Time-space tradeoffs for set operations

Abstract

This paper considers time-space tradeoffs for various set operations. Denoting the time requirement of an algorithm by T and its space requirement by S, it is shown that TS=Ω(n 2) for set complementation and TS=Ω(n 3 2 ) for set intersection, in the R-way branching program model. In the more restricted model of comparison branching programs, the paper provides two additional types of results. A tradeoff of TS=Ω(n 2-ε(n)) , derived from Yao's lower bound for element distinctness, is shown for set disjointness, set union and set intersection [where ε(n)=O(( logn) − 1 2 ) ]. A bound of TS=Ω(n 3 2 ) is shown for deciding set equality and set inclusion. Finally, a classification of set operations is presented, and it is shown that all problems of a large naturally arising class are as hard as the problems bounded in this paper.