introduces a class of d-dimensional bussed hypercubes that can perform simultaneous bidirectional communication across any dimension using d+1, rather than 2d, ports per node. Each network Q/sub d/(T) is based on a tree T, which specifies the 'shape' of the busses, and can perform d(d+1)/2 permutations pi /sub ij/(x)=x(+)c/sub ij/ via a simple global command. This construction is then generalized to any d permutations II=( pi /sub 1/,..., pi /sub d/) of any set of nodes X. Given any edge-labeled directed tree T, whose kth arc is associated with the permutation pi /sub k/, a bussed network N(II,T) is constructed that can-in one clock tick-perform any of the O(d/sup 2/) permutations arising from the paths in the tree T.

We introduce an efficient method for computing matrix products of the formY=AXB, whereA andB are sparse and constant. We analyze the complexity of the method, develop quantitative criteria for determining when it can be used effectively, and demonstrate its use in a Kalman filter.

The solutions to a scalar, homogeneous, constant-coefficient, linear recurrence are expressible in terms of the powers of a companion matrix. We show how to compute these powers efficiently via polynomial multiplication. The result is a simple expression for the solution, which does not involve the characteristic roots and which is valid for any module over any commutative ring. The formula yields the nth term of the solution to a kth order recurrence with $O(\mu (k) \cdot \log n)$ arithmetic operations, where $\mu (k)$ is the total number of arithmetic operations required to multiply two polynomials of degree $k - 1$. Thus if the ring supports a fast Fourier transform, then $O(k \cdot \log k \cdot \log n)$ operations are sufficient to compute the nth term.