An improved algorithm for transitive closure on acyclic digraphs

Abstract

In [6] Geralcikova, Koubek describe an algorithm for finding the transitive closure of an acyclic digraph G with worst case runtime O(n·e _red), where n is the number of nodes and e _red 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·e _red) and space O(n·k), where k is the width of a chain decomposition. For the expected values in the G _n,p model of a random acyclic digraph with 0 < p < 1 we have:

$$\begin{gathered}E(k) = O(\frac{{\ln (p \cdot n)}}{p}) \hfill \\E(e_{red} ) = O(\min (n \cdot |lnp|,p \cdot n^2 )) = O(n \cdot \ln n) \hfill \\E(k \cdot e_{red} ) = \left\{ {\begin{array}{*{20}c}{O(n^2 )for\frac{{ln^2 n}}{n} \leqslant p < 1} \\{O(n^2 \cdot \ln \ln n)otherwise} \\\end{array} } \right. \hfill \\\end{gathered}$$