Abstract
Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x=(x/sub 1/,x/sub 2/,...,x/sub n/)/spl isin/M/sup n/, initially (1,1,...,1) with two kinds of operations, for each i/spl isin/{1,...,n}, a/spl isin/M, an operation change/sub i,a/ which changes x/sub i/ to a and a single operation product returning /spl Pi//sub i=1//sup n/x/sub i/. This is the dynamic word problem. If we in addition for each j/spl isin/{1,...,n} have an operation prefix/sub j/ returning /spl Pi//sub i=1//sup j/x/sub i/, we talk about the dynamic prefix problem. We analyze the complexity of these problems in the cell probe or decision assignment tree model for two natural cell sizes, 1 bit and log n bits. We obtain a classification of the complexity based on algebraic properties of M. >
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