# Faster algorithms for the nonemptiness of streett automata and for communication protocol pruning

## Abstract

This paper shows how a general technique, called *lock-step search*, used in dynamic graph algorithms, can be used to improve the running time of two problems arising in program verification and communication protocol design.

- (1)
We consider the

*nonemptiness problem for Streett automata*: We are given a directed graph*G*= (*V*,*E*) with*n*= ¦*V*¦ and*m*= ¦*E*¦, and a collection of pairs of subsets of vertices, called*Streett pairs*, 〈*L*_{ i },*U*_{ i }〉,*i*= 1.*k*. The question is whether*G*has a cycle (not necessarily simple) which, for each 1 ≤*i*≤*k*, if it contains a vertex from*L*_{ i }then it also contains a vertex of*U*_{ i }. Let*b*=Σ_{ i=1..k }|*L*_{ i }|+|*U*_{ i }|. The previously best algorithm takes time*O*((*m*+*b*) min{*n*,*k*}). We present an algorithm that takes time \(O(m\min \{ \sqrt {m log n, } k,n\} + b min\{ log n, k\} )\). - (2)
In

*communication protocol pruning*we are given a directed graph*G*= (*V*,*E*) with*l*special vertices. The problem is to efficiently maintain the strongly-connected components of the special vertices on a restricted set of edge deletions. Let*m*_{ i }be the number of edges in the strongly connected component of the*i*th special vertex. The previously best algorithm repeatedly recomputes the strongly-connected components which leads to a running time of*O*(Σ_{ i }*m*_{ i }^{2}). We present an algorithm with time \(O(\sqrt l \sum _i m_i^{1.5} )\).

## Keywords

Model Check Source Component Edge Deletion Special Vertex Current Graph## Preview

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