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.
