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 k 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 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.