Locality, Communication, and Interconnect Length in Multicomputers

Abstract

We derive a lower bound on the average interconnect (edge) length in d-dimensional embeddings of arbitrary graphs, expressed in terms of diameter and symmetry. It is optimal for all graph topologies we have examined, including complete graph, star, binary n-cube, cube-connected cycles, complete binary tree, and mesh with wraparound (e.g., torus, ring). The lower bound is technology independent, and shows that many interconnection topologies of today's multicomputers do not scale well in the physical world (d 3). The new proof technique is simple, geometrical, and works for wires with zero volume, e.g., for optical (fibre) or photonic (fibreless, laser) communication networks. Apparently, while getting rid of the "yon Neumann" bottleneck in the shift from sequential to nonsequential computation, a new communication bottleneck arises because of the interplay between locality of computation, communication, and the number of dimensions of physical space. As a consequence, realistic models for nonsequential computation should charge extra for communication, in terms of time and space.

Systolic Routing in an Optical Fat Tree

“…The results are only speculative because of a 1 We use whp, with high probability to mean with probability at least § ¥ © § ! [14,15]. The length of wires between routing nodes increases with respect to the physical space required.…”

The Full Cost of Cryptanalytic Attacks

“…As the number of processors and memory blocks gets into the thousands, making all the high-speed connections becomes costly. To address the objection that wires are thin, Vitányi treated wires as having no volume in showing that exponential computations cannot be completed in polynomial time with parallelism [22]. In our case if wires are assumed to have zero thickness but non-zero cost per unit length, the wiring costs in Section 2.1 drop from Θ(n 3/2 ) to Θ(n 4/3 ), the number of components in Theorem 1 drops to Θ(p + m + pr(p + m) 1/3 ), the full cost in Corollary 1 drops to Θ((T /p)(p + m + pr(p + m) 1/3 )), and Corollary 2 is unaffected.…”

Periodically regular chordal rings

“…Whereas an optimized network with small diameter and large bisection width is desirable, its cost/performance ratio suffers from dramatically increased area and wire length [6], [32]. For optimal PRC rings, this leads to the layout area yx P alog P x and the length of the longest wire yxalogx, which are of the same orders of magnitude as the respective requirements of CCC networks [28].…”