Recursive Star-Tree Parallel Data Structure

• Published in: SIAM journal on computing Vol. 22; no. 2; pp. 221 - 242
• Main Authors: Berkman, Omer, Vishkin, Uzi
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.1993
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer science; Computer science; control theory; systems; Exact sciences and technology; Leaves; Theoretical computing; Tree; Parallel algorithm; Parallel computation; Data structure; Optimal algorithm; Recursivity
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/0222017

This paper introduces a novel parallel data structure called the recursive star-tree (denoted "^ * $-tree"). For its definition a generalization of the $ * $ functional is used (where for a function $f * f(n) = \min \{ {i|f^{(i)} (n) \leqslant 1} \}$ and $f^{(i)} $ is the $i$th iterate of $f$). Recursive ^ * $-trees are derived by using recursion in the spirit of the inverse Ackermann function. The recursive ^ * $-tree data structure leads to a new design paradigm for parallel algorithms. This paradigm allows for extremely fast parallel computations, specifically, $O(\alpha (n))$ time (where $\alpha (n)$ is the inverse of the Ackermann function), using an optimal number of processors on the (weakest) concurrent-read, concurrent-write parallel random-access machine (CRCW PRAM). These computations need only constant time, and use an optimal number of processors if the following nonstandard assumption about the model of parallel computation is added to the CRCW PRAM: an extremely small number of processors each can write simultaneously into different bits of the same word. Applications include finding lowest common ancestors in trees by a new algorithm that is considerably simpler than the known algorithms for the problem, restricted domain merging, parentheses matching, and a new parallel reducibility.