[601] | 1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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| 2 | * |
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| 3 | * This file is a part of LEMON, a generic C++ optimization library. |
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| 4 | * |
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| 5 | * Copyright (C) 2003-2009 |
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| 6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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| 7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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| 8 | * |
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| 9 | * Permission to use, modify and distribute this software is granted |
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| 10 | * provided that this copyright notice appears in all copies. For |
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| 11 | * precise terms see the accompanying LICENSE file. |
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| 12 | * |
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| 13 | * This software is provided "AS IS" with no warranty of any kind, |
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| 14 | * express or implied, and with no claim as to its suitability for any |
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| 15 | * purpose. |
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| 16 | * |
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| 17 | */ |
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| 18 | |
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| 19 | #ifndef LEMON_NETWORK_SIMPLEX_H |
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| 20 | #define LEMON_NETWORK_SIMPLEX_H |
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| 21 | |
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| 22 | /// \ingroup min_cost_flow |
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| 23 | /// |
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| 24 | /// \file |
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[605] | 25 | /// \brief Network Simplex algorithm for finding a minimum cost flow. |
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[601] | 26 | |
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| 27 | #include <vector> |
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| 28 | #include <limits> |
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| 29 | #include <algorithm> |
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| 30 | |
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[603] | 31 | #include <lemon/core.h> |
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[601] | 32 | #include <lemon/math.h> |
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| 33 | |
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| 34 | namespace lemon { |
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| 35 | |
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| 36 | /// \addtogroup min_cost_flow |
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| 37 | /// @{ |
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| 38 | |
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[605] | 39 | /// \brief Implementation of the primal Network Simplex algorithm |
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[601] | 40 | /// for finding a \ref min_cost_flow "minimum cost flow". |
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| 41 | /// |
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[605] | 42 | /// \ref NetworkSimplex implements the primal Network Simplex algorithm |
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[601] | 43 | /// for finding a \ref min_cost_flow "minimum cost flow". |
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[606] | 44 | /// This algorithm is a specialized version of the linear programming |
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| 45 | /// simplex method directly for the minimum cost flow problem. |
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| 46 | /// It is one of the most efficient solution methods. |
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| 47 | /// |
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| 48 | /// In general this class is the fastest implementation available |
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| 49 | /// in LEMON for the minimum cost flow problem. |
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[640] | 50 | /// Moreover it supports both directions of the supply/demand inequality |
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| 51 | /// constraints. For more information see \ref SupplyType. |
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| 52 | /// |
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| 53 | /// Most of the parameters of the problem (except for the digraph) |
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| 54 | /// can be given using separate functions, and the algorithm can be |
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| 55 | /// executed using the \ref run() function. If some parameters are not |
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| 56 | /// specified, then default values will be used. |
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[601] | 57 | /// |
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[605] | 58 | /// \tparam GR The digraph type the algorithm runs on. |
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[641] | 59 | /// \tparam V The value type used for flow amounts, capacity bounds |
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[607] | 60 | /// and supply values in the algorithm. By default it is \c int. |
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| 61 | /// \tparam C The value type used for costs and potentials in the |
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[641] | 62 | /// algorithm. By default it is the same as \c V. |
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[601] | 63 | /// |
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[608] | 64 | /// \warning Both value types must be signed and all input data must |
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| 65 | /// be integer. |
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[601] | 66 | /// |
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[605] | 67 | /// \note %NetworkSimplex provides five different pivot rule |
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[609] | 68 | /// implementations, from which the most efficient one is used |
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| 69 | /// by default. For more information see \ref PivotRule. |
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[641] | 70 | template <typename GR, typename V = int, typename C = V> |
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[601] | 71 | class NetworkSimplex |
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| 72 | { |
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[605] | 73 | public: |
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[601] | 74 | |
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[642] | 75 | /// The type of the flow amounts, capacity bounds and supply values |
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[641] | 76 | typedef V Value; |
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[642] | 77 | /// The type of the arc costs |
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[607] | 78 | typedef C Cost; |
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[605] | 79 | |
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| 80 | public: |
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| 81 | |
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[640] | 82 | /// \brief Problem type constants for the \c run() function. |
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[605] | 83 | /// |
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[640] | 84 | /// Enum type containing the problem type constants that can be |
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| 85 | /// returned by the \ref run() function of the algorithm. |
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| 86 | enum ProblemType { |
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| 87 | /// The problem has no feasible solution (flow). |
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| 88 | INFEASIBLE, |
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| 89 | /// The problem has optimal solution (i.e. it is feasible and |
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| 90 | /// bounded), and the algorithm has found optimal flow and node |
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| 91 | /// potentials (primal and dual solutions). |
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| 92 | OPTIMAL, |
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| 93 | /// The objective function of the problem is unbounded, i.e. |
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| 94 | /// there is a directed cycle having negative total cost and |
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| 95 | /// infinite upper bound. |
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| 96 | UNBOUNDED |
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| 97 | }; |
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| 98 | |
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| 99 | /// \brief Constants for selecting the type of the supply constraints. |
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| 100 | /// |
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| 101 | /// Enum type containing constants for selecting the supply type, |
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| 102 | /// i.e. the direction of the inequalities in the supply/demand |
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| 103 | /// constraints of the \ref min_cost_flow "minimum cost flow problem". |
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| 104 | /// |
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| 105 | /// The default supply type is \c GEQ, since this form is supported |
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| 106 | /// by other minimum cost flow algorithms and the \ref Circulation |
