A Parallel Solution to the Extended Set Union Problem with Unlimited Backtracking

In this paper, we study on the EREW-PRAM model a parallel solution to the extended set union problem with unlimited backtracking which maintains a dynamic partition Π of an n-element set S subject to the usual operations Find, Union, Backtrack and Restore as well as the new operations SetUnion, MultiUnion. The SetUnion operation is a special case of the commonly known Union operation aimed to unify two prespecified set-names, while MultiUnion operation deals with a batch of Union operations. A new data structure, called k-Parallel Union Find (or, k-PUF) trees, is introduced to represent a disjoint set in Π. The structure is defined for a wide range for the parameter k, but the log n more interesting results are achieved for k = log log. In n this case, using the k-PUF trees, both SetUnion and Restore operations are performed in constant parallel time requiring optimal work O�k�. This constant-time performance is not achievable parallelizing the existing data structures. Moreover, using p = O�k � processors, MultiUnion for a batch of p operations is performed in O�k � time, requiring optimal work O�pk�.