Author(s):

Chatterjee, Krishnendu; Henzinger, Monika

Title: 
An O(n2) time algorithm for alternating Büchi games

Affiliation 
IST Austria 
Abstract: 
Computing the winning set for Büchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications. The long standing best known upper bound for solving the problem is Õ(n·m), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the Õ(n·m) boundary by presenting a new technique that reduces the running time to O(n 2). This bound also leads to O(n 2) time algorithms for computing the set of almostsure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of Õ(n·m)), (2) in concurrent graph games with constant actions (improving an earlier bound of O(n 3)), and (3) in Markov decision processes (improving for m > n 4/3 an earlier bound of O(min(m 1.5, m·n 2/3)). We also show that the same technique can be used to compute the maximal endcomponent decomposition of a graph in time O(n 2), which is an improvement over earlier bounds for m > n 4/3. Finally, we show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per operation. This is the first dynamic algorithm for this problem.

Keywords: 
Computeraided verification; Graph games; Büchi objectives; Dynamic graph algorithms; Graph algorithms

Conference Title:

SODA: Symposium on Discrete Algorithms

Conference Dates:

January 1719, 2012

Conference Location:

Kyoto, Japan

ISBN:

15579468

Publisher:

SIAM

Date Published:

20120101

Start Page: 
1386

End Page:

1399

URL: 

Notes: 
The research was supported by Austrian Science Fund (FWF) Grant No P 23499N23 on Modern Graph Algorithmic Techniques in Formal Verification, Vienna Science and Technology Fund (WWTF) Grant ICT10002, FWF NFN Grant No S11407N23 (RiSE), ERC Start grant (279307: Graph Games), and Microsoft faculty fellows award.

Open access: 
yes (repository) 