Fringe analysis of synchronized parallel algorithms on 2-3 trees

Abstract

We are interested in the fringe analysis of synchronized parallel insertion algorithms on 2–3 trees, namely the algorithm of W. Paul, U. Vishkin and H. Wagener (PVW). This algorithm inserts keys into a tree of size n with parallel time O(log n + log k).

Fringe analysis studies the distribution of the bottom subtrees and it is still an open problem for parallel algorithms on search trees. To tackle this problem we introduce a new kind of algorithms whose two extreme cases seems to upper and lower bounds the performance of the PVW algorithm.

We extend the fringe analysis to parallel algorithms and we get a rich mathematical structure giving new interpretations even in the sequential case. The process of insertions is modeled by a Markov chain and the coefficients of the transition matrix are related with the expected local behavior of our algorithm. Finally, we show that this matrix has a power expansion over (n+1) -1 where the coefficients are the binomial transform of the expected local behavior. This expansion shows that the parallel case can be approximated by iterating the sequential case.