1015 lines
32 KiB
C++
Executable File
1015 lines
32 KiB
C++
Executable File
/* -*- mode: C++; indent-tabs-mode: nil; -*-
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*
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* This file is a part of LEMON, a generic C++ optimization library.
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*
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* Copyright (C) 2003-2013
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* Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
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* (Egervary Research Group on Combinatorial Optimization, EGRES).
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*
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* Permission to use, modify and distribute this software is granted
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* provided that this copyright notice appears in all copies. For
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* precise terms see the accompanying LICENSE file.
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*
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* This software is provided "AS IS" with no warranty of any kind,
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* express or implied, and with no claim as to its suitability for any
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* purpose.
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*
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*/
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#ifndef LEMON_CAPACITY_SCALING_H
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#define LEMON_CAPACITY_SCALING_H
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/// \ingroup min_cost_flow_algs
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///
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/// \file
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/// \brief Capacity Scaling algorithm for finding a minimum cost flow.
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#include <vector>
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#include <limits>
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#include <lemon/core.h>
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#include <lemon/bin_heap.h>
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namespace lemon {
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/// \brief Default traits class of CapacityScaling algorithm.
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///
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/// Default traits class of CapacityScaling algorithm.
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/// \tparam GR Digraph type.
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/// \tparam V The number type used for flow amounts, capacity bounds
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/// and supply values. By default it is \c int.
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/// \tparam C The number type used for costs and potentials.
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/// By default it is the same as \c V.
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template <typename GR, typename V = int, typename C = V>
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struct CapacityScalingDefaultTraits
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{
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/// The type of the digraph
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typedef GR Digraph;
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/// The type of the flow amounts, capacity bounds and supply values
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typedef V Value;
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/// The type of the arc costs
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typedef C Cost;
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/// \brief The type of the heap used for internal Dijkstra computations.
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///
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/// The type of the heap used for internal Dijkstra computations.
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/// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
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/// its priority type must be \c Cost and its cross reference type
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/// must be \ref RangeMap "RangeMap<int>".
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typedef BinHeap<Cost, RangeMap<int> > Heap;
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};
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/// \addtogroup min_cost_flow_algs
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/// @{
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/// \brief Implementation of the Capacity Scaling algorithm for
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/// finding a \ref min_cost_flow "minimum cost flow".
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///
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/// \ref CapacityScaling implements the capacity scaling version
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/// of the successive shortest path algorithm for finding a
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/// \ref min_cost_flow "minimum cost flow" \cite amo93networkflows,
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/// \cite edmondskarp72theoretical. It is an efficient dual
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/// solution method, which runs in polynomial time
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/// \f$O(m\log U (n+m)\log n)\f$, where <i>U</i> denotes the maximum
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/// of node supply and arc capacity values.
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///
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/// This algorithm is typically slower than \ref CostScaling and
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/// \ref NetworkSimplex, but in special cases, it can be more
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/// efficient than them.
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/// (For more information, see \ref min_cost_flow_algs "the module page".)
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///
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/// Most of the parameters of the problem (except for the digraph)
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/// can be given using separate functions, and the algorithm can be
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/// executed using the \ref run() function. If some parameters are not
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/// specified, then default values will be used.
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///
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/// \tparam GR The digraph type the algorithm runs on.
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/// \tparam V The number type used for flow amounts, capacity bounds
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/// and supply values in the algorithm. By default, it is \c int.
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/// \tparam C The number type used for costs and potentials in the
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/// algorithm. By default, it is the same as \c V.
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/// \tparam TR The traits class that defines various types used by the
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/// algorithm. By default, it is \ref CapacityScalingDefaultTraits
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/// "CapacityScalingDefaultTraits<GR, V, C>".
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/// In most cases, this parameter should not be set directly,
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/// consider to use the named template parameters instead.
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///
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/// \warning Both \c V and \c C must be signed number types.
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/// \warning Capacity bounds and supply values must be integer, but
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/// arc costs can be arbitrary real numbers.
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/// \warning This algorithm does not support negative costs for
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/// arcs having infinite upper bound.
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#ifdef DOXYGEN
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template <typename GR, typename V, typename C, typename TR>
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#else
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template < typename GR, typename V = int, typename C = V,
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typename TR = CapacityScalingDefaultTraits<GR, V, C> >
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#endif
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class CapacityScaling
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{
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public:
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/// The type of the digraph
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typedef typename TR::Digraph Digraph;
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/// The type of the flow amounts, capacity bounds and supply values
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typedef typename TR::Value Value;
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/// The type of the arc costs
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typedef typename TR::Cost Cost;
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/// The type of the heap used for internal Dijkstra computations
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typedef typename TR::Heap Heap;
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/// \brief The \ref lemon::CapacityScalingDefaultTraits "traits class"
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/// of the algorithm
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typedef TR Traits;
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public:
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/// \brief Problem type constants for the \c run() function.
