We present a fast and practical algorithm to compute the transitive closure (TC) of a directed graph. It is based on computing a reachability indexing scheme of a directed acyclic graph (DAG), G = (V, E). Given any path/chain decomposition of G we show how to compute in parameterized linear time such a reachability scheme that can answer reachability queries in constant time. The experimental results reveal that our method is significantly faster in practice than the theoretical bounds imply, indicating that path/chain decomposition algorithms can be applied to obtain fast and practical solutions to the transitive closure (TC) problem. Furthermore, we show that the number of non-transitive edges of a DAG G is ≤ width*|V| and that we can find a substantially large subset of the transitive edges of G in linear time using a path/chain decomposition. Our extensive experimental results show the interplay between these concepts in various models of DAGs.
@InProceedings{kritikakis_et_al:LIPIcs.SEA.2023.2, author = {Kritikakis, Giorgos and Tollis, Ioannis G.}, title = {{Fast Reachability Using DAG Decomposition}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {2:1--2:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-279-2}, ISSN = {1868-8969}, year = {2023}, volume = {265}, editor = {Georgiadis, Loukas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.2}, URN = {urn:nbn:de:0030-drops-183526}, doi = {10.4230/LIPIcs.SEA.2023.2}, annote = {Keywords: graph algorithms, hierarchy, directed acyclic graphs (DAG), path/chain decomposition, transitive closure, transitive reduction, reachability, reachability indexing scheme} }
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