Systematic Derivation of Tree Contraction Algorithms

Abstract

While tree contraction algorithms play an important role in efficient tree computation in parallel, it is difficult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving efficient tree contraction algorithms from recursive functions on trees. We identify a general recursive form that can be parallelized into efficient tree contraction algorithms, and present a derivation strategy for transforming general recursive functions to the parallelizable form. We illustrate our approach by deriving a novel parallel algorithm for the maximum connected-set sum problem on arbitrary trees, the tree-version of the well-known maximum segment sum problem.

Parallel Tree Contraction with Fewer Types of Primitive Contraction Operations and Its Application to Trees of Unbounded Degree

“…At least, absence of Compress R L makes it no more complicated to work out the corresponding calculations. This is the reason Theorem 4 is easier to use than that by Matsuzaki et al [11], [12].…”

“…In particular, this intuitive understanding makes it easier to develop ρ ⊗,⊕ from ⊕ and ⊗. Matsuzaki et al [11], [12] proposed a theorem similar to Theorem 4. Their theorem also requires a function similar to ρ ⊗,⊕ for implementing tree reduction by parallel tree contraction.…”

“…We consider tree reductions [11], [12], [18]. Given an associative binary operator, (⊕) :: A → A → A, and a (possibly non-associative) binary operator, (⊗) :: B → A → A, a tree reduction, reduce ⊗,⊕ :: RTree B → A, is identified as follows.…”

“…However, most studies ignored these issues. A notable exception is that by Matsuzaki et al [11], [12]. They showed that parallel rose-tree reductions can be translated to parallel binary-tree processing if the premise of Theorem 4 is fulfilled.…”

“…Most studies [1], [3], [4], [11], [12], [14] were based on binarization, namely, transforming the rose tree to a binary tree then applying a parallel tree contraction algorithm for binary trees. Bi- 1 The University of Tokyo, Meguro, Tokyo 153-8902, Japan matsuzaki.kiminori@kochi-tech.ac.jp narization is effective in theory, since it only affects the constant factor -it at most doubles the number of nodes.…”