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| 107 | /// algorithm, as well. |
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| 108 | /// The \c LEQ problem type can be selected using the \ref supplyType() |
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[605] | 109 | /// function. |
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| 110 | /// |
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[640] | 111 | /// Note that the equality form is a special case of both supply types. |
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| 112 | enum SupplyType { |
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| 113 | |
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| 114 | /// This option means that there are <em>"greater or equal"</em> |
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| 115 | /// supply/demand constraints in the definition, i.e. the exact |
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| 116 | /// formulation of the problem is the following. |
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| 117 | /** |
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| 118 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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| 119 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq |
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| 120 | sup(u) \quad \forall u\in V \f] |
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| 121 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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| 122 | */ |
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| 123 | /// It means that the total demand must be greater or equal to the |
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| 124 | /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or |
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| 125 | /// negative) and all the supplies have to be carried out from |
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| 126 | /// the supply nodes, but there could be demands that are not |
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| 127 | /// satisfied. |
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| 128 | GEQ, |
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| 129 | /// It is just an alias for the \c GEQ option. |
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| 130 | CARRY_SUPPLIES = GEQ, |
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| 131 | |
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| 132 | /// This option means that there are <em>"less or equal"</em> |
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| 133 | /// supply/demand constraints in the definition, i.e. the exact |
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| 134 | /// formulation of the problem is the following. |
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| 135 | /** |
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| 136 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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| 137 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq |
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| 138 | sup(u) \quad \forall u\in V \f] |
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| 139 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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| 140 | */ |
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| 141 | /// It means that the total demand must be less or equal to the |
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| 142 | /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or |
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| 143 | /// positive) and all the demands have to be satisfied, but there |
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| 144 | /// could be supplies that are not carried out from the supply |
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| 145 | /// nodes. |
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| 146 | LEQ, |
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| 147 | /// It is just an alias for the \c LEQ option. |
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| 148 | SATISFY_DEMANDS = LEQ |
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| 149 | }; |
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| 150 | |
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| 151 | /// \brief Constants for selecting the pivot rule. |
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| 152 | /// |
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| 153 | /// Enum type containing constants for selecting the pivot rule for |
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| 154 | /// the \ref run() function. |
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| 155 | /// |
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[605] | 156 | /// \ref NetworkSimplex provides five different pivot rule |
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| 157 | /// implementations that significantly affect the running time |
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| 158 | /// of the algorithm. |
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| 159 | /// By default \ref BLOCK_SEARCH "Block Search" is used, which |
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| 160 | /// proved to be the most efficient and the most robust on various |
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| 161 | /// test inputs according to our benchmark tests. |
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| 162 | /// However another pivot rule can be selected using the \ref run() |
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| 163 | /// function with the proper parameter. |
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| 164 | enum PivotRule { |
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| 165 | |
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| 166 | /// The First Eligible pivot rule. |
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| 167 | /// The next eligible arc is selected in a wraparound fashion |
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| 168 | /// in every iteration. |
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| 169 | FIRST_ELIGIBLE, |
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| 170 | |
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| 171 | /// The Best Eligible pivot rule. |
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| 172 | /// The best eligible arc is selected in every iteration. |
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| 173 | BEST_ELIGIBLE, |
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| 174 | |
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| 175 | /// The Block Search pivot rule. |
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| 176 | /// A specified number of arcs are examined in every iteration |
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| 177 | /// in a wraparound fashion and the best eligible arc is selected |
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| 178 | /// from this block. |
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| 179 | BLOCK_SEARCH, |
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| 180 | |
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| 181 | /// The Candidate List pivot rule. |
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| 182 | /// In a major iteration a candidate list is built from eligible arcs |
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| 183 | /// in a wraparound fashion and in the following minor iterations |
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| 184 | /// the best eligible arc is selected from this list. |
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| 185 | CANDIDATE_LIST, |
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| 186 | |
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| 187 | /// The Altering Candidate List pivot rule. |
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| 188 | /// It is a modified version of the Candidate List method. |
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| 189 | /// It keeps only the several best eligible arcs from the former |
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| 190 | /// candidate list and extends this list in every iteration. |
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| 191 | ALTERING_LIST |
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| 192 | }; |
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[609] | 193 | |
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[605] | 194 | private: |
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| 195 | |
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| 196 | TEMPLATE_DIGRAPH_TYPEDEFS(GR); |
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| 197 | |
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[601] | 198 | typedef std::vector<Arc> ArcVector; |
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| 199 | typedef std::vector<Node> NodeVector; |
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| 200 | typedef std::vector<int> IntVector; |
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| 201 | typedef std::vector<bool> BoolVector; |
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[642] | 202 | typedef std::vector<Value> ValueVector; |
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[607] | 203 | typedef std::vector<Cost> CostVector; |
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[601] | 204 | |
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| 205 | // State constants for arcs |
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| 206 | enum ArcStateEnum { |
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| 207 | STATE_UPPER = -1, |
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| 208 | STATE_TREE = 0, |
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| 209 | STATE_LOWER = 1 |
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| 210 | }; |
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| 211 | |
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| 212 | private: |
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| 213 | |
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[605] | 214 | // Data related to the underlying digraph |
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| 215 | const GR &_graph; |
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| 216 | int _node_num; |
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| 217 | int _arc_num; |
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| 218 | |
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| 219 | // Parameters of the problem |
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[642] | 220 | bool _have_lower; |
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[640] | 221 | SupplyType _stype; |
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[641] | 222 | Value _sum_supply; |
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[601] | 223 | |
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[605] | 224 | // Data structures for storing the digraph |
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[603] | 225 | IntNodeMap _node_id; |
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[642] | 226 | IntArcMap _arc_id; |
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[603] | 227 | IntVector _source; |
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| 228 | IntVector _target; |
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| 229 | |
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[605] | 230 | // Node and arc data |
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[642] | 231 | ValueVector _lower; |
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| 232 | ValueVector _upper; |
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| 233 | ValueVector _cap; |
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[607] | 234 | CostVector _cost; |
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[642] | 235 | ValueVector _supply; |
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| 236 | ValueVector _flow; |
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[607] | 237 | CostVector _pi; |
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[601] | 238 | |
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[603] | 239 | // Data for storing the spanning tree structure |
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[601] | 240 | IntVector _parent; |
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| 241 | IntVector _pred; |
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| 242 | IntVector _thread; |
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[604] | 243 | IntVector _rev_thread; |
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| 244 | IntVector _succ_num; |
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| 245 | IntVector _last_succ; |
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| 246 | IntVector _dirty_revs; |
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[601] | 247 | BoolVector _forward; |
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| 248 | IntVector _state; |
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| 249 | int _root; |
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| 250 | |
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| 251 | // Temporary data used in the current pivot iteration |
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[603] | 252 | int in_arc, join, u_in, v_in, u_out, v_out; |
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| 253 | int first, second, right, last; |
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[601] | 254 | int stem, par_stem, new_stem; |
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[641] | 255 | Value delta; |
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[601] | 256 | |
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[640] | 257 | public: |
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| 258 | |
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| 259 | /// \brief Constant for infinite upper bounds (capacities). |
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| 260 | /// |
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| 261 | /// Constant for infinite upper bounds (capacities). |
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[641] | 262 | /// It is \c std::numeric_limits<Value>::infinity() if available, |
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| 263 | /// \c std::numeric_limits<Value>::max() otherwise. |
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| 264 | const Value INF; |
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[640] | 265 | |
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[601] | 266 | private: |
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| 267 | |
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[605] | 268 | // Implementation of the First Eligible pivot rule |
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[601] | 269 | class FirstEligiblePivotRule |
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| 270 | { |
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| 271 | private: |
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| 272 | |
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| 273 | // References to the NetworkSimplex class |
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| 274 | const IntVector &_source; |
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| 275 | const IntVector &_target; |
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[607] | 276 | const CostVector &_cost; |
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[601] | 277 | const IntVector &_state; |
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[607] | 278 | const CostVector &_pi; |
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[601] | 279 | int &_in_arc; |
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| 280 | int _arc_num; |
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| 281 | |
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| 282 | // Pivot rule data |
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| 283 | int _next_arc; |
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| 284 | |
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| 285 | public: |
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| 286 | |
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[605] | 287 | // Constructor |
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[601] | 288 | FirstEligiblePivotRule(NetworkSimplex &ns) : |
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[603] | 289 | _source(ns._source), _target(ns._target), |
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[601] | 290 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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[603] | 291 | _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0) |
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[601] | 292 | {} |
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| 293 | |
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[605] | 294 | // Find next entering arc |
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[601] | 295 | bool findEnteringArc() { |
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[607] | 296 | Cost c; |
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[601] | 297 | for (int e = _next_arc; e < _arc_num; ++e) { |
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| 298 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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| 299 | if (c < 0) { |
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| 300 | _in_arc = e; |
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| 301 | _next_arc = e + 1; |
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| 302 | return true; |
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| 303 | } |
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| 304 | } |
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| 305 | for (int e = 0; e < _next_arc; ++e) { |
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| 306 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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| 307 | if (c < 0) { |
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| 308 | _in_arc = e; |
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| 309 | _next_arc = e + 1; |
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| 310 | return true; |
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| 311 | } |
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| 312 | } |
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| 313 | return false; |
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| 314 | } |
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| 315 | |
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| 316 | }; //class FirstEligiblePivotRule |
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| 317 | |
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| 318 | |
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[605] | 319 | // Implementation of the Best Eligible pivot rule |
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[601] | 320 | class BestEligiblePivotRule |
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| 321 | { |
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| 322 | private: |
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| 323 | |
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| 324 | // References to the NetworkSimplex class |
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| 325 | const IntVector &_source; |
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| 326 | const IntVector &_target; |
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[607] | 327 | const CostVector &_cost; |
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[601] | 328 | const IntVector &_state; |
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[607] | 329 | const CostVector &_pi; |
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[601] | 330 | int &_in_arc; |
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| 331 | int _arc_num; |
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| 332 | |
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| 333 | public: |
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| 334 | |
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[605] | 335 | // Constructor |
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[601] | 336 | BestEligiblePivotRule(NetworkSimplex &ns) : |
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[603] | 337 | _source(ns._source), _target(ns._target), |
