Optimal Folding of Data Flow Graphs Based on Finite Projective Geometry Using Vector Space Partitioning

Abstract

A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry (PG) based balanced bipartite graphs. Many of these applications of finite projective geometry are actively being researched upon, especially in coding theory. Almost all these applications need large bipartite graphs, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization while designing systems modeled using PG-based graphs. In such systems, the number of processing units is equal to the number of vertices, each performing an atomic computation. We present a novel way of partitioning the vertex set assigned to various atomic computations, into blocks. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. The symmetric properties of projective space lattices enable us to develop a conflict-free communication schedule. We employed the technique of coset decomposition of a finite field for partitioning. The folding scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. We first provide a scheme for a finite projective space of dimension five, and the corresponding schedules. This specific scheme is then generalized for arbitrary finite projective spaces. Both the folding schemes have been verified by both simulation as well as hardware prototyping. For example, a semi-parallel decoder architecture for a new class of expander codes was designed and implemented using this scheme, with potential deployment in DVD-R/CD-ROM drives.

A design methodology for optimally folded, pipelined architectures in VLSI applications using projective space lattices

“…It is a well-known fact that the lattice of subspaces in any projective space is a modular, geometric lattice [9]. A projective space of dimension 2 is shown in figure 1.…”

“…The computational folding can be implemented after (balanced) graph partitioning in two dual ways. In the first way, that is used in [8], [9], the within-fold computation is done sequentially, and across-fold computation is done parallely. Such a scheme is generally called a supernode-based folded design, since a logical supernode is held responsible for operating over a fold.…”

“…The scheme is based on simple concepts of modulo arithmetic, and circulance of PGbased balanced bipartite graphs. It is an engineering-oriented, practical alternative to another scheme based on vector space partitioning [9]. The core of that scheme is based on adapting the method of vector space partitioning [5] to projective spaces, and hence involves fair amount of mathematical rigor.…”

Expander-like Codes based on Finite Projective Geometry

“…• As detailed in a companion paper [5], the mapping of vertices to points and hyperplanes enables us to use several projective geometry properties for disproving the existence of certain bipartite subgraphs of a fixed minimum degree. This strategy leads us to finding the minimum number of vertices required to form a complete bipartite subgraph of a given minimum degree.…”

“…Also, we wanted the codes to be practically useful. Hence, in a companion paper, we present throughput-optimal VLSI design of decoder for such codes [5]. For decoding, we employ a variation of Zemor's algorithm.…”