Automata, Languages and Programming
Volume 1853 of the series Lecture Notes in Computer Science pp 809820
Testing Acyclicity of Directed Graphs in Sublinear Time
 Michael A. BenderAffiliated withDepartment of Computer Science, State University of New York at Stony Brook
 , Dana RonAffiliated withDepartment of Electrical Engineering  Systems, Tel Aviv University
Abstract
This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs  acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of O(l/∈^{2}), where ∈ is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least e fraction of its entries so that it become acyclic. For the incidence list representation, most appropriate for sparse graphs, an Ω(V 1/3) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. These results stand in contrast to what is known about testing acyclicity in undirected graphs.
 Title
 Testing Acyclicity of Directed Graphs in Sublinear Time
 Book Title
 Automata, Languages and Programming
 Book Subtitle
 27th International Colloquium, ICALP 2000 Geneva, Switzerland, July 9–15, 2000 Proceedings
 Pages
 pp 809820
 Copyright
 2000
 DOI
 10.1007/354045022X_68
 Print ISBN
 9783540677154
 Online ISBN
 9783540450221
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1853
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Ugo Montanari ^{(4)}
 José D. P. Rolim ^{(5)}
 Emo Welzl ^{(6)}
 Editor Affiliations

 4. Department of Computer Sciences, University of Pisa
 5. Center for Computer Sciences, University of Geneva
 6. Department of Computer Sciences, ETH Zurich
 Authors

 Michael A. Bender ^{(7)}
 Dana Ron ^{(8)}
 Author Affiliations

 7. Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY, 117944400, USA
 8. Department of Electrical Engineering  Systems, Tel Aviv University, Ramat Aviv, Israel
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