9 research outputs found
Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension
International audienceUnder the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes-where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| â 1. âą Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time Ă(k · mn 1âΔ_d), where Δ_d â (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. âą Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. âą Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublin-ear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of Δ-nets, region decomposition and other partition techniques
Sensor Coverage Strategy in Underwater Wireless Sensor Networks
This paper mainly describes studies hydrophone placement strategy in a complex underwater environment model to compute a set of "good" locations where data sampling will be most effective. Throughout this paper it is assumed that a 3-D underwater topographic map of a workspace is given as input.Since the negative gradient direction is the fastest descent direction, we fit a complex underwater terrain to a differentiable function and find the minimum value of the function to determine the low-lying area of the underwater terrain.The hydrophone placement strategy relies on gradient direction algorithm that solves a problem of maximize underwater coverage: Find the maximize coverage set of hydrophone inside a 3-D workspace. After finding the maximize underwater coverage set, to better take into account the optimal solution to the problem of data sampling, the finite VC-dimension algorithm computes a set of hydrophone that satisfies hydroacoustic signal energy loss constraints. We use the principle of the maximize splitting subset of the coverage set and the âdualâ set of the coverage covering set, so as to find the hitting set, and finally find the suboptimal set (i.e., the sensor suboptimal coverage set).Compared with the random deployment algorithm, although the computed set of hydrophone is not guaranteed to have minimum size, the algorithm does compute with high network coverage quality
Packing and covering balls in graphs excluding a minor
We prove that for every integer there exists a constant such
that for every -minor-free graph , and every set of balls in ,
the minimum size of a set of vertices of intersecting all the balls of
is at most times the maximum number of vertex-disjoint balls in . This
was conjectured by Chepoi, Estellon, and Vax\`es in 2007 in the special case of
planar graphs and of balls having the same radius.Comment: v3: final versio
Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension
International audienceUnder the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes-where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| â 1. âą Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time Ă(k · mn 1âΔ_d), where Δ_d â (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. âą Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. âą Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublin-ear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of Δ-nets, region decomposition and other partition techniques
Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension
International audienceUnder the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes-where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| â 1. âą Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time Ă(k · mn 1âΔ_d), where Δ_d â (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. âą Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. âą Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublin-ear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of Δ-nets, region decomposition and other partition techniques
Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension
International audienceUnder the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes-where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| â 1. âą Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time Ă(k · mn 1âΔ_d), where Δ_d â (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. âą Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. âą Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublin-ear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of Δ-nets, region decomposition and other partition techniques
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively