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About 10,000 papers for “maximum flow”

Maximum Flow

2008

Imagine that you are a courier service, and you want to deliver some cargo from one city to another. You can deliver them using various flights from cities to cities, but each flight has a limited amount of space that you can use. An important question is, how much of our cargo can be shipped to the destination using the different flights available? To answer this question, we explore what is called a network flow graph, and show how we can model different problems using such a graph. A network flow graph G=(V,E) is a directed graph with two special vertices: the source vertex s, and the sink (destination) vertex t. Each vertex represents a city where we can send or receive cargo. An edge (u,v) in the graph means that there is a flight that flies directly from u to v. Each edge has an associated capacity, always finite, representing the amount of space available on this flight. For simplicity, we assume there can only be one edge (u,v) for vertices u and v, but we do allow reverse edges (v,u). With this graph, we now want to know how much cargo we can ship from s to t. Since the cargo "flows " through the graph from s to t, we call this the maximum flow problem. A straightforward solution is to do the following: keep finding paths from s to t where we can send flow along, send as much flow as possible along each path, and update the flow graph afterwards to account for the used space. The following shows an arbitrary selection of a path on the above graph.

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[Maximum flow meters].

Jordi Giner, M Calpena, A Blanc · PubMed · 1998

The use of maximum fluid measuring instruments is a simple technique which enables one to monitor the state of air passages in patients suffering from respiratory diseases, and from asthma in particular. The use of these instruments, although simple, requires a special technique in order to evaluate changes observed over a long period of time. Graphing the numerical values obtained through observation aids in the interpretation of these values. These measurements can become a useful factor in the diagnosis, classification and treatment of asthmatic patients. Proper maintenance of these instruments is of vital importance so that they function correctly. All of these factors come into play to contribute to making maximum fluid measuring instruments become "respiratory thermometers".

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Maximum flow problem

Jelena Piplica · Repository of Faculty of Science · 2023

Dijkstra algoritam i Bellman-Ford algoritam su algoritmi koji se koriste za pronalaženje najkraćeg puta u težinskom grafu. Dijkstra algoritam djeluje na pozitivnim težinama bridova i koristi se za pronalaženje najkraćeg puta od jednog početnog vrha do svih ostalih vrhova u grafu. Bellman-Ford algoritam, s druge strane, može se koristiti čak i kada postoje negativne težine bridova, ali ima složenost vremena veću od Dijkstra algoritma. Ford-Fulkerson algoritam koristi se za pronalaženje maksimalnog mrežnog protoka, a temelji se na uvećavajućim putevima. Ovaj algoritam iterativno pronalazi uvećavajuće puteve i uvećava protok dok ne postigne maksimalni protok. Ovi algoritmi su važan alat za rješavanje različitih problema u području grafova i optimizacije.

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Maximum Flow Problem

Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin · Encyclopedia of Optimization · 2001

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Maximum Flow Algorithms

Zeynep Sargut, Ravindra K. Ahuja · Wiley Encyclopedia of Operations Research and Management Science · 2011

Abstract In this article, we will explain the maximum flow problem and the solution algorithms classified under augmenting path, blocking flow, preflow‐push, and other algorithms. For the first three classes, we first present the generic version of the algorithm and then present some elaborate versions in the literature. We also give other approaches and the more efficient algorithms for unit capacity, bipartite, and planar networks. Although the binary blocking flow algorithm has the best theoretical worst‐case running time, the preflow‐push type algorithms are shown to be the most efficient algorithms in practice.

