An Efficient Algorithm for the Computation of Optimum Paths in Weighted Graphs

Marius Bozga (1) , Radu Iosif (1) , Vassiliki Sfyrla (1)

1 VERIMAG - IMAG - VERIMAG

The goal of this paper is to identify a more efficient algorithm for the computation of the path of minimum ratio (i.e. the quotient of the weight divided by the length) in a weighted graph. The main application of this technique is to improve the efficiency of reachability analysis for flat counter automata with Difference Bound Matrix constraints on transitions. A previous result showed that these paths could be defined using Presburger arithmetic. However, this method is costly, as the complexity of deciding satisfiability of Presburger formulae has a double exponential lower bound. Our solution avoids the use of Presburger arithmetic, by computing the function between any $n \in \Nat$ and the weight of the minimal path of length $n$. The function is computed iteratively, by computing the minimal fixed point of a system of set constraints, involving semilinear sets. This requires a min operator on linear sets, which is implemented using rewrite rules.

Mots clés: weighted graphs; minimal paths; Presburger arithmetic; policy iteration

Domaines: Systèmes embarqués

Marius Bozga, Radu Iosif, Vassiliki Sfyrla. An Efficient Algorithm for the Computation of Optimum Paths in Weighted Graphs. MEMICS 2007: Third Doctoral Workshop on Mathematical and Engineering Methods in Computer Science, Oct 2007, Znojmo, Czech Republic. pp.19-27. ⟨hal-00374984⟩