Dynamic word problems

Abstract

Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector 2 = (21,22,. . .,z,,) E initially (1,1,. . ., 1) wifh two kinds of operations, for each i E { 1,. . . , n}, a E M, an operation changei,a which changes xi to a and a single operation product returning n:==, xi. This i s the dynamic word problem. If we in addition for each j E { 1, . . . , n} have an operation pref ixj returning ni=, xi, 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 logn bits. We obtain a classification of the Complexity based on algebraic properties of M.