Optimal search for rationals

We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff property, the ratio property, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let n denote the number of nodes of a graph, m the number of edges (for constant treewidth graphs m = O(n)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a multiplicative factor of in time O(n · log(n/ )) and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time O(n · log(|a · b|)) = O(n · log(n · W )), when the output is a b , as compared to the previously best known algorithm with running time O(n 2 · log(n · W )). Third, for the minimum initial credit problem we show that (i) for general graphs the problem can be solved in O(n 2 · m) time and the associated decision problem can be solved in O(n · m) time, improving the previous known O(n 3 · m · log(n · W )) and O(n 2 · m) bounds, respectively; and (ii) for constant treewidth graphs we present an algorithm that requires O(n · log n) time, improving the previous known O(n 4 · log(n · W )) bound. We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.

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“…Then ν * can be obtained by using continued fractions, e.g. as in [36]. We rely in the work of Papadimitriou [41] to obtain a tighter bound.…”

Section: Exact Solutionmentioning

confidence: 99%

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

Chatterjee

Ibsen-Jensen

Pavlogiannis

2015

Computer Aided Verification

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“…Such unique rational is val Γ (v) and can be easily found in time logarithmic w.r.t. n [16]. Thus, we have an algorithm A to solve the value problem on Γ in time polynomial w.r.t.…”

Section: Approximate Solutions For Mpgmentioning

confidence: 99%

“…= O(n · log(n) + log(W )), T is also polynomial w.r. ] and can be easily found in time logarithmic w.r.t. n [16].…”

Section: Approximate Solutions For Mpgmentioning

confidence: 99%

A note on the approximation of mean-payoff games

Gentilini

2014

Information Processing Letters

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Abstract. We consider the problem of designing approximation schemes for the values of mean-payoff games. It was recently shown that (1) mean-payoff with rational weights scaled on [−1, 1] admit additive fully-polynomial approximation schemes, and (2) mean-payoff games with positive weights admit relative fully-polynomial approximation schemes. We show that the problem of designing additive/relative approximation schemes for general mean-payoff games (i.e. with no constraint on their edge-weights) is P-time equivalent to determining their exact solution.

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“…In the rational case, Papadimitriou [18], Reiss [22], Kwek and Mehlhorn [11], and Fori²ek [6] give similar solutions, which all use O(log n) queries.…”

Section: Algorithms For the Farey Sequencementioning

confidence: 99%

Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

Pawlewicz

Pătraşcu

2008

Algorithmica

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We present the rst sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3 ), which is a signicant improvement over the previous algorithms taking time O(n).We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7 , where D is the diameter of the polygon.

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Optimal search for rationals

Cited by 36 publications

References 4 publications

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

A note on the approximation of mean-payoff games

Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

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