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[601] | 338 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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[603] | 339 | _in_arc(ns.in_arc), _arc_num(ns._arc_num) |
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[601] | 340 | {} |
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| 341 | |
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[605] | 342 | // Find next entering arc |
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[601] | 343 | bool findEnteringArc() { |
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[607] | 344 | Cost c, min = 0; |
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[601] | 345 | for (int e = 0; e < _arc_num; ++e) { |
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| 346 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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| 347 | if (c < min) { |
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| 348 | min = c; |
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| 349 | _in_arc = e; |
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| 350 | } |
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| 351 | } |
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| 352 | return min < 0; |
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| 353 | } |
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| 354 | |
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| 355 | }; //class BestEligiblePivotRule |
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| 356 | |
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| 357 | |
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[605] | 358 | // Implementation of the Block Search pivot rule |
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[601] | 359 | class BlockSearchPivotRule |
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| 360 | { |
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| 361 | private: |
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| 362 | |
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| 363 | // References to the NetworkSimplex class |
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| 364 | const IntVector &_source; |
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| 365 | const IntVector &_target; |
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[607] | 366 | const CostVector &_cost; |
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[601] | 367 | const IntVector &_state; |
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[607] | 368 | const CostVector &_pi; |
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[601] | 369 | int &_in_arc; |
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| 370 | int _arc_num; |
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| 371 | |
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| 372 | // Pivot rule data |
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| 373 | int _block_size; |
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| 374 | int _next_arc; |
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| 375 | |
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| 376 | public: |
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| 377 | |
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[605] | 378 | // Constructor |
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[601] | 379 | BlockSearchPivotRule(NetworkSimplex &ns) : |
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[603] | 380 | _source(ns._source), _target(ns._target), |
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[601] | 381 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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[603] | 382 | _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0) |
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[601] | 383 | { |
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| 384 | // The main parameters of the pivot rule |
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| 385 | const double BLOCK_SIZE_FACTOR = 2.0; |
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| 386 | const int MIN_BLOCK_SIZE = 10; |
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| 387 | |
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[612] | 388 | _block_size = std::max( int(BLOCK_SIZE_FACTOR * |
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| 389 | std::sqrt(double(_arc_num))), |
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[601] | 390 | MIN_BLOCK_SIZE ); |
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| 391 | } |
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| 392 | |
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[605] | 393 | // Find next entering arc |
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[601] | 394 | bool findEnteringArc() { |
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[607] | 395 | Cost c, min = 0; |
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[601] | 396 | int cnt = _block_size; |
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| 397 | int e, min_arc = _next_arc; |
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| 398 | for (e = _next_arc; e < _arc_num; ++e) { |
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| 399 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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| 400 | if (c < min) { |
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| 401 | min = c; |
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| 402 | min_arc = e; |
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| 403 | } |
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| 404 | if (--cnt == 0) { |
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| 405 | if (min < 0) break; |
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| 406 | cnt = _block_size; |
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| 407 | } |
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| 408 | } |
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| 409 | if (min == 0 || cnt > 0) { |
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| 410 | for (e = 0; e < _next_arc; ++e) { |
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| 411 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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| 412 | if (c < min) { |
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| 413 | min = c; |
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| 414 | min_arc = e; |
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| 415 | } |
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| 416 | if (--cnt == 0) { |
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| 417 | if (min < 0) break; |
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| 418 | cnt = _block_size; |
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| 419 | } |
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| 420 | } |
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| 421 | } |
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| 422 | if (min >= 0) return false; |
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| 423 | _in_arc = min_arc; |
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| 424 | _next_arc = e; |
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| 425 | return true; |
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| 426 | } |
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| 427 | |
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| 428 | }; //class BlockSearchPivotRule |
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| 429 | |
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| 430 | |
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[605] | 431 | // Implementation of the Candidate List pivot rule |
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[601] | 432 | class CandidateListPivotRule |
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| 433 | { |
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| 434 | private: |
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| 435 | |
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| 436 | // References to the NetworkSimplex class |
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| 437 | const IntVector &_source; |
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| 438 | const IntVector &_target; |
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[607] | 439 | const CostVector &_cost; |
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[601] | 440 | const IntVector &_state; |
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[607] | 441 | const CostVector &_pi; |
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[601] | 442 | int &_in_arc; |
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| 443 | int _arc_num; |
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| 444 | |
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| 445 | // Pivot rule data |
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| 446 | IntVector _candidates; |
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| 447 | int _list_length, _minor_limit; |
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| 448 | int _curr_length, _minor_count; |
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| 449 | int _next_arc; |
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| 450 | |
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| 451 | public: |
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| 452 | |
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| 453 | /// Constructor |
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| 454 | CandidateListPivotRule(NetworkSimplex &ns) : |
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[603] | 455 | _source(ns._source), _target(ns._target), |
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[601] | 456 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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[603] | 457 | _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0) |
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[601] | 458 | { |
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| 459 | // The main parameters of the pivot rule |
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| 460 | const double LIST_LENGTH_FACTOR = 1.0; |
