Mapping Computation with No Memory

Abstract

We investigate the computation of mappings from a set S ⁿ to itself with in situ programs, that is using no extra variables than the input, and performing modifications of one component at a time. We consider several types of mappings and obtain effective computation and decomposition methods, together with upper bounds on the program length (number of assignments). Our technique is combinatorial and algebraic (graph coloration, partition ordering, modular arithmetics).

For general mappings, we build a program with maximal length 5n − 4, or 2n − 1 for bijective mappings. The length is reducible to 4n − 3 when |S| is a power of 2. This is the main combinatorial result of the paper, which can be stated equivalently in terms of multistage interconnection networks as: any mapping of {0,1}ⁿ can be performed by a routing in a double n-dimensional Beneš network. Moreover, the maximal length is 2n − 1 for linear mappings when S is any field, or a quotient of an Euclidean domain (e.g. ℤ/sℤ). In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint.

The in situ trait of the programs constructed here applies to optimization of program and chip design with respect to the number of variables, since no extra writing memory is used. In a non formal way, our approach is to perform an arbitrary transformation of objects by successive elementary local transformations inside these objects only with respect to their successive states.