Fringe analysis of synchronized parallel insertion algorithms in 2-3 Trees

Fringe analysis uses the distribution of bottom subtrees or fringe of search trees under the assumption of random insertion of keys, yielding an average case analysis of the fringe. The results in the fringe give upper and lower bounds for several measures for the whole tree. We are interested in the fringe analysis of the synchronized parallel insertion algorithms of Paul, Vishkin, and Wagener (PVW) on 2–3 trees. This algorithm inserts k keys with k processors into a tree of size n with time O(log n + log k). As the direct analysis of this algorithm is very di:cult we tackle this problem by introducing a new family of algorithms, denoted by MacroSplit algorithms, and our main theorem proves that two algorithms of this family, denoted MaxMacroSplit and MinMacroSplit, bound the behavior of the fringe in the PVW algorithm. Previous work deals with the fringe analysis of sequential algorithms, but this type of analysis was still an open problem for parallel algorithms on search trees. We extend fringe analysis to parallel algorithms and we get a rich mathematical structure giving new interpretations even in the sequential case. We prove that random insertion of keys generates a binomial distribution, that the synchronized insertion of keys can be modeled by a Markov chain, and that the coe:cients of the transition matrix of the Markov chain are related to the expected local behavior of our algorithm. Finally, we show that the coe:cients of the power expansion of this matrix over Partially supported by: (i) ACI-CONICYT grant through the Catalunya–Chile Cooperation Program (DOG 2320-30.1.1997), (ii) AMYRI VII.13 project (CYTED), (iii) Fondecyt grant 199-0627 (CONICYT), (iv) the IST program of the EU under contract number IST-1999-14186 (ALCOM-FT), (v) Spanish CICYT under grant TIC1999-0754-C03-02 (Mallba), and (vi) Spanish CICYT under grant TIC-2000-1970. ∗Corresponding author. E-mail address: [email protected] (R. Baeza-Yates). 0304-3975/03/$ see front matter c © 2002 Elsevier Science B.V. All rights reserved. PII: S0304 -3975(02)00318 -3 232 R. Baeza-Yates et al. / Theoretical Computer Science 299 (2003) 231–271 (n+1)−1 are the binomial transform of the expected local behavior of the algorithm. We Inally show that the fringe of the PVW algorithm asymptotically converges to the sequential case. c © 2002 Elsevier Science B.V. All rights reserved.