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Maximum Flows and Parametric Shortest Paths in Planar Graphs
"... We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously ..."
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We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously increase the parameter, each change in the shortest path tree can be effected in O(log n) time using standard dynamic tree data structures, and the special structure of the parametrization implies that each directed edge enters the evolving shortest path tree at most once. The resulting maximumflow algorithm is identical to the recent algorithm of Borradaile and Klein [J. ACM 2009], but our new formulation allows a simpler presentation and analysis. On the other hand, we demonstrate that for a similarly structured parametric shortest path problem on the torus, the shortest path tree can change Ω(n²) times in the worst case, suggesting that a different method may be required to efficiently compute maximum flows in highergenus graphs.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 7 (5 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
SubpathOptimality of MultiCriteria Shortest Paths in Time and EventDependent Networks
"... We consider the problem of finding all Paretooptimal (s, t)paths in discrete timedependent graphs with respect to several criteria. Label setting or label correcting algorithms can be used to compute all Paretooptima in static multidimensional weighted digraphs. These are generalizations of the ..."
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Cited by 1 (0 self)
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We consider the problem of finding all Paretooptimal (s, t)paths in discrete timedependent graphs with respect to several criteria. Label setting or label correcting algorithms can be used to compute all Paretooptima in static multidimensional weighted digraphs. These are generalizations of the wellknown algorithms for the single criterion shortest paths problem. Their correctness is based on the subpathoptimality. Unfortunately, this property vanishes in the classical notions of timedependent networks. Therefore, Kostreva et al. (1993), Getachew et al. (2000) and Hamacher et al. (2006) solved the multicriteria timedependent problem only for special cases. We give some new definitions for discrete timedependent graphs which differ in some aspects from previous works. These new notions allow us to establish the subpathoptimality in discrete timedependent graphs. Our concept can be interpreted as an implicit time expansion. Furthermore, we extend this concept to eventdependent graphs. This is useful for many practical applications in which weights of paths do not only depend on time but also on other attributes.
BAPTISTE Hervé
"... Recherche de chemins multiobjectifs pour la conception et la réalisation d’une centrale de mobilité destinée aux cyclistes THÈSE dirigée par: ..."
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Recherche de chemins multiobjectifs pour la conception et la réalisation d’une centrale de mobilité destinée aux cyclistes THÈSE dirigée par:
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 A Combinatorial Solution for Modelbased Image Segmentation and Realtime Tracking
"... Abstract—We propose a combinatorial solution to determine the optimal elastic matching of a deformable template to an image. The central idea is to cast the optimal matching of each template point to a corresponding image pixel as a problem of finding a minimum cost cyclic path in the threedimensio ..."
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Abstract—We propose a combinatorial solution to determine the optimal elastic matching of a deformable template to an image. The central idea is to cast the optimal matching of each template point to a corresponding image pixel as a problem of finding a minimum cost cyclic path in the threedimensional product space spanned by the template and the input image. We introduce a cost functional associated with each cycle which consists of three terms: a data fidelity term favoring strong intensity gradients, a shape consistency term favoring similarity of tangent angles of corresponding points and an elastic penalty for stretching or shrinking. The functional is normalized with respect to the total length to avoid a bias toward shorter curves. Optimization is performed by Lawler’s Minimum Ratio Cycle algorithm parallelized on stateoftheart graphics cards. The algorithm provides the optimal segmentation and point correspondence between template and segmented curve in computation times which are essentially linear in the number of pixels. To the best of our knowledge this is the only existing globally optimal algorithm for realtime tracking of deformable shapes. I.
1 The Elastic Ratio: Introducing Curvature into Ratiobased Image Segmentation
"... Abstract—We present the first ratiobased image segmentation method which allows to impose curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature ..."
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Abstract—We present the first ratiobased image segmentation method which allows to impose curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature of the curve. The key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment. Among ratios whose discrete counterparts can be globally minimized with our approach, we focus in particular on the elastic ratio L(C)
MultipleSource Shortest Paths in Embedded Graphs ∗
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g 2 n log n) time.