Article

An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees

Authors:

Patrick EugsterAuthors Info & Claims

ESOP '09: Proceedings of the 18th European Symposium on Programming Languages and Systems: Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009

Pages 175 - 189

https://doi.org/10.1007/978-3-642-00590-9_13

Published: 28 March 2009 Publication History

Abstract

The context-free language (CFL) reachability problem is well known and studied in computer science, as a fundamental problem underlying many important static analyses such as points-to-analysis. Solving the CFL reachability problem in the general case is very hard. Popular solutions resorting to a graph traversal exhibit a time complexity of <em>O</em> (<em>k</em> ³ <em>n</em> ³) for a grammar of size <em>k</em> . For Dyck CFLs, a particular class of CFLs, this complexity can be reduced to <em>O</em> (<em>kn</em> ³). Only recently the first subcubic algorithm was proposed by Chaudhuri, dividing the complexity of predating solutions by a factor of log<em>n</em> .

In this paper we propose an effective algorithm for solving the CFL reachability problem for Dyck languages when the considered graph is a bidirected tree with specific constraints. Our solution pre-processes the graph in <em>O</em> (<em>n</em> log<em>n</em> log<em>k</em> ) time in a space of <em>O</em> (<em>n</em> log<em>n</em> ), after which any Dyck-CFL reachability query can be answered in <em>O</em> (1) time, while a naïve online algorithm will require <em>O</em> (<em>n</em> ) time to answer a query or require <em>O</em> (<em>n</em> ²) to store the pre-computed results for all pairs of nodes.

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Cited By

Xu PLei YSui YXue J(2024)Iterative-Epoch Online Cycle Elimination for Context-Free Language ReachabilityProceedings of the ACM on Programming Languages10.1145/36498628:OOPSLA1(1437-1462)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649862
Pavlogiannis A(2023)CFL/Dyck ReachabilityACM SIGLOG News10.1145/3583660.35836649:4(5-25)Online publication date: 6-Feb-2023
https://dl.acm.org/doi/10.1145/3583660.3583664
Lei YSui YDing SZhang Q(2022)Taming transitive redundancy for context-free language reachabilityProceedings of the ACM on Programming Languages10.1145/35633436:OOPSLA2(1556-1582)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563343
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Index Terms

An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees
1. Theory of computation
  1. Formal languages and automata theory
    1. Grammars and context-free languages

Index terms have been assigned to the content through auto-classification.

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cover image Guide Proceedings

ESOP '09: Proceedings of the 18th European Symposium on Programming Languages and Systems: Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009

March 2009

409 pages

ISBN:9783642005893

Editor:
Giuseppe Castagna
CNRS, Université Denis Diderot, Paris 7, France

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 28 March 2009

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Cited By

Xu PLei YSui YXue J(2024)Iterative-Epoch Online Cycle Elimination for Context-Free Language ReachabilityProceedings of the ACM on Programming Languages10.1145/36498628:OOPSLA1(1437-1462)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649862
Pavlogiannis A(2023)CFL/Dyck ReachabilityACM SIGLOG News10.1145/3583660.35836649:4(5-25)Online publication date: 6-Feb-2023
https://dl.acm.org/doi/10.1145/3583660.3583664
Lei YSui YDing SZhang Q(2022)Taming transitive redundancy for context-free language reachabilityProceedings of the ACM on Programming Languages10.1145/35633436:OOPSLA2(1556-1582)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563343
Shi QWang YYao PZhang C(2022)Indexing the extended Dyck-CFL reachability for context-sensitive program analysisProceedings of the ACM on Programming Languages10.1145/35633396:OOPSLA2(1438-1468)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563339
Christakis MCottenier TFilieri ALuo LMansur MPike LRosner NSchäf MSengupta AVisser WRoychoudhury ACadar CKim M(2022)Input splitting for cloud-based static application security testing platformsProceedings of the 30th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering10.1145/3540250.3558944(1367-1378)Online publication date: 7-Nov-2022
https://dl.acm.org/doi/10.1145/3540250.3558944
Kjelstrøm APavlogiannis A(2022)The decidability and complexity of interleaved bidirected Dyck reachabilityProceedings of the ACM on Programming Languages10.1145/34986736:POPL(1-26)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3498673
Cai CZhang QZuo ZNguyen KXu GSu Z(2018)Calling-to-reference context translation via constraint-guided CFL-reachabilityACM SIGPLAN Notices10.1145/3296979.319237853:4(196-210)Online publication date: 11-Jun-2018
https://dl.acm.org/doi/10.1145/3296979.3192378
Cai CZhang QZuo ZNguyen KXu GSu ZFoster JGrossman D(2018)Calling-to-reference context translation via constraint-guided CFL-reachabilityProceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation10.1145/3192366.3192378(196-210)Online publication date: 11-Jun-2018
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Chatterjee KChoudhary BPavlogiannis A(2017)Optimal Dyck reachability for data-dependence and alias analysisProceedings of the ACM on Programming Languages10.1145/31581182:POPL(1-30)Online publication date: 27-Dec-2017
https://dl.acm.org/doi/10.1145/3158118
Zhang QLyu MYuan HSu Z(2013)Fast algorithms for Dyck-CFL-reachability with applications to alias analysisACM SIGPLAN Notices10.1145/2499370.246215948:6(435-446)Online publication date: 16-Jun-2013
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