Fast Point-To-Point Dyck Constrained Shortest Paths On A Dag

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(n(3) / 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(n(2) 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(n(3)). 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.