ICALP 1986: Automata, Languages and Programming pp 376-386

# An improved algorithm for transitive closure on acyclic digraphs

• Klaus Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)

## Abstract

In [6] Geralcikova, Koubek describe an algorithm for finding the transitive closure of an acyclic digraph G with worst case runtime O(n·ered), where n is the number of nodes and ered is the number of edges in the transitive reduction of G. We present an improvement on their algorithm which runs in worst case time O(k·ered) and space O(n·k), where k is the width of a chain decomposition. For the expected values in the Gn,p model of a random acyclic digraph with 0 < p < 1 we have:
$$\begin{gathered}E(k) = O(\frac{{\ln (p \cdot n)}}{p}) \hfill \\E(e_{red} ) = O(\min (n \cdot |lnp|,p \cdot n^2 )) = O(n \cdot \ln n) \hfill \\E(k \cdot e_{red} ) = \left\{ {\begin{array}{*{20}c}{O(n^2 )for\frac{{ln^2 n}}{n} \leqslant p < 1} \\{O(n^2 \cdot \ln \ln n)otherwise} \\\end{array} } \right. \hfill \\\end{gathered}$$

## Keywords

Random Graph Transitive Closure Improve Algorithm Random Graph Model Topological Sorting
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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