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| 461 | const int MIN_LIST_LENGTH = 10; |
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| 462 | const double MINOR_LIMIT_FACTOR = 0.1; |
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| 463 | const int MIN_MINOR_LIMIT = 3; |
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| 464 | |
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[612] | 465 | _list_length = std::max( int(LIST_LENGTH_FACTOR * |
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| 466 | std::sqrt(double(_arc_num))), |
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[601] | 467 | MIN_LIST_LENGTH ); |
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| 468 | _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length), |
---|
| 469 | MIN_MINOR_LIMIT ); |
---|
| 470 | _curr_length = _minor_count = 0; |
---|
| 471 | _candidates.resize(_list_length); |
---|
| 472 | } |
---|
| 473 | |
---|
| 474 | /// Find next entering arc |
---|
| 475 | bool findEnteringArc() { |
---|
[607] | 476 | Cost min, c; |
---|
[601] | 477 | int e, min_arc = _next_arc; |
---|
| 478 | if (_curr_length > 0 && _minor_count < _minor_limit) { |
---|
| 479 | // Minor iteration: select the best eligible arc from the |
---|
| 480 | // current candidate list |
---|
| 481 | ++_minor_count; |
---|
| 482 | min = 0; |
---|
| 483 | for (int i = 0; i < _curr_length; ++i) { |
---|
| 484 | e = _candidates[i]; |
---|
| 485 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 486 | if (c < min) { |
---|
| 487 | min = c; |
---|
| 488 | min_arc = e; |
---|
| 489 | } |
---|
| 490 | if (c >= 0) { |
---|
| 491 | _candidates[i--] = _candidates[--_curr_length]; |
---|
| 492 | } |
---|
| 493 | } |
---|
| 494 | if (min < 0) { |
---|
| 495 | _in_arc = min_arc; |
---|
| 496 | return true; |
---|
| 497 | } |
---|
| 498 | } |
---|
| 499 | |
---|
| 500 | // Major iteration: build a new candidate list |
---|
| 501 | min = 0; |
---|
| 502 | _curr_length = 0; |
---|
| 503 | for (e = _next_arc; e < _arc_num; ++e) { |
---|
| 504 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 505 | if (c < 0) { |
---|
| 506 | _candidates[_curr_length++] = e; |
---|
| 507 | if (c < min) { |
---|
| 508 | min = c; |
---|
| 509 | min_arc = e; |
---|
| 510 | } |
---|
| 511 | if (_curr_length == _list_length) break; |
---|
| 512 | } |
---|
| 513 | } |
---|
| 514 | if (_curr_length < _list_length) { |
---|
| 515 | for (e = 0; e < _next_arc; ++e) { |
---|
| 516 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 517 | if (c < 0) { |
---|
| 518 | _candidates[_curr_length++] = e; |
---|
| 519 | if (c < min) { |
---|
| 520 | min = c; |
---|
| 521 | min_arc = e; |
---|
| 522 | } |
---|
| 523 | if (_curr_length == _list_length) break; |
---|
| 524 | } |
---|
| 525 | } |
---|
| 526 | } |
---|
| 527 | if (_curr_length == 0) return false; |
---|
| 528 | _minor_count = 1; |
---|
| 529 | _in_arc = min_arc; |
---|
| 530 | _next_arc = e; |
---|
| 531 | return true; |
---|
| 532 | } |
---|
| 533 | |
---|
| 534 | }; //class CandidateListPivotRule |
---|
| 535 | |
---|
| 536 | |
---|
[605] | 537 | // Implementation of the Altering Candidate List pivot rule |
---|
[601] | 538 | class AlteringListPivotRule |
---|
| 539 | { |
---|
| 540 | private: |
---|
| 541 | |
---|
| 542 | // References to the NetworkSimplex class |
---|
| 543 | const IntVector &_source; |
---|
| 544 | const IntVector &_target; |
---|
[607] | 545 | const CostVector &_cost; |
---|
[601] | 546 | const IntVector &_state; |
---|
[607] | 547 | const CostVector &_pi; |
---|
[601] | 548 | int &_in_arc; |
---|
| 549 | int _arc_num; |
---|
| 550 | |
---|
| 551 | // Pivot rule data |
---|
| 552 | int _block_size, _head_length, _curr_length; |
---|
| 553 | int _next_arc; |
---|
| 554 | IntVector _candidates; |
---|
[607] | 555 | CostVector _cand_cost; |
---|
[601] | 556 | |
---|
| 557 | // Functor class to compare arcs during sort of the candidate list |
---|
| 558 | class SortFunc |
---|
| 559 | { |
---|
| 560 | private: |
---|
[607] | 561 | const CostVector &_map; |
---|
[601] | 562 | public: |
---|
[607] | 563 | SortFunc(const CostVector &map) : _map(map) {} |
---|
[601] | 564 | bool operator()(int left, int right) { |
---|
| 565 | return _map[left] > _map[right]; |
---|
| 566 | } |
---|
| 567 | }; |
---|
| 568 | |
---|
| 569 | SortFunc _sort_func; |
---|
| 570 | |
---|
| 571 | public: |
---|
| 572 | |
---|
[605] | 573 | // Constructor |
---|
[601] | 574 | AlteringListPivotRule(NetworkSimplex &ns) : |
---|
[603] | 575 | _source(ns._source), _target(ns._target), |
---|
[601] | 576 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
---|
[603] | 577 | _in_arc(ns.in_arc), _arc_num(ns._arc_num), |
---|
[601] | 578 | _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost) |
---|
| 579 | { |
---|
| 580 | // The main parameters of the pivot rule |
---|
| 581 | const double BLOCK_SIZE_FACTOR = 1.5; |
---|
| 582 | const int MIN_BLOCK_SIZE = 10; |
---|
| 583 | const double HEAD_LENGTH_FACTOR = 0.1; |
---|
| 584 | const int MIN_HEAD_LENGTH = 3; |
---|
| 585 | |
---|
[612] | 586 | _block_size = std::max( int(BLOCK_SIZE_FACTOR * |
---|
| 587 | std::sqrt(double(_arc_num))), |
---|
[601] | 588 | MIN_BLOCK_SIZE ); |
---|
| 589 | _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size), |
---|
| 590 | MIN_HEAD_LENGTH ); |
---|
| 591 | _candidates.resize(_head_length + _block_size); |
---|
| 592 | _curr_length = 0; |
---|
| 593 | } |
---|
| 594 | |
---|
[605] | 595 | // Find next entering arc |
---|
[601] | 596 | bool findEnteringArc() { |
---|
| 597 | // Check the current candidate list |
---|
| 598 | int e; |
---|
| 599 | for (int i = 0; i < _curr_length; ++i) { |
---|
| 600 | e = _candidates[i]; |
---|
| 601 | _cand_cost[e] = _state[e] * |
---|
| 602 | (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 603 | if (_cand_cost[e] >= 0) { |
---|
| 604 | _candidates[i--] = _candidates[--_curr_length]; |
---|
| 605 | } |
---|
| 606 | } |
---|
| 607 | |
---|
| 608 | // Extend the list |
---|
| 609 | int cnt = _block_size; |
---|
[603] | 610 | int last_arc = 0; |
---|
[601] | 611 | int limit = _head_length; |
---|
| 612 | |
---|
| 613 | for (int e = _next_arc; e < _arc_num; ++e) { |
---|
| 614 | _cand_cost[e] = _state[e] * |
---|
| 615 | (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 616 | if (_cand_cost[e] < 0) { |
---|
| 617 | _candidates[_curr_length++] = e; |
---|
[603] | 618 | last_arc = e; |
---|
[601] | 619 | } |
---|
| 620 | if (--cnt == 0) { |
---|
| 621 | if (_curr_length > limit) break; |
---|
| 622 | limit = 0; |
---|
| 623 | cnt = _block_size; |
---|
| 624 | } |
---|
| 625 | } |
---|
| 626 | if (_curr_length <= limit) { |
---|
| 627 | for (int e = 0; e < _next_arc; ++e) { |
---|
| 628 | _cand_cost[e] = _state[e] * |
---|
| 629 | (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
| 630 | if (_cand_cost[e] < 0) { |
---|
| 631 | _candidates[_curr_length++] = e; |
---|
[603] | 632 | last_arc = e; |
---|
[601] | 633 | } |
---|
| 634 | if (--cnt == 0) { |
---|
| 635 | if (_curr_length > limit) break; |
---|
| 636 | limit = 0; |
---|
| 637 | cnt = _block_size; |
---|
| 638 | } |
---|
| 639 | } |
---|
| 640 | } |
---|
| 641 | if (_curr_length == 0) return false; |
---|
[603] | 642 | _next_arc = last_arc + 1; |
---|
[601] | 643 | |
---|
| 644 | // Make heap of the candidate list (approximating a partial sort) |
---|
| 645 | make_heap( _candidates.begin(), _candidates.begin() + _curr_length, |
---|
| 646 | _sort_func ); |
---|
| 647 | |
---|
| 648 | // Pop the first element of the heap |
---|
| 649 | _in_arc = _candidates[0]; |
---|
| 650 | pop_heap( _candidates.begin(), _candidates.begin() + _curr_length, |
---|
| 651 | _sort_func ); |
---|
| 652 | _curr_length = std::min(_head_length, _curr_length - 1); |
---|
| 653 | return true; |
---|
| 654 | } |
---|
| 655 | |
---|
| 656 | }; //class AlteringListPivotRule |
---|
| 657 | |
---|
| 658 | public: |
---|
| 659 | |
---|
[605] | 660 | /// \brief Constructor. |
---|
[601] | 661 | /// |
---|
[609] | 662 | /// The constructor of the class. |
---|
[601] | 663 | /// |
---|
[603] | 664 | /// \param graph The digraph the algorithm runs on. |
---|
[605] | 665 | NetworkSimplex(const GR& graph) : |
---|
[642] | 666 | _graph(graph), _node_id(graph), _arc_id(graph), |
---|
[641] | 667 | INF(std::numeric_limits<Value>::has_infinity ? |
---|
| 668 | std::numeric_limits<Value>::infinity() : |
---|
| 669 | std::numeric_limits<Value>::max()) |
---|
[605] | 670 | { |
---|
[640] | 671 | // Check the value types |
---|
[641] | 672 | LEMON_ASSERT(std::numeric_limits<Value>::is_signed, |
---|
[640] | 673 | "The flow type of NetworkSimplex must be signed"); |
---|
| 674 | LEMON_ASSERT(std::numeric_limits<Cost>::is_signed, |
---|
| 675 | "The cost type of NetworkSimplex must be signed"); |
---|
[642] | 676 | |
---|
| 677 | // Resize vectors |
---|
| 678 | _node_num = countNodes(_graph); |
---|
| 679 | _arc_num = countArcs(_graph); |
---|
| 680 | int all_node_num = _node_num + 1; |
---|
| 681 | int all_arc_num = _arc_num + _node_num; |
---|
[601] | 682 | |
---|
[642] | 683 | _source.resize(all_arc_num); |
---|
| 684 | _target.resize(all_arc_num); |
---|
| 685 | |
---|
| 686 | _lower.resize(all_arc_num); |
---|
| 687 | _upper.resize(all_arc_num); |
---|
| 688 | _cap.resize(all_arc_num); |
---|
| 689 | _cost.resize(all_arc_num); |
---|
| 690 | _supply.resize(all_node_num); |
---|
| 691 | _flow.resize(all_arc_num); |
---|
| 692 | _pi.resize(all_node_num); |
---|
| 693 | |
---|
| 694 | _parent.resize(all_node_num); |
---|
| 695 | _pred.resize(all_node_num); |
---|
| 696 | _forward.resize(all_node_num); |
---|
| 697 | _thread.resize(all_node_num); |
---|
| 698 | _rev_thread.resize(all_node_num); |
---|
| 699 | _succ_num.resize(all_node_num); |
---|
| 700 | _last_succ.resize(all_node_num); |
---|
| 701 | _state.resize(all_arc_num); |
---|
| 702 | |
---|
| 703 | // Copy the graph (store the arcs in a mixed order) |
---|
| 704 | int i = 0; |
---|
| 705 | for (NodeIt n(_graph); n != INVALID; ++n, ++i) { |
---|
| 706 | _node_id[n] = i; |
---|
| 707 | } |
---|
| 708 | int k = std::max(int(std::sqrt(double(_arc_num))), 10); |
---|
| 709 | i = 0; |
---|
| 710 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
| 711 | _arc_id[a] = i; |
---|
| 712 | _source[i] = _node_id[_graph.source(a)]; |
---|
| 713 | _target[i] = _node_id[_graph.target(a)]; |
---|
| 714 | if ((i += k) >= _arc_num) i = (i % k) + 1; |
---|
