TY - JOUR

T1 - Testing Eulerianity and connectivity in directed sparse graphs

AU - Orenstein, Yaron

AU - Ron, Dana

N1 - Funding Information:
We would like to thank Yuichi Yoshida for a helpful correspondence, as well as the anonymous reviewers of this paper for their comments and suggestions for improvements. This work was supported by the Israel Science Foundation (grant number 89/05).

PY - 2011/10/21

Y1 - 2011/10/21

N2 - Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespecified property or is ∈-far from the property (for a given distance parameter ∈, and for a prespecified distance measure). The tester is given query access to the input, and is required to run in sublinear time. In this paper, we focus on testing properties of directed graphs (digraphs). In particular, we present the following results (where n is the number of vertices in the graph, d is the maximum degree, and d avg is the average degree). • We present a testing algorithm for the property of Eulerianity in bounded-degree digraphs, which runs in time 1 Õ (1/∈). For unbounded-degree digraphs, we show a lower bound of Ω (√n/∈), and give a testing algorithm that runs in time Õ (√n/∈3/2). • We study the property of k-vertex-connectivity, and give testing algorithms for both bounded-degree and unbounded-degree digraphs that run in time Õ((ck/∈d)k d) and Õ ((ck/∈davg)k+1), respectively (where c > 1 is a constant). In addition, we give a simpler analysis of the testing algorithm for k-vertex-connectivity in bounded-degree undirected graphs that was shown by Yoshida and Ito [Y. Yoshida, H. Ito, Property testing on k-vertex-connectivity of graphs, in: ICALP'08: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, Springer-Verlag, Berlin, Heidelberg, 2008, pp. 539-550] and extend the result to unbounded-degree undirected graphs. • We consider the property of k-edge-connectivity in digraphs, and simplify the analysis of the algorithm of Yoshida and Ito [Y. Yoshida, H. Ito, Testing k-edge-connectivity of digraphs, Journal of System Science and Complexity 23 (1) (2010) 91-101] for this property. In addition, we give a simpler analysis for the correctness of the testing algorithm for k-edge-connectivity in undirected graphs that was introduced by Goldreich and Ron [O. Goldreich, D. Ron, Property testing in bounded degree graphs, Algorithmica (2002) 302-343].

AB - Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespecified property or is ∈-far from the property (for a given distance parameter ∈, and for a prespecified distance measure). The tester is given query access to the input, and is required to run in sublinear time. In this paper, we focus on testing properties of directed graphs (digraphs). In particular, we present the following results (where n is the number of vertices in the graph, d is the maximum degree, and d avg is the average degree). • We present a testing algorithm for the property of Eulerianity in bounded-degree digraphs, which runs in time 1 Õ (1/∈). For unbounded-degree digraphs, we show a lower bound of Ω (√n/∈), and give a testing algorithm that runs in time Õ (√n/∈3/2). • We study the property of k-vertex-connectivity, and give testing algorithms for both bounded-degree and unbounded-degree digraphs that run in time Õ((ck/∈d)k d) and Õ ((ck/∈davg)k+1), respectively (where c > 1 is a constant). In addition, we give a simpler analysis of the testing algorithm for k-vertex-connectivity in bounded-degree undirected graphs that was shown by Yoshida and Ito [Y. Yoshida, H. Ito, Property testing on k-vertex-connectivity of graphs, in: ICALP'08: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, Springer-Verlag, Berlin, Heidelberg, 2008, pp. 539-550] and extend the result to unbounded-degree undirected graphs. • We consider the property of k-edge-connectivity in digraphs, and simplify the analysis of the algorithm of Yoshida and Ito [Y. Yoshida, H. Ito, Testing k-edge-connectivity of digraphs, Journal of System Science and Complexity 23 (1) (2010) 91-101] for this property. In addition, we give a simpler analysis for the correctness of the testing algorithm for k-edge-connectivity in undirected graphs that was introduced by Goldreich and Ron [O. Goldreich, D. Ron, Property testing in bounded degree graphs, Algorithmica (2002) 302-343].

KW - Directed graphs

KW - Property testing

UR - http://www.scopus.com/inward/record.url?scp=84865778783&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2011.06.038

DO - 10.1016/j.tcs.2011.06.038

M3 - Article

AN - SCOPUS:84865778783

SN - 0304-3975

VL - 412

SP - 6390

EP - 6408

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 45

ER -