A Fast Parametric Maximum Flow Algorithm and Applications

Gallo, Giorgio; Grigoriadis, Michael D.; Tarjan, Robert E.

doi:10.1137/0218003

Cited by 533 publications

(490 citation statements)

References 33 publications

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“…A maximum non-empty closed set in G(θ) can be computed, 19,10,23 as follows. Define a new directed graph G st (θ) = (V ∪ {s, t},E ∪ E st ).…”

Section: Lemma 8 For a Given θ The Region R * (θ) Specified By A Mamentioning

confidence: 99%

“…(4) It is well-known that computing a maximum-cost closed set in G(θ) is equivalent to compute a minimum s-t cut in G st (θ). 19,10 In G st (θ), the source s has a directed edge to every vertex υ(x z ) in G(θ) with a cost of w(υ(x z )), the sink t has a directed edge with a cost of 0 from every vertex in G(θ), and the cost of all other edges is +∞. Obviously, based on the vertex-cost assignment scheme (4), the cost of every edge from source s is a non-increasing function of θ and all other edges in G st (θ) have a constant cost with respect to θ.…”

Section: Computing An Optimal-ratio Smooth Lower-half Regionmentioning

confidence: 99%

“…Thus, G st (θ) is a monotone parametric flow network. 13,10 To compute an optimal-ratio smooth lower-half region in Γ, as in Section 2.2, we need to compute an optimal smooth lower-half region with a maximized parametric net-cost for each of a sequence of parameters {θ 1 , θ 2 ,…, θ m } generated by the hand probing process (see Section 2.1), where m = O(n). Due to the monotonicity of G st (θ), we can apply Gusfield and Martel's parametric minimum cut algorithm 13 to compute all those O(n) optimal smooth lower-half regions in Γ in the complexity of solving a single maximum flow problem.…”

Section: Computing An Optimal-ratio Smooth Lower-half Regionmentioning

confidence: 99%

“…We notice that the O(n) calls to the minimum st cut algorithm are on a sequence of weighted directed graphs, which forms a monotone parametric flow network. 13,10 Hence, the optimal-ratio smooth lower-half region in Γ can be detected in the complexity of computing a single maximum flow using Gusfield and Martel's algorithm. 13 Furthermore, for the case of using the normalization function g(R) = | R|, we establish a connection between our convex hull model for the problem and the traditional Newton based approach for the fractional programming problem (see, e.g., Gondran and Minous 12 ).…”

Section: Introductionmentioning

confidence: 99%

“…13 Furthermore, for the case of using the normalization function g(R) = | R|, we establish a connection between our convex hull model for the problem and the traditional Newton based approach for the fractional programming problem (see, e.g., Gondran and Minous 12 ). This connection enables us to apply Gallo et al 's simple parametric minimum s-t cut algorithm, 10 yielding an O(n 2 log n) time algorithm for computing an optimal-ratio smooth lower-half region.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

2006

Algorithms and Computation

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In this paper, we study several interesting optimal-ratio region detection (ORD) problems in d-D (d ≥ 3) discrete geometric spaces, which arise in high dimensional medical image segmentation. Given a d-D voxel grid of n cells, two classes of geometric regions that are enclosed by a single or two coupled smooth heighfield surfaces defined on the entire grid domain are considered. The objective functions are normalized by a function of the desired regions, which avoids a bias to produce an overly large or small region resulting from data noise. The normalization functions that we employ are used in real medical image segmentation. To our best knowledge, no previous results on these problems in high dimensions are known. We develop a unified algorithmic framework based on a careful characterization of the intrinsic geometric structures and a nontrivial graph transformation scheme, yielding efficient polynomial time algorithms for solving these ORD problems. Our main ideas include the following. We observe that the optimal solution to the ORD problems can be obtained via the construction of a convex hull for a set of O(n) unknown 2-D points using the hand probing technique. The probing oracles are implemented by computing a minimum s-t cut in a weighted directed graph. The ORD problems are then solved by O(n) calls to the minimum s-t cut algorithm. For the class of regions bounded by a single heighfield surface, our further investigation shows that the O(n) calls to the minimum s-t cut algorithm are on a monotone parametric flow network, which enables to detect the optimal-ratio region in the complexity of computing a single maximum flow.

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“…A maximum non-empty closed set in G(θ) can be computed, 19,10,23 as follows. Define a new directed graph G st (θ) = (V ∪ {s, t},E ∪ E st ).…”

Section: Lemma 8 For a Given θ The Region R * (θ) Specified By A Mamentioning

confidence: 99%

Section: Computing An Optimal-ratio Smooth Lower-half Regionmentioning

confidence: 99%

Section: Computing An Optimal-ratio Smooth Lower-half Regionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

2006

Algorithms and Computation

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Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems

Ahuja

Orlin²

1991

Naval Research Logistics

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Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distance‐directed algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distance‐directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n2m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distance‐directed algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems.

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On constructing DAG‐schedules with large areas

Roche

Rosenberg

Rajaraman

2015

Concurrency and Computation

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Abstract. The Area of a schedule Σ for a DAG G is a quality metric that measures the rate at which Σ renders G's nodes eligible for execution. Specifically, AREA(Σ) is the average number of nodes of G that are eligible for execution as Σ executes G node by node. Extensive simulations suggest that, for many distributions of processor availability and power, DAG-schedules having larger Areas execute DAGs faster on platforms that are dynamically heterogeneous: the platform's processors change power and availability status in unpredictable ways and at unpredictable times. (Clouds and desktop grids exemplify such platforms.) While Area-maximal schedules can provably be found for every DAG, efficient generators of such schedules are known only for families of well-structured DAGs. Our first result shows that the problem of crafting Area-maximal schedules for general DAGs is NP-complete, hence likely computationally intractable. The lack of efficient Area-maximizing schedulers for general DAGs has instigated the development of several heuristics for producing DAG-schedules that have large Areas. We propose a novel polynomial-time heuristic that produces schedules having quite large Areas; the heuristic is based on the Sidney decomposition of a DAG. (1) Simulations on DAGs having random structure yield the following results. The Sidney heuristic produces schedules whose Areas: (a) are at least 85% of maximal; (b) are at least 1.25 times greater than previously known heuristics. (2) Simulations on DAGs having the structure of random "LEGO R " DAGs (as formulated in earlier studies) indicate that the schedules produced by the Sidney heuristic have Areas that are at least 1.5 times greater than previously known heuristics. The "85%" result is obtained from formulating the Areamaximization problem as a Linear Program (LP); the Areas of DAG-schedules produced by the Sidney heuristic are at least 85% of the Area-value produced by the (unrounded) LP. (3) The reported results on random DAGs are essentially matched by a second heuristic, which produces DAG-schedules by rounding the results of the LP formulation.

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A Fast Parametric Maximum Flow Algorithm and Applications

Cited by 533 publications

References 33 publications

Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems

On constructing DAG‐schedules with large areas

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