| 715 | } |
---|
| 716 | |
---|
| 717 | // Initialize maps |
---|
| 718 | for (int i = 0; i != _node_num; ++i) { |
---|
| 719 | _supply[i] = 0; |
---|
| 720 | } |
---|
| 721 | for (int i = 0; i != _arc_num; ++i) { |
---|
| 722 | _lower[i] = 0; |
---|
| 723 | _upper[i] = INF; |
---|
| 724 | _cost[i] = 1; |
---|
| 725 | } |
---|
| 726 | _have_lower = false; |
---|
| 727 | _stype = GEQ; |
---|
[601] | 728 | } |
---|
| 729 | |
---|
[609] | 730 | /// \name Parameters |
---|
| 731 | /// The parameters of the algorithm can be specified using these |
---|
| 732 | /// functions. |
---|
| 733 | |
---|
| 734 | /// @{ |
---|
| 735 | |
---|
[605] | 736 | /// \brief Set the lower bounds on the arcs. |
---|
| 737 | /// |
---|
| 738 | /// This function sets the lower bounds on the arcs. |
---|
[640] | 739 | /// If it is not used before calling \ref run(), the lower bounds |
---|
| 740 | /// will be set to zero on all arcs. |
---|
[605] | 741 | /// |
---|
| 742 | /// \param map An arc map storing the lower bounds. |
---|
[641] | 743 | /// Its \c Value type must be convertible to the \c Value type |
---|
[605] | 744 | /// of the algorithm. |
---|
| 745 | /// |
---|
| 746 | /// \return <tt>(*this)</tt> |
---|
[640] | 747 | template <typename LowerMap> |
---|
| 748 | NetworkSimplex& lowerMap(const LowerMap& map) { |
---|
[642] | 749 | _have_lower = true; |
---|
[605] | 750 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
[642] | 751 | _lower[_arc_id[a]] = map[a]; |
---|
[605] | 752 | } |
---|
| 753 | return *this; |
---|
| 754 | } |
---|
| 755 | |
---|
| 756 | /// \brief Set the upper bounds (capacities) on the arcs. |
---|
| 757 | /// |
---|
| 758 | /// This function sets the upper bounds (capacities) on the arcs. |
---|
[640] | 759 | /// If it is not used before calling \ref run(), the upper bounds |
---|
| 760 | /// will be set to \ref INF on all arcs (i.e. the flow value will be |
---|
| 761 | /// unbounded from above on each arc). |
---|
[605] | 762 | /// |
---|
| 763 | /// \param map An arc map storing the upper bounds. |
---|
[641] | 764 | /// Its \c Value type must be convertible to the \c Value type |
---|
[605] | 765 | /// of the algorithm. |
---|
| 766 | /// |
---|
| 767 | /// \return <tt>(*this)</tt> |
---|
[640] | 768 | template<typename UpperMap> |
---|
| 769 | NetworkSimplex& upperMap(const UpperMap& map) { |
---|
[605] | 770 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
[642] | 771 | _upper[_arc_id[a]] = map[a]; |
---|
[605] | 772 | } |
---|
| 773 | return *this; |
---|
| 774 | } |
---|
| 775 | |
---|
| 776 | /// \brief Set the costs of the arcs. |
---|
| 777 | /// |
---|
| 778 | /// This function sets the costs of the arcs. |
---|
| 779 | /// If it is not used before calling \ref run(), the costs |
---|
| 780 | /// will be set to \c 1 on all arcs. |
---|
| 781 | /// |
---|
| 782 | /// \param map An arc map storing the costs. |
---|
[607] | 783 | /// Its \c Value type must be convertible to the \c Cost type |
---|
[605] | 784 | /// of the algorithm. |
---|
| 785 | /// |
---|
| 786 | /// \return <tt>(*this)</tt> |
---|
[640] | 787 | template<typename CostMap> |
---|
| 788 | NetworkSimplex& costMap(const CostMap& map) { |
---|
[605] | 789 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
[642] | 790 | _cost[_arc_id[a]] = map[a]; |
---|
[605] | 791 | } |
---|
| 792 | return *this; |
---|
| 793 | } |
---|
| 794 | |
---|
| 795 | /// \brief Set the supply values of the nodes. |
---|
| 796 | /// |
---|
| 797 | /// This function sets the supply values of the nodes. |
---|
| 798 | /// If neither this function nor \ref stSupply() is used before |
---|
| 799 | /// calling \ref run(), the supply of each node will be set to zero. |
---|
| 800 | /// (It makes sense only if non-zero lower bounds are given.) |
---|
| 801 | /// |
---|
| 802 | /// \param map A node map storing the supply values. |
---|
[641] | 803 | /// Its \c Value type must be convertible to the \c Value type |
---|
[605] | 804 | /// of the algorithm. |
---|
| 805 | /// |
---|
| 806 | /// \return <tt>(*this)</tt> |
---|
[640] | 807 | template<typename SupplyMap> |
---|
| 808 | NetworkSimplex& supplyMap(const SupplyMap& map) { |
---|
[605] | 809 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
[642] | 810 | _supply[_node_id[n]] = map[n]; |
---|
[605] | 811 | } |
---|
| 812 | return *this; |
---|
| 813 | } |
---|
| 814 | |
---|
| 815 | /// \brief Set single source and target nodes and a supply value. |
---|
| 816 | /// |
---|
| 817 | /// This function sets a single source node and a single target node |
---|
| 818 | /// and the required flow value. |
---|
| 819 | /// If neither this function nor \ref supplyMap() is used before |
---|
| 820 | /// calling \ref run(), the supply of each node will be set to zero. |
---|
| 821 | /// (It makes sense only if non-zero lower bounds are given.) |
---|
| 822 | /// |
---|
[640] | 823 | /// Using this function has the same effect as using \ref supplyMap() |
---|
| 824 | /// with such a map in which \c k is assigned to \c s, \c -k is |
---|
| 825 | /// assigned to \c t and all other nodes have zero supply value. |
---|
| 826 | /// |
---|
[605] | 827 | /// \param s The source node. |
---|
| 828 | /// \param t The target node. |
---|
| 829 | /// \param k The required amount of flow from node \c s to node \c t |
---|
| 830 | /// (i.e. the supply of \c s and the demand of \c t). |
---|
| 831 | /// |
---|
| 832 | /// \return <tt>(*this)</tt> |
---|
[641] | 833 | NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) { |
---|
[642] | 834 | for (int i = 0; i != _node_num; ++i) { |
---|
| 835 | _supply[i] = 0; |
---|
| 836 | } |
---|
| 837 | _supply[_node_id[s]] = k; |
---|
| 838 | _supply[_node_id[t]] = -k; |
---|
[605] | 839 | return *this; |
---|
| 840 | } |
---|
[609] | 841 | |
---|
[640] | 842 | /// \brief Set the type of the supply constraints. |
---|
[609] | 843 | /// |
---|
[640] | 844 | /// This function sets the type of the supply/demand constraints. |
---|
| 845 | /// If it is not used before calling \ref run(), the \ref GEQ supply |
---|
[609] | 846 | /// type will be used. |
---|
| 847 | /// |
---|
[640] | 848 | /// For more information see \ref SupplyType. |
---|
[609] | 849 | /// |
---|
| 850 | /// \return <tt>(*this)</tt> |
---|
[640] | 851 | NetworkSimplex& supplyType(SupplyType supply_type) { |
---|
| 852 | _stype = supply_type; |
---|
[609] | 853 | return *this; |
---|
| 854 | } |
---|
[605] | 855 | |
---|
[609] | 856 | /// @} |
---|
[601] | 857 | |
---|
[605] | 858 | /// \name Execution Control |
---|
| 859 | /// The algorithm can be executed using \ref run(). |
---|
| 860 | |
---|
[601] | 861 | /// @{ |
---|
| 862 | |
---|
| 863 | /// \brief Run the algorithm. |
---|
| 864 | /// |
---|
| 865 | /// This function runs the algorithm. |
---|
[609] | 866 | /// The paramters can be specified using functions \ref lowerMap(), |
---|
[640] | 867 | /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), |
---|
[642] | 868 | /// \ref supplyType(). |
---|
[609] | 869 | /// For example, |
---|
[605] | 870 | /// \code |
---|
| 871 | /// NetworkSimplex<ListDigraph> ns(graph); |
---|
[640] | 872 | /// ns.lowerMap(lower).upperMap(upper).costMap(cost) |
---|
[605] | 873 | /// .supplyMap(sup).run(); |
---|
| 874 | /// \endcode |
---|
[601] | 875 | /// |
---|
[606] | 876 | /// This function can be called more than once. All the parameters |
---|
| 877 | /// that have been given are kept for the next call, unless |
---|
| 878 | /// \ref reset() is called, thus only the modified parameters |
---|
| 879 | /// have to be set again. See \ref reset() for examples. |
---|
[642] | 880 | /// However the underlying digraph must not be modified after this |
---|
| 881 | /// class have been constructed, since it copies and extends the graph. |
---|
[606] | 882 | /// |
---|
[605] | 883 | /// \param pivot_rule The pivot rule that will be used during the |
---|
| 884 | /// algorithm. For more information see \ref PivotRule. |
---|
[601] | 885 | /// |
---|
[640] | 886 | /// \return \c INFEASIBLE if no feasible flow exists, |
---|
| 887 | /// \n \c OPTIMAL if the problem has optimal solution |
---|
| 888 | /// (i.e. it is feasible and bounded), and the algorithm has found |
---|
| 889 | /// optimal flow and node potentials (primal and dual solutions), |
---|
| 890 | /// \n \c UNBOUNDED if the objective function of the problem is |
---|
| 891 | /// unbounded, i.e. there is a directed cycle having negative total |
---|
| 892 | /// cost and infinite upper bound. |
---|
| 893 | /// |
---|
| 894 | /// \see ProblemType, PivotRule |
---|
| 895 | ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) { |
---|
| 896 | if (!init()) return INFEASIBLE; |
---|
| 897 | return start(pivot_rule); |
---|
[601] | 898 | } |
---|
| 899 | |
---|
[606] | 900 | /// \brief Reset all the parameters that have been given before. |
---|
| 901 | /// |
---|
| 902 | /// This function resets all the paramaters that have been given |
---|
[609] | 903 | /// before using functions \ref lowerMap(), \ref upperMap(), |
---|
[642] | 904 | /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType(). |
---|
[606] | 905 | /// |
---|
| 906 | /// It is useful for multiple run() calls. If this function is not |
---|
| 907 | /// used, all the parameters given before are kept for the next |
---|
| 908 | /// \ref run() call. |
---|
[642] | 909 | /// However the underlying digraph must not be modified after this |
---|
| 910 | /// class have been constructed, since it copies and extends the graph. |
---|
[606] | 911 | /// |
---|
| 912 | /// For example, |
---|
| 913 | /// \code |
---|
| 914 | /// NetworkSimplex<ListDigraph> ns(graph); |
---|
| 915 | /// |
---|
| 916 | /// // First run |
---|
[640] | 917 | /// ns.lowerMap(lower).upperMap(upper).costMap(cost) |
---|
[606] | 918 | /// .supplyMap(sup).run(); |
---|
| 919 | /// |
---|
| 920 | /// // Run again with modified cost map (reset() is not called, |
---|
| 921 | /// // so only the cost map have to be set again) |
---|
| 922 | /// cost[e] += 100; |
---|
| 923 | /// ns.costMap(cost).run(); |
---|
| 924 | /// |
---|
| 925 | /// // Run again from scratch using reset() |
---|
| 926 | /// // (the lower bounds will be set to zero on all arcs) |
---|
| 927 | /// ns.reset(); |
---|
[640] | 928 | /// ns.upperMap(capacity).costMap(cost) |
---|
[606] | 929 | /// .supplyMap(sup).run(); |
---|
| 930 | /// \endcode |
---|
| 931 | /// |
---|
| 932 | /// \return <tt>(*this)</tt> |
