Efficient exact paths for dyck and semi-dyck labeled path reachability (extended abstract)

Abstract

The exact path length problem is to determine if there is a path of a given fixed cost between two vertices. This paper focuses on the exact path problem for costs −1, 0 or +1 between all pairs of vertices in an edge-weighted digraph. The edge weights are from {−1, +1}. In this case, this paper gives an O(n ω ) exact path solution.Here ω is the best exponent for matrix multiplication and O is the asymptotic upper-bound mod polylog factors.Variations of this algorithm determine which pairs of digraph nodes have Dyck or semi-Dyck labeled paths between them, assuming two parenthesis. Therefore, determining digraph reachability for Dyck or semi-Dyck labeled paths costs O(n ω ). A path label is made by concatenating all symbols along the path's edges.The exact path length problem has many applications. These applications include the labeled path problems given here, which in turn, also have numerous applications. * This current paper uses standard definitions of Dyck and semi-Dyck languages.Context-free language constrained graph problems are fundamental to a plethora of challenges. This paper gives algorithms for determining Dyck (semi-Dyck) constrained paths on digraphs based on the exact path problem. Dyck and semi-Dyck contextfree languages are important. A central application for the exact path problem is for determining Dyck and semi-Dyck constrained paths in digraphs. Here these languages have a single parenthesis type.Definition 1 (Exact path length problem [2]) Consider an integer edge weighted digraph G. Given an integer κ, the EPL (exact path length problem) is to determine whether there is a path between a given pair vertices costing exactly κ. Nykänen and Ukkonen [2]show the general EPL is N P-Complete. They also give a pseudo-polynomial algorithm for the EPL. The current paper uses a special case of the EPL where κ ∈ {−1, 0, +1} and edge costs are from the set {−1, +1}.Given these restricted edge costs, and for κ = 0, applying Nykänen and Ukkonen's algorithm costs O(n 3 + n ω log |κ|) time 1 , see [2]. For κ = 0, their algorithm costs O(n 3 ).Solving this Dyck (semi-Dyck) labeled path problem is interesting due to the close relationship between transitive closure, Boolean and algebraic matrix multiplication, and context-free grammar recognition. For example, Lee [3] gives an equivalence between Context-free parsers and Boolean matrix multiplication algorithms. Semi-Dyck and Dyck Constrained GraphsDyck and semi-Dyck languages are parenthesis languages. Dyck or semi-Dyck languages with two parenthesis symbols and n total parentheses can be parsed in O(n) time and space. However, efficiently computing Dyck (and semi-Dyck) constrained reachability on digraphs seems more challenging.Let D be a Dyck language of one open-parentheses symbol a and one closeparentheses symbol a −1 . A sentence w ∈ D iff w can be reduced using right-inverse reduction, e.g. a a −1 = , to the empty string . The Dyck language D is derivable from the grammar:Semi-Dyck languages allow reductions using both right-inverses and left-invers...

The fine-grained and parallel complexity of andersen’s pointer analysis

“…When ≥ 2, the problem is solvable in ( 3 ) time, and this bound is believed to be tight (wrt polynomial improvements) [Heintze and McAllester 1997]. The case of = 1 was recently solved independently in [Bradford 2018]. However, our algorithm behind Theorem 3.5 is more straightforward: it establishes a purely combinatorial reduction of the problem to (log 2 ) many transitive-closure operations.…”

“…From there, it relies on algebraic, fast-matrix multiplication for performing each transitive closure in ( ) time. In contrast, the algorithm in [Bradford 2018] is considerably longer and relies on intricate algebraic transformations. In addition, our algorithm is a log -factor faster.…”

“…The desired result follows. □A comparison note with[Bradford 2018].A sub-cubic bound for D 1 -Reachability was recently established independently in [Bradford 2018]. The crux of that algorithm is an elegant algebraic matrix encoding of flat D 1 -Reachability to (log ) AGMY matrix multiplications, each performed in ( • log ) time [Alon et al 1997].…”

Subcubic certificates for CFL reachability

“…Although Boolean matrix multiplication reduces to this problem, which diminishes the hope for faster combinatorial algorithms (avoiding fast matrix multiplication), the best known conditional lower bound for the problem has order |𝑉 | 𝜔 , where 𝜔 is the matrix multiplication exponent. On the other hand, Dyck-1 reachability (the language of balanced parentheses with one kind of parentheses) can be solved in time Õ (|𝑉 | 𝜔 ) [Bradford 2018;Bringmann 2018;Mathiasen and Pavlogiannis 2021;Vyalyi 2019], matching best conditional lower bounds.…”

Shortest Paths in One-Counter Systems

“…(4) Our results prove the existence of paths of length O(n 2 ); how efficiently can such paths be found? Bradford [8] gives a sketch of an approach more efficient than a search in the cubic-size graph of all configurations with counter value at most n 2 , achieving running time n ω · polylog(n), where ω < 2.373 is the matrix multiplication exponent. (5) Is there a quadratic upper bound for the pumping constant for one-counter languages?…”