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Applications of parametric maxflow in computer vision
"... The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for ..."
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Cited by 39 (7 self)
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The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for different λ’s. An example is a weighting between data and regularization terms in image segmentation or stereo: it is desirable to vary it both during training (to learn λ from ground truth data) and testing (to select best λ using highknowledge constraints, e.g. user input). We review algorithmic aspects of this parametric maximum flow problem previously unknown in vision, such as the ability to compute all breakpoints of λ and corresponding optimal configurations in finite time. These results allow, in particular, to minimize the ratio of some geometric functionals, such as flux of a vector field over length (or area). Previously, such functionals were tackled with shortest path techniques applicable only in 2D. We give theoretical improvements for “PDE cuts ” [5]. We present experimental results for image segmentation, 3D reconstruction, and the cosegmentation problem. 1.
Stochastic shortest paths via quasiconvex maximization
 PROCEEDINGS OF EUROPEAN SYMPOSIUM OF ALGORITHMS
, 2006
"... We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally ..."
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Cited by 15 (7 self)
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We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally distributed edge lengths, which is based on quasiconvex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general nonconvex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.
Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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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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Cited by 6 (1 self)
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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.
A Lower Bound for the Shortest Path Problem
"... We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring al ..."
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Cited by 5 (0 self)
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We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring algorithm for the Shortest Path Problem is optimal in the unbounded fanin PRAM model without bit operations. 1
Partial LeastSquares Point Matching under Translations
"... We consider the problem of translating a given pattern set B of size m, and matching every point of B to some point of a larger ground set A of size n in an injective way, minimizing the sum of the squared distances between matched points. We show that when B can only be translated along a line, the ..."
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Cited by 2 (2 self)
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We consider the problem of translating a given pattern set B of size m, and matching every point of B to some point of a larger ground set A of size n in an injective way, minimizing the sum of the squared distances between matched points. We show that when B can only be translated along a line, there can be at most m(n − m) + 1 different matchings as B moves along the line, and hence the optimal translation can be found in polynomial time. 1
Shortest Paths in TimeDependent FIFO Networks Using Edge Load Forecasts
, 2009
"... We study the problem of finding shortest paths in timedependent networks with edge load forecasts where the behavior of each edge is modeled as a timedependent arrival function with FIFO property. Here, we present a new algorithm that computes for a given start node s and destination node d, the sh ..."
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Cited by 2 (0 self)
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We study the problem of finding shortest paths in timedependent networks with edge load forecasts where the behavior of each edge is modeled as a timedependent arrival function with FIFO property. Here, we present a new algorithm that computes for a given start node s and destination node d, the shortest paths and earliest arrival times for all possible starting times. Our algorithm runs in time O((Fd + λ)(E  + V log V )) where Fd is the output size (number of linear pieces needed to represent the earliest arrival time function) and λ is the input size (number of linear pieces needed to represent the local earliest arrival time functions for all edges in the network). Our method improves significantly on the best previously known algorithm which requires time O(FmaxV E) where Fmax ≥ Fd is the maximum number of linear pieces needed to represent the earliest arrival time function between the start node s to any node in the network. It has been conjectured that there are cases where Fmax is of superpolynomial size; however, even in such cases, Fd might still be of linear size. In such cases, our algorithm would take polynomial time to find the solution, while other methods require superpolynomial time. Both of the above methods are not useful in practice for graphs where Fd is of superpolynomial size. For such graphs, we present the first approximation method to compute for all possible starting times at s, the earliest arrival times at d within error at most ǫ. Our algorithm runs in time O ( ∆ (E  + V log V )) where ∆ is the difference beǫ tween the earliest arrival times at d for the latest and earliest starting times at s.
Twophase algorithms for the parametric shortest path problem
"... Abstract. A parametric weighted graph is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edgeweighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, an ..."
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Abstract. A parametric weighted graph is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edgeweighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, and arise in various natural scenarios. Parametric weighted graph algorithms consist of two phases. A preprocessing phase whose input is a parametric weighted graph, and whose output is a data structure, the advice, that is later used by the instantiation phase, where a specific value for the variable is given. The instantiation phase outputs the solution to the (standard) weighted graph problem that arises from the instantiation. The goal is to have the running time of the instantiation phase supersede the running time of any algorithm that solves the weighted graph problem from scratch, by taking advantage of the advice. In this paper we construct several parametric algorithms for the shortest path problem. For the case of linear function weights we present an algorithm for the single source shortest path problem. Its preprocessing phase runs in Õ(V 4) time, while its instantiation phase runs in only O(E + V log V) time. The fastest standard algorithm for single source shortest path runs in O(V E) time. For the case of weight functions defined by degree d polynomials, we present an algorithm with quasipolynomial
A Lower Bound on Computing Blocking Flows in Graphs
"... We show that the Maximum Blocking Flow problem on a graph of n vertices cannot be computed in time o(n 1/8 ) with polynomially many processors on an unbounded fanin PRAM without bit operations, even when the bitlengths of the inputs are restricted to be of size O(n 2 ). 1 ..."
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We show that the Maximum Blocking Flow problem on a graph of n vertices cannot be computed in time o(n 1/8 ) with polynomially many processors on an unbounded fanin PRAM without bit operations, even when the bitlengths of the inputs are restricted to be of size O(n 2 ). 1