---|
| 933 | NetworkSimplex& reset() { |
---|
[642] | 934 | for (int i = 0; i != _node_num; ++i) { |
---|
| 935 | _supply[i] = 0; |
---|
| 936 | } |
---|
| 937 | for (int i = 0; i != _arc_num; ++i) { |
---|
| 938 | _lower[i] = 0; |
---|
| 939 | _upper[i] = INF; |
---|
| 940 | _cost[i] = 1; |
---|
| 941 | } |
---|
| 942 | _have_lower = false; |
---|
[640] | 943 | _stype = GEQ; |
---|
[606] | 944 | return *this; |
---|
| 945 | } |
---|
| 946 | |
---|
[601] | 947 | /// @} |
---|
| 948 | |
---|
| 949 | /// \name Query Functions |
---|
| 950 | /// The results of the algorithm can be obtained using these |
---|
| 951 | /// functions.\n |
---|
[605] | 952 | /// The \ref run() function must be called before using them. |
---|
| 953 | |
---|
[601] | 954 | /// @{ |
---|
| 955 | |
---|
[605] | 956 | /// \brief Return the total cost of the found flow. |
---|
| 957 | /// |
---|
| 958 | /// This function returns the total cost of the found flow. |
---|
[640] | 959 | /// Its complexity is O(e). |
---|
[605] | 960 | /// |
---|
| 961 | /// \note The return type of the function can be specified as a |
---|
| 962 | /// template parameter. For example, |
---|
| 963 | /// \code |
---|
| 964 | /// ns.totalCost<double>(); |
---|
| 965 | /// \endcode |
---|
[607] | 966 | /// It is useful if the total cost cannot be stored in the \c Cost |
---|
[605] | 967 | /// type of the algorithm, which is the default return type of the |
---|
| 968 | /// function. |
---|
| 969 | /// |
---|
| 970 | /// \pre \ref run() must be called before using this function. |
---|
[642] | 971 | template <typename Number> |
---|
| 972 | Number totalCost() const { |
---|
| 973 | Number c = 0; |
---|
| 974 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
| 975 | int i = _arc_id[a]; |
---|
| 976 | c += Number(_flow[i]) * Number(_cost[i]); |
---|
[605] | 977 | } |
---|
| 978 | return c; |
---|
| 979 | } |
---|
| 980 | |
---|
| 981 | #ifndef DOXYGEN |
---|
[607] | 982 | Cost totalCost() const { |
---|
| 983 | return totalCost<Cost>(); |
---|
[605] | 984 | } |
---|
| 985 | #endif |
---|
| 986 | |
---|
| 987 | /// \brief Return the flow on the given arc. |
---|
| 988 | /// |
---|
| 989 | /// This function returns the flow on the given arc. |
---|
| 990 | /// |
---|
| 991 | /// \pre \ref run() must be called before using this function. |
---|
[641] | 992 | Value flow(const Arc& a) const { |
---|
[642] | 993 | return _flow[_arc_id[a]]; |
---|
[605] | 994 | } |
---|
| 995 | |
---|
[642] | 996 | /// \brief Return the flow map (the primal solution). |
---|
[601] | 997 | /// |
---|
[642] | 998 | /// This function copies the flow value on each arc into the given |
---|
| 999 | /// map. The \c Value type of the algorithm must be convertible to |
---|
| 1000 | /// the \c Value type of the map. |
---|
[601] | 1001 | /// |
---|
| 1002 | /// \pre \ref run() must be called before using this function. |
---|
[642] | 1003 | template <typename FlowMap> |
---|
| 1004 | void flowMap(FlowMap &map) const { |
---|
| 1005 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
| 1006 | map.set(a, _flow[_arc_id[a]]); |
---|
| 1007 | } |
---|
[601] | 1008 | } |
---|
| 1009 | |
---|
[605] | 1010 | /// \brief Return the potential (dual value) of the given node. |
---|
| 1011 | /// |
---|
| 1012 | /// This function returns the potential (dual value) of the |
---|
| 1013 | /// given node. |
---|
| 1014 | /// |
---|
| 1015 | /// \pre \ref run() must be called before using this function. |
---|
[607] | 1016 | Cost potential(const Node& n) const { |
---|
[642] | 1017 | return _pi[_node_id[n]]; |
---|
[605] | 1018 | } |
---|
| 1019 | |
---|
[642] | 1020 | /// \brief Return the potential map (the dual solution). |
---|
[601] | 1021 | /// |
---|
[642] | 1022 | /// This function copies the potential (dual value) of each node |
---|
| 1023 | /// into the given map. |
---|
| 1024 | /// The \c Cost type of the algorithm must be convertible to the |
---|
| 1025 | /// \c Value type of the map. |
---|
[601] | 1026 | /// |
---|
| 1027 | /// \pre \ref run() must be called before using this function. |
---|
[642] | 1028 | template <typename PotentialMap> |
---|
| 1029 | void potentialMap(PotentialMap &map) const { |
---|
| 1030 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
| 1031 | map.set(n, _pi[_node_id[n]]); |
---|
| 1032 | } |
---|
[601] | 1033 | } |
---|
| 1034 | |
---|
| 1035 | /// @} |
---|
| 1036 | |
---|
| 1037 | private: |
---|
| 1038 | |
---|
| 1039 | // Initialize internal data structures |
---|
| 1040 | bool init() { |
---|
[605] | 1041 | if (_node_num == 0) return false; |
---|
[601] | 1042 | |
---|
[642] | 1043 | // Check the sum of supply values |
---|
| 1044 | _sum_supply = 0; |
---|
| 1045 | for (int i = 0; i != _node_num; ++i) { |
---|
| 1046 | _sum_supply += _supply[i]; |
---|
| 1047 | } |
---|
[643] | 1048 | if ( !((_stype == GEQ && _sum_supply <= 0) || |
---|
| 1049 | (_stype == LEQ && _sum_supply >= 0)) ) return false; |
---|
[601] | 1050 | |
---|
[642] | 1051 | // Remove non-zero lower bounds |
---|
| 1052 | if (_have_lower) { |
---|
| 1053 | for (int i = 0; i != _arc_num; ++i) { |
---|
| 1054 | Value c = _lower[i]; |
---|
| 1055 | if (c >= 0) { |
---|
| 1056 | _cap[i] = _upper[i] < INF ? _upper[i] - c : INF; |
---|
| 1057 | } else { |
---|
| 1058 | _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF; |
---|
| 1059 | } |
---|
| 1060 | _supply[_source[i]] -= c; |
---|
| 1061 | _supply[_target[i]] += c; |
---|
| 1062 | } |
---|
| 1063 | } else { |
---|
| 1064 | for (int i = 0; i != _arc_num; ++i) { |
---|
| 1065 | _cap[i] = _upper[i]; |
---|
| 1066 | } |
---|
[605] | 1067 | } |
---|
[601] | 1068 | |
---|
[609] | 1069 | // Initialize artifical cost |
---|
[640] | 1070 | Cost ART_COST; |
---|
[609] | 1071 | if (std::numeric_limits<Cost>::is_exact) { |
---|
[640] | 1072 | ART_COST = std::numeric_limits<Cost>::max() / 4 + 1; |
---|
[609] | 1073 | } else { |
---|
[640] | 1074 | ART_COST = std::numeric_limits<Cost>::min(); |
---|
[609] | 1075 | for (int i = 0; i != _arc_num; ++i) { |
---|
[640] | 1076 | if (_cost[i] > ART_COST) ART_COST = _cost[i]; |
---|
[609] | 1077 | } |
---|
[640] | 1078 | ART_COST = (ART_COST + 1) * _node_num; |
---|
[609] | 1079 | } |
---|
| 1080 | |
---|
[642] | 1081 | // Initialize arc maps |
---|
| 1082 | for (int i = 0; i != _arc_num; ++i) { |
---|
| 1083 | _flow[i] = 0; |
---|
| 1084 | _state[i] = STATE_LOWER; |
---|
| 1085 | } |
---|
| 1086 | |
---|
[601] | 1087 | // Set data for the artificial root node |
---|
| 1088 | _root = _node_num; |
---|
| 1089 | _parent[_root] = -1; |
---|
| 1090 | _pred[_root] = -1; |
---|
| 1091 | _thread[_root] = 0; |
---|
[604] | 1092 | _rev_thread[0] = _root; |
---|
[642] | 1093 | _succ_num[_root] = _node_num + 1; |
---|
[604] | 1094 | _last_succ[_root] = _root - 1; |
---|
[640] | 1095 | _supply[_root] = -_sum_supply; |
---|
[642] | 1096 | _pi[_root] = _sum_supply < 0 ? -ART_COST : ART_COST; |
---|
[601] | 1097 | |
---|
| 1098 | // Add artificial arcs and initialize the spanning tree data structure |
---|
| 1099 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
[642] | 1100 | _parent[u] = _root; |
---|
| 1101 | _pred[u] = e; |
---|
[601] | 1102 | _thread[u] = u + 1; |
---|
[604] | 1103 | _rev_thread[u + 1] = u; |
---|
| 1104 | _succ_num[u] = 1; |
---|
| 1105 | _last_succ[u] = u; |
---|
[640] | 1106 | _cost[e] = ART_COST; |
---|
| 1107 | _cap[e] = INF; |
---|
[606] | 1108 | _state[e] = STATE_TREE; |
---|
[640] | 1109 | if (_supply[u] > 0 || (_supply[u] == 0 && _sum_supply <= 0)) { |
---|
[601] | 1110 | _flow[e] = _supply[u]; |
---|
| 1111 | _forward[u] = true; |
---|
[640] | 1112 | _pi[u] = -ART_COST + _pi[_root]; |
---|
[601] | 1113 | } else { |
---|
| 1114 | _flow[e] = -_supply[u]; |
---|
| 1115 | _forward[u] = false; |
---|
[640] | 1116 | _pi[u] = ART_COST + _pi[_root]; |
---|
[601] | 1117 | } |
---|
| 1118 | } |
---|
| 1119 | |
---|
| 1120 | return true; |
---|
| 1121 | } |
---|
| 1122 | |
---|
| 1123 | // Find the join node |
---|
| 1124 | void findJoinNode() { |
---|
[603] | 1125 | int u = _source[in_arc]; |
---|
| 1126 | int v = _target[in_arc]; |
---|
[601] | 1127 | while (u != v) { |
---|
[604] | 1128 | if (_succ_num[u] < _succ_num[v]) { |
---|
| 1129 | u = _parent[u]; |
---|
| 1130 | } else { |
---|
| 1131 | v = _parent[v]; |
---|
| 1132 | } |
---|
[601] | 1133 | } |
---|
| 1134 | join = u; |
---|
| 1135 | } |
---|
| 1136 | |
---|
| 1137 | // Find the leaving arc of the cycle and returns true if the |
---|
| 1138 | // leaving arc is not the same as the entering arc |
---|
| 1139 | bool findLeavingArc() { |
---|
| 1140 | // Initialize first and second nodes according to the direction |
---|
| 1141 | // of the cycle |
---|
[603] | 1142 | if (_state[in_arc] == STATE_LOWER) { |
---|
| 1143 | first = _source[in_arc]; |
---|
| 1144 | second = _target[in_arc]; |
---|
[601] | 1145 | } else { |
---|
[603] | 1146 | first = _target[in_arc]; |
---|
| 1147 | second = _source[in_arc]; |
---|
[601] | 1148 | } |
---|
[603] | 1149 | delta = _cap[in_arc]; |
---|
[601] | 1150 | int result = 0; |
---|
[641] | 1151 | Value d; |
---|
[601] | 1152 | int e; |
---|
| 1153 | |
---|
| 1154 | // Search the cycle along the path form the first node to the root |
---|
| 1155 | for (int u = first; u != join; u = _parent[u]) { |
---|
| 1156 | e = _pred[u]; |
---|
[640] | 1157 | d = _forward[u] ? |
---|
| 1158 | _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]); |
---|
[601] | 1159 | if (d < delta) { |
---|
| 1160 | delta = d; |
---|
| 1161 | u_out = u; |
---|
| 1162 | result = 1; |
---|
| 1163 | } |
---|
| 1164 | } |
---|
| 1165 | // Search the cycle along the path form the second node to the root |
---|
| 1166 | for (int u = second; u != join; u = _parent[u]) { |
---|
| 1167 | e = _pred[u]; |
---|
[640] | 1168 | d = _forward[u] ? |
---|
| 1169 | (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e]; |
---|
[601] | 1170 | if (d <= delta) { |
---|
| 1171 | delta = d; |
---|
| 1172 | u_out = u; |
---|
| 1173 | result = 2; |
---|
| 1174 | } |
---|
| 1175 | } |
---|
| 1176 | |
---|
| 1177 | if (result == 1) { |
---|
| 1178 | u_in = first; |
---|
| 1179 | v_in = second; |
---|
| 1180 | } else { |
---|
| 1181 | u_in = second; |
---|
| 1182 | v_in = first; |
---|
| 1183 | } |
---|
| 1184 | return result != 0; |
---|
| 1185 | } |
---|
| 1186 | |
---|
| 1187 | // Change _flow and _state vectors |
---|
| 1188 | void changeFlow(bool change) { |
---|
