Realizing all Index Generation Functions by the Row-Shift Method

Abstract

We propose a method that allows the realization of all index generation functions using flexible decomposition charts. It is based on the first-fit decreasing heuristic used by Tarjan and Yao to store sparse matrices. We show that the first-fit-decreasing heuristic can yield nonminimal tables in the case of functions that do not satisfy the harmonic decay property. We show that an index generation function representation that just satisfies the harmonic decay property, called the perfect harmonic decay sequence, allows a simple matrix approach for calculating an error matrix, that describes the degree to which a given function representation departs from a perfect harmonic decay sequence. This gives insight into how function representations can be changed to realize the harmonic decay criteria. We also show the existence of sparse function representations for which no compression is possible. In such a case, we can still implement the corresponding index generation function, but it requires the largest resources possible.