Comments on "Minimum number of adders for implementing a multiplier and its application to the design of multiplierless digital filters"

Abstract
For the original paper see ibid., vol. 42 p. 453-460 (July 1995). In the aforementioned paper by D. Li, a method of designing integer multipliers that uses fewer adders than canonic signed-digit (CSD) coding was described. This method is claimed by the author to be optimal, i.e., it produces a multiplier with the least possible adders. The commenters claim that they have presented a similar method, implemented using the MAG algorithm, to perform the same task, which was shown to be optimal. Comparing the two techniques shows that Li's method is not necessarily optimal. Both methods perform an exhaustive search over a set of multiplier configurations. In the commenters' paper, they described these configurations in terms of graphs. It is said that this method proves to be quite useful in that it becomes easy to describe the cases not covered by Li's algorithm. It is stated that the algorithm proposed by Li is suboptimal because: 1) graphs produced using fundamentals (intermediate vertices) of lower cost graphs are not considered; 2) tree-structured graphs are not considered; 3) right shifts (divide-by-two) are not allowed, causing some multipliers to require too many adders.

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