Asynchronous Shared Memory Search Structures

We study the problem of storing an ordered set on an asynchronous shared memory parallel computer. We examine the case where we want to perform successor (least upper bound) queries efficiently on the set members that are stored. We also examine the case where processors insert and delete members of the set. Due to asynchrony, we require processors to perform queries and to maintain the structure independently. Although several such structures have been proposed, the analysis of these structures has been very limited. We here use the recently proposed QRQW PRAM model to provide upper and lower bounds on the performance of such data structures. In the asynchronous QRQW PRAM, the problem of processors concurrently and independently searching a shared data structure is very similar to the problem of routing packets through a network. Using this as a guide, we introduce the Search-Butterfly, a search structure that combines the efficient packet routing properties of the butterfly graph with the efficient search structure properties of the B-Tree. We analyze the behavior of the Search-Butterfly when the following operations are performed: arbitrary searches, random searches, and random searches, insertions, and deletions. We also provide lower bounds that show that the results are within a factor of O(\log n) of optimal where n is the number of keys in the structure. When the searches are random, the results are within a constant factor of optimal. Many of the proofs are derived from closely related results for packet routing. Others are of independent interest, most notably a method of adding queues to any network belonging to a large class of queuing networks with non-Markovian routing in a manner that allows us to bound the delay experienced by packets in the augmented network. Received October 1996, and in final form July 1997.