A simple parallel tree contraction algorithm

Efficient sequential and parallel algorithms for the negative cycle problem

“…The tree T i is obtained by T i−1 by applying a local operation, called SHUNT, to a subset of the leaves of T i−1 . The SHUNT operation consists in turn by two other operations, called prune and bypass [1]. Let l be a leaf in a tree T i−1 .…”

“…The main goal is to evaluate the root face of T . The parallel tree-contraction algorithm [1] evaluates the root of a tree T processing a logarithmic number of binary trees T 0 , T 1 , ..., T k , k = O(log |T |), T 0 = T and T k contains only one node. Also, |T i | ≤ ε|T i−1 |, 0 < ε < 1.…”

A Faster Implementation of a Parallel Tree Contraction Scheme and Its Application on Distance-Hereditary Graphs

“…If a rehanging occurs, as discussed in the paragraph before Lemma 11, we make an assumption on the reduction order of Algorithm SCD that only the nodes in the subtree rooted at ␣ are reduced before the rehanging, where ␣ is the node processed in Case 1.1. 1.…”

“…The above result implies several other problems related to the minimum ␥-dominating clique problem can be solved with the same parallel complexities. ᮊ 2000 Academic Press 1. INTRODUCTION w x A graph is distance-hereditary 2, 25 if the distance stays the same between any of two vertices in every connected induced subgraph contain-Ž ing both where the distance between two vertices is the length of a .…”

Deciding whether graph G has page number one is in NC

“…Step 3 can be computed in $O(logl)$ time with $O(l/logl)$ processors, where $l$ is the number of vertices in the parse tree, by applying tree contraction $al$ gorithm in [1] (see also [7].) As $l$ is $O(n)$ , this step Step 4.…”