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

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(𝑚min{𝑚𝑙𝑜𝑔𝑛,‾‾‾‾‾‾√𝑘,𝑛}+𝑏𝑚𝑖𝑛{𝑙𝑜𝑔𝑛,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 ith special vertex. The previously best algorithm repeatedly recomputes the strongly-connected components which leads to a running time of O(Σ i m 2i). We present an algorithm with time 𝑂(𝑙√∑𝑖𝑚1.5𝑖).