| 1189 | // Augment along the cycle |
---|
| 1190 | if (delta > 0) { |
---|
[641] | 1191 | Value val = _state[in_arc] * delta; |
---|
[603] | 1192 | _flow[in_arc] += val; |
---|
| 1193 | for (int u = _source[in_arc]; u != join; u = _parent[u]) { |
---|
[601] | 1194 | _flow[_pred[u]] += _forward[u] ? -val : val; |
---|
| 1195 | } |
---|
[603] | 1196 | for (int u = _target[in_arc]; u != join; u = _parent[u]) { |
---|
[601] | 1197 | _flow[_pred[u]] += _forward[u] ? val : -val; |
---|
| 1198 | } |
---|
| 1199 | } |
---|
| 1200 | // Update the state of the entering and leaving arcs |
---|
| 1201 | if (change) { |
---|
[603] | 1202 | _state[in_arc] = STATE_TREE; |
---|
[601] | 1203 | _state[_pred[u_out]] = |
---|
| 1204 | (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER; |
---|
| 1205 | } else { |
---|
[603] | 1206 | _state[in_arc] = -_state[in_arc]; |
---|
[601] | 1207 | } |
---|
| 1208 | } |
---|
| 1209 | |
---|
[604] | 1210 | // Update the tree structure |
---|
| 1211 | void updateTreeStructure() { |
---|
| 1212 | int u, w; |
---|
| 1213 | int old_rev_thread = _rev_thread[u_out]; |
---|
| 1214 | int old_succ_num = _succ_num[u_out]; |
---|
| 1215 | int old_last_succ = _last_succ[u_out]; |
---|
[601] | 1216 | v_out = _parent[u_out]; |
---|
| 1217 | |
---|
[604] | 1218 | u = _last_succ[u_in]; // the last successor of u_in |
---|
| 1219 | right = _thread[u]; // the node after it |
---|
| 1220 | |
---|
| 1221 | // Handle the case when old_rev_thread equals to v_in |
---|
| 1222 | // (it also means that join and v_out coincide) |
---|
| 1223 | if (old_rev_thread == v_in) { |
---|
| 1224 | last = _thread[_last_succ[u_out]]; |
---|
| 1225 | } else { |
---|
| 1226 | last = _thread[v_in]; |
---|
[601] | 1227 | } |
---|
| 1228 | |
---|
[604] | 1229 | // Update _thread and _parent along the stem nodes (i.e. the nodes |
---|
| 1230 | // between u_in and u_out, whose parent have to be changed) |
---|
[601] | 1231 | _thread[v_in] = stem = u_in; |
---|
[604] | 1232 | _dirty_revs.clear(); |
---|
| 1233 | _dirty_revs.push_back(v_in); |
---|
[601] | 1234 | par_stem = v_in; |
---|
| 1235 | while (stem != u_out) { |
---|
[604] | 1236 | // Insert the next stem node into the thread list |
---|
| 1237 | new_stem = _parent[stem]; |
---|
| 1238 | _thread[u] = new_stem; |
---|
| 1239 | _dirty_revs.push_back(u); |
---|
[601] | 1240 | |
---|
[604] | 1241 | // Remove the subtree of stem from the thread list |
---|
| 1242 | w = _rev_thread[stem]; |
---|
| 1243 | _thread[w] = right; |
---|
| 1244 | _rev_thread[right] = w; |
---|
[601] | 1245 | |
---|
[604] | 1246 | // Change the parent node and shift stem nodes |
---|
[601] | 1247 | _parent[stem] = par_stem; |
---|
| 1248 | par_stem = stem; |
---|
| 1249 | stem = new_stem; |
---|
| 1250 | |
---|
[604] | 1251 | // Update u and right |
---|
| 1252 | u = _last_succ[stem] == _last_succ[par_stem] ? |
---|
| 1253 | _rev_thread[par_stem] : _last_succ[stem]; |
---|
[601] | 1254 | right = _thread[u]; |
---|
| 1255 | } |
---|
| 1256 | _parent[u_out] = par_stem; |
---|
| 1257 | _thread[u] = last; |
---|
[604] | 1258 | _rev_thread[last] = u; |
---|
| 1259 | _last_succ[u_out] = u; |
---|
[601] | 1260 | |
---|
[604] | 1261 | // Remove the subtree of u_out from the thread list except for |
---|
| 1262 | // the case when old_rev_thread equals to v_in |
---|
| 1263 | // (it also means that join and v_out coincide) |
---|
| 1264 | if (old_rev_thread != v_in) { |
---|
| 1265 | _thread[old_rev_thread] = right; |
---|
| 1266 | _rev_thread[right] = old_rev_thread; |
---|
| 1267 | } |
---|
| 1268 | |
---|
| 1269 | // Update _rev_thread using the new _thread values |
---|
| 1270 | for (int i = 0; i < int(_dirty_revs.size()); ++i) { |
---|
| 1271 | u = _dirty_revs[i]; |
---|
| 1272 | _rev_thread[_thread[u]] = u; |
---|
| 1273 | } |
---|
| 1274 | |
---|
| 1275 | // Update _pred, _forward, _last_succ and _succ_num for the |
---|
| 1276 | // stem nodes from u_out to u_in |
---|
| 1277 | int tmp_sc = 0, tmp_ls = _last_succ[u_out]; |
---|
| 1278 | u = u_out; |
---|
| 1279 | while (u != u_in) { |
---|
| 1280 | w = _parent[u]; |
---|
| 1281 | _pred[u] = _pred[w]; |
---|
| 1282 | _forward[u] = !_forward[w]; |
---|
| 1283 | tmp_sc += _succ_num[u] - _succ_num[w]; |
---|
| 1284 | _succ_num[u] = tmp_sc; |
---|
| 1285 | _last_succ[w] = tmp_ls; |
---|
| 1286 | u = w; |
---|
| 1287 | } |
---|
| 1288 | _pred[u_in] = in_arc; |
---|
| 1289 | _forward[u_in] = (u_in == _source[in_arc]); |
---|
| 1290 | _succ_num[u_in] = old_succ_num; |
---|
| 1291 | |
---|
| 1292 | // Set limits for updating _last_succ form v_in and v_out |
---|
| 1293 | // towards the root |
---|
| 1294 | int up_limit_in = -1; |
---|
| 1295 | int up_limit_out = -1; |
---|
| 1296 | if (_last_succ[join] == v_in) { |
---|
| 1297 | up_limit_out = join; |
---|
[601] | 1298 | } else { |
---|
[604] | 1299 | up_limit_in = join; |
---|
| 1300 | } |
---|
| 1301 | |
---|
| 1302 | // Update _last_succ from v_in towards the root |
---|
| 1303 | for (u = v_in; u != up_limit_in && _last_succ[u] == v_in; |
---|
| 1304 | u = _parent[u]) { |
---|
| 1305 | _last_succ[u] = _last_succ[u_out]; |
---|
| 1306 | } |
---|
| 1307 | // Update _last_succ from v_out towards the root |
---|
| 1308 | if (join != old_rev_thread && v_in != old_rev_thread) { |
---|
| 1309 | for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; |
---|
| 1310 | u = _parent[u]) { |
---|
| 1311 | _last_succ[u] = old_rev_thread; |
---|
| 1312 | } |
---|
| 1313 | } else { |
---|
| 1314 | for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; |
---|
| 1315 | u = _parent[u]) { |
---|
| 1316 | _last_succ[u] = _last_succ[u_out]; |
---|
| 1317 | } |
---|
| 1318 | } |
---|
| 1319 | |
---|
| 1320 | // Update _succ_num from v_in to join |
---|
| 1321 | for (u = v_in; u != join; u = _parent[u]) { |
---|
| 1322 | _succ_num[u] += old_succ_num; |
---|
| 1323 | } |
---|
| 1324 | // Update _succ_num from v_out to join |
---|
| 1325 | for (u = v_out; u != join; u = _parent[u]) { |
---|
| 1326 | _succ_num[u] -= old_succ_num; |
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[601] | 1327 | } |
---|
| 1328 | } |
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| 1329 | |
---|
[604] | 1330 | // Update potentials |
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| 1331 | void updatePotential() { |
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[607] | 1332 | Cost sigma = _forward[u_in] ? |
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[601] | 1333 | _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] : |
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| 1334 | _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]]; |
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[608] | 1335 | // Update potentials in the subtree, which has been moved |
---|
| 1336 | int end = _thread[_last_succ[u_in]]; |
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| 1337 | for (int u = u_in; u != end; u = _thread[u]) { |
---|
| 1338 | _pi[u] += sigma; |
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[601] | 1339 | } |
---|
| 1340 | } |
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| 1341 | |
---|
| 1342 | // Execute the algorithm |
---|
[640] | 1343 | ProblemType start(PivotRule pivot_rule) { |
---|
[601] | 1344 | // Select the pivot rule implementation |
---|
| 1345 | switch (pivot_rule) { |
---|
[605] | 1346 | case FIRST_ELIGIBLE: |
---|
[601] | 1347 | return start<FirstEligiblePivotRule>(); |
---|
[605] | 1348 | case BEST_ELIGIBLE: |
---|
[601] | 1349 | return start<BestEligiblePivotRule>(); |
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[605] | 1350 | case BLOCK_SEARCH: |
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[601] | 1351 | return start<BlockSearchPivotRule>(); |
---|
[605] | 1352 | case CANDIDATE_LIST: |
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[601] | 1353 | return start<CandidateListPivotRule>(); |
---|
[605] | 1354 | case ALTERING_LIST: |
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[601] | 1355 | return start<AlteringListPivotRule>(); |
---|
| 1356 | } |
---|
[640] | 1357 | return INFEASIBLE; // avoid warning |
---|
[601] | 1358 | } |
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| 1359 | |
---|
[605] | 1360 | template <typename PivotRuleImpl> |
---|
[640] | 1361 | ProblemType start() { |
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[605] | 1362 | PivotRuleImpl pivot(*this); |
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[601] | 1363 | |
---|
[605] | 1364 | // Execute the Network Simplex algorithm |
---|
[601] | 1365 | while (pivot.findEnteringArc()) { |
---|
| 1366 | findJoinNode(); |
---|
| 1367 | bool change = findLeavingArc(); |
---|
[640] | 1368 | if (delta >= INF) return UNBOUNDED; |
---|
[601] | 1369 | changeFlow(change); |
---|
| 1370 | if (change) { |
---|
[604] | 1371 | updateTreeStructure(); |
---|
| 1372 | updatePotential(); |
---|
[601] | 1373 | } |
---|
| 1374 | } |
---|
[640] | 1375 | |
---|
| 1376 | // Check feasibility |
---|
| 1377 | if (_sum_supply < 0) { |
---|
| 1378 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
| 1379 | if (_supply[u] >= 0 && _flow[e] != 0) return INFEASIBLE; |
---|
| 1380 | } |
---|
| 1381 | } |
---|
| 1382 | else if (_sum_supply > 0) { |
---|
| 1383 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
| 1384 | if (_supply[u] <= 0 && _flow[e] != 0) return INFEASIBLE; |
---|
| 1385 | } |
---|
| 1386 | } |
---|
| 1387 | else { |
---|
| 1388 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
| 1389 | if (_flow[e] != 0) return INFEASIBLE; |
---|
| 1390 | } |
---|
| 1391 | } |
---|
[601] | 1392 | |
---|
[642] | 1393 | // Transform the solution and the supply map to the original form |
---|
| 1394 | if (_have_lower) { |
---|
[601] | 1395 | for (int i = 0; i != _arc_num; ++i) { |
---|
[642] | 1396 | Value c = _lower[i]; |
---|
| 1397 | if (c != 0) { |
---|
| 1398 | _flow[i] += c; |
---|
| 1399 | _supply[_source[i]] += c; |
---|
| 1400 | _supply[_target[i]] -= c; |
---|
| 1401 | } |
---|
[601] | 1402 | } |
---|
| 1403 | } |
---|
| 1404 | |
---|
[640] | 1405 | return OPTIMAL; |
---|
[601] | 1406 | } |
---|
| 1407 | |
---|
| 1408 | }; //class NetworkSimplex |
---|
| 1409 | |
---|
| 1410 | ///@} |
---|
| 1411 | |
---|
| 1412 | } //namespace lemon |
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| 1413 | |
---|
| 1414 | #endif //LEMON_NETWORK_SIMPLEX_H |
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