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Rate maximum flow and average flow

McMahon, Glenda, Shaw, Rod · Loughborough University Research Repository (Loughborough University) · 2019

Rate maximum flow and average flow

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Rate maximum flow and average flow

Chatterton, Ken, Shaw, Rod · Loughborough University Research Repository (Loughborough University) · 2025

Rate maximum flow and average flow

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The maximum-flow problem

Gaja Velkavrh · 2012

Maximum flow problem is one of the fundamental problems in network flow theory and has been extensively investigated. In this work we describe the development of efficient algorithms for the maximum flow problem and the current situation in this area. For each algorithm, we describe the main idea, which is used to achieve a better time complexity and the relationship with previous algorithms. We did not describe algorithms in more detail, because some, especially more recent, are very complicated. Therefore, we mentioned literature where algorithms are described in more detail. The problem was first defined by Dantzig (1951) and solved by Ford and Fulkerson (1956), using the augmenting paths method. Since then has been an increasing number of algorithms. Edmonds and Karp (1968) showed that the algorithm of Ford - Fulkerson has time complexity O(nm^2), if the flow is increased firstly on shortest paths. Independently Dinitz (1970) demonstrated the concept of networks with the shortest paths - multi-layered networks - and reached the time complexity O(n^2 m). This limit was improved by Karzanov (1974), who introduced the concept of preflows in multi-layer networks. This time complexity was later improved by Goldberg and Tarjan (1986). Their algorithm sends preflow from the nodes that are closest to the sink. Information about the distance is better than a multi-layered network, because it is simpler to understand and easier to use in parallel algorithms. By using excess scaling Orlin and Ahuja (1988) improved Goldberg - Tarjan's algorithm. With progressive combination of blocking backflow and preflow Waisse (1992) showed the fastest algorithm for the maximum flow problem. With storage of the list of neighbor nodes heights for each node Pham, Lavallee, Bui and Do (2005) improved Goldberg - Tarjan's algorithm. Currently the fastest approximation algorithm is envisioned by Christiano, Kelner, Mandry and Spielman (2010).

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Preprocessing Maximum Flow Algorithms

Liers, Frauke, Pardella, Gregor · Computer Science Publication Server (University of Cologne) · 2010

Maximum-flow problems occur in a wide range of applications. Although already well-studied, they are still an area of active research. The fastest available implementations for determining maximum flows in graphs are either based on augmenting-path or on push-relabel algorithms. In this work, we present two ingredients that, appropriately used, can considerably speed up these methods. On the one hand, we present flow-conserving conditions under which subgraphs can be contracted to a single node. These rules are in the same spirit as presented by Padberg and Rinaldi (Math. Programming (47), 1990) for the minimum cut problem in graphs. On the other hand, we propose a two-step max-flow algorithm for solving the problem on instances coming from physics and computer vision. In the two-step algorithm flow is first sent along augmenting paths of restricted lengths only. Starting from this flow, the problem is then solved to optimality using some known max-flow methods. By extensive experiments on random instances and on instances coming from applications in theoretical physics and in computer vision, we show that a suitable combination of the proposed techniques speeds up traditionally used methods.

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Maximum Flow Efficiency Hypothesis

Vijay P. Singh · Cambridge University Press eBooks · 2022

Design of a stable alluvial channel is based on the hypothesis that the equilibrium state of a channel corresponds to maximum flow. The channel design can then be accomplished by employing the continuity equation, resistance law, sediment transport equation, and the channel cross-section shape. This chapter derives the channel hydraulic geometry for primarily three cross-sections, namely trapezoidal, rectangular, and triangular.

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Maximum Flow and Applications

Sandeep Sen, Amit Kumar · Cambridge University Press eBooks · 2019

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Maximum Network Flow Algorithms

Mohan Chandra Adhikari, Umila Pyakurel · Journal of Advanced College of Engineering and Management · 2021

The aim of the maximum network flow problem is to push as much flow as possible between two special vertices, the source and the sink satisfying the capacity constraints. For the solution of the maximum flow problem, there exists a number of algorithms. The existing algorithms can be divided into two families. First, augmenting path algorithms that satisfy the conservation constraints at intermediate vertices and the second preflow push relabel algorithms that violates the conservation constraints at the intermediate vertices resulting incoming flow more than outgoing flow.In this paper, we study different algorithms that determine the maximum flow in the static and dynamic networks.

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