Fast point-to-point Dyck constrained shortest paths on a DAG (Extended abstract)

Many aspects of program analysis are related to CFL (context-free language) constrained path problems on graphs. A path is constrained by requiring its list of edge labels to form a string that is a member of the associated CFL. Constrained shortest path problems give O(n3/ log n)-bottlenecks for program analysis, where n is the number of nodes. Labeled path problems also have many other applications. Assume two terminals (parenthesis) in a Dyck CFL. Given any two vertices s and t and the output of Nykanen and Ukkonen's exact integer path length algorithm. then this paper finds a minimal-cost point-to-point Dyck path cost in such edge-labeled digraphs in O(n2 log n) additional operations. This paper is on the DAG (directed acyclic graph) case. Our result depends on a special case of the exact integer path length problem of Nykanen and Ukkonen that costs O(n3). A new algorithm is introduced that joins special edges output by Nykanen and Ukkonen's algorithm. Nykanen and Ukkonen's algorithm along with our basic algorithm gives a minimal-cost point-to-point Dyck shortest path cost between two nodes.

[1]  Phillip G. Bradford,et al.  Quickest path distances on context-free labeled graphs , 2007 .

[2]  Mihalis Yannakakis,et al.  Graph-theoretic methods in database theory , 1990, PODS.

[3]  William W. Wadge,et al.  Preferential Regular Path Queries , 2009, Fundam. Informaticae.

[4]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[5]  Phillip G. Bradford,et al.  Labeled shortest paths in digraphs with negative and positive edge weights , 2009, RAIRO Theor. Informatics Appl..

[6]  H. James Hoover,et al.  Limits to Parallel Computation: P-Completeness Theory , 1995 .

[7]  Lillian Lee,et al.  Fast context-free grammar parsing requires fast boolean matrix multiplication , 2001, JACM.

[8]  Madhav V. Marathe,et al.  Formal Language Constrained Path Problems , 1998, SWAT.

[9]  Wojciech Rytter,et al.  Fast Recognition of Pushdown Automaton and Context-free Languages , 1986, Inf. Control..

[10]  Phillip G. Bradford,et al.  A Distributed Context-Free Language Constrained Shortest Path Algorithm , 2008, 2008 37th International Conference on Parallel Processing.

[11]  Paul Walton Purdom,et al.  A transitive closure algorithm , 1970, BIT.

[12]  Swarat Chaudhuri,et al.  Subcubic algorithms for recursive state machines , 2008, POPL '08.

[13]  Graham Cormode,et al.  Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[14]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[15]  Ehsan Khamespanah,et al.  Efficient TCTL Model Checking Algorithm for Timed Actors , 2014, AGERE!@SPLASH.

[16]  Matti Nykänen,et al.  The Exact Path Length Problem , 2002, J. Algorithms.

[17]  Venkatesh Choppella,et al.  Source-Tracking Unification , 2003, CADE.