Lower bounds for symbolic computation on graphs: Strongly connected components, liveness, safety and diameter

K. Chatterjee, W. Dvorák, M. Henzinger, V. Loitzenbauer, in:, ACM, 2018, pp. 2341–2356.


Conference Paper | Published | English
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; ; ;
Department
Series Title
SODA
Abstract
A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and coliveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with O(D) symbolic operations, where D is the diameter of the graph. Additionally we present an approximation algorithm for the graph diameter which requires Õ(n/D) symbolic steps to achieve a (1 +ϵ)-approximation for any constant > 0. This compares to O(n/D) symbolic steps for the (naive) exact algorithm and O(D) symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of [15], showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components.
Publishing Year
Date Published
2018-01-01
Page
2341 - 2356
Conference
SODA: Symposium on Discrete Algorithms
Conference Location
New Orleans, Louisiana, USA
Conference Date
2018-01-07 – 2018-01-10
IST-REx-ID

Cite this

Chatterjee K, Dvorák W, Henzinger M, Loitzenbauer V. Lower bounds for symbolic computation on graphs: Strongly connected components, liveness, safety and diameter. In: ACM; 2018:2341-2356. doi:10.1137/1.9781611975031.151
Chatterjee, K., Dvorák, W., Henzinger, M., & Loitzenbauer, V. (2018). Lower bounds for symbolic computation on graphs: Strongly connected components, liveness, safety and diameter (pp. 2341–2356). Presented at the SODA: Symposium on Discrete Algorithms, New Orleans, Louisiana, USA: ACM. https://doi.org/10.1137/1.9781611975031.151
Chatterjee, Krishnendu, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. “Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety and Diameter,” 2341–56. ACM, 2018. https://doi.org/10.1137/1.9781611975031.151.
K. Chatterjee, W. Dvorák, M. Henzinger, and V. Loitzenbauer, “Lower bounds for symbolic computation on graphs: Strongly connected components, liveness, safety and diameter,” presented at the SODA: Symposium on Discrete Algorithms, New Orleans, Louisiana, USA, 2018, pp. 2341–2356.
Chatterjee K, Dvorák W, Henzinger M, Loitzenbauer V. 2018. Lower bounds for symbolic computation on graphs: Strongly connected components, liveness, safety and diameter. SODA: Symposium on Discrete Algorithms, SODA, 2341–2356.
Chatterjee, Krishnendu, et al. Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety and Diameter. ACM, 2018, pp. 2341–56, doi:10.1137/1.9781611975031.151.

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