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///
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/// Enum type containing the problem type constants that can be
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/// returned by the \ref run() function of the algorithm.
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enum ProblemType {
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/// The problem has no feasible solution (flow).
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INFEASIBLE,
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/// The problem has optimal solution (i.e. it is feasible and
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/// bounded), and the algorithm has found optimal flow and node
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/// potentials (primal and dual solutions).
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OPTIMAL,
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/// The digraph contains an arc of negative cost and infinite
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/// upper bound. It means that the objective function is unbounded
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/// on that arc, however, note that it could actually be bounded
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/// over the feasible flows, but this algroithm cannot handle
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/// these cases.
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UNBOUNDED
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};
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private:
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TEMPLATE_DIGRAPH_TYPEDEFS(GR);
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typedef std::vector<int> IntVector;
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typedef std::vector<Value> ValueVector;
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typedef std::vector<Cost> CostVector;
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typedef std::vector<char> BoolVector;
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// Note: vector<char> is used instead of vector<bool> for efficiency reasons
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private:
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// Data related to the underlying digraph
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const GR &_graph;
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int _node_num;
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int _arc_num;
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int _res_arc_num;
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int _root;
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// Parameters of the problem
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bool _has_lower;
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Value _sum_supply;
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// Data structures for storing the digraph
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IntNodeMap _node_id;
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IntArcMap _arc_idf;
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IntArcMap _arc_idb;
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IntVector _first_out;
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BoolVector _forward;
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IntVector _source;
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IntVector _target;
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IntVector _reverse;
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// Node and arc data
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ValueVector _lower;
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ValueVector _upper;
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CostVector _cost;
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ValueVector _supply;
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ValueVector _res_cap;
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CostVector _pi;
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ValueVector _excess;
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IntVector _excess_nodes;
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IntVector _deficit_nodes;
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Value _delta;
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int _factor;
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IntVector _pred;
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public:
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/// \brief Constant for infinite upper bounds (capacities).
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///
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/// Constant for infinite upper bounds (capacities).
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/// It is \c std::numeric_limits<Value>::infinity() if available,
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/// \c std::numeric_limits<Value>::max() otherwise.
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const Value INF;
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private:
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// Special implementation of the Dijkstra algorithm for finding
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// shortest paths in the residual network of the digraph with
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// respect to the reduced arc costs and modifying the node
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// potentials according to the found distance labels.
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class ResidualDijkstra
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{
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private:
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int _node_num;
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bool _geq;
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const IntVector &_first_out;
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const IntVector &_target;
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const CostVector &_cost;
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const ValueVector &_res_cap;
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const ValueVector &_excess;
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CostVector &_pi;
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IntVector &_pred;
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IntVector _proc_nodes;
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CostVector _dist;
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public:
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ResidualDijkstra(CapacityScaling& cs) :
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_node_num(cs._node_num), _geq(cs._sum_supply < 0),
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_first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
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_res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
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_pred(cs._pred), _dist(cs._node_num)
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{}
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int run(int s, Value delta = 1) {
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RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
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Heap heap(heap_cross_ref);
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heap.push(s, 0);
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_pred[s] = -1;
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_proc_nodes.clear();
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// Process nodes
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while (!heap.empty() && _excess[heap.top()] > -delta) {
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int u = heap.top(), v;
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Cost d = heap.prio() + _pi[u], dn;
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_dist[u] = heap.prio();
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_proc_nodes.push_back(u);
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heap.pop();
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// Traverse outgoing residual arcs
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int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
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for (int a = _first_out[u]; a != last_out; ++a) {
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if (_res_cap[a] < delta) continue;
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v = _target[a];
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switch (heap.state(v)) {
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case Heap::PRE_HEAP:
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heap.push(v, d + _cost[a] - _pi[v]);
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_pred[v] = a;
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break;
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case Heap::IN_HEAP:
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dn = d + _cost[a] - _pi[v];
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if (dn < heap[v]) {
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heap.decrease(v, dn);
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_pred[v] = a;
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}
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break;
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case Heap::POST_HEAP:
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break;
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}
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}
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}
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if (heap.empty()) return -1;
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// Update potentials of processed nodes
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int t = heap.top();
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Cost dt = heap.prio();
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for (int i = 0; i < int(_proc_nodes.size()); ++i) {
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_pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
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}
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return t;
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}
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}; //class ResidualDijkstra
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public:
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/// \name Named Template Parameters
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/// @{
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template <typename T>
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struct SetHeapTraits : public Traits {
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typedef T Heap;
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};
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/// \brief \ref named-templ-param "Named parameter" for setting
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/// \c Heap type.
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///
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/// \ref named-templ-param "Named parameter" for setting \c Heap
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/// type, which is used for internal Dijkstra computations.
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/// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
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/// its priority type must be \c Cost and its cross reference type
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/// must be \ref RangeMap "RangeMap<int>".
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template <typename T>
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struct SetHeap
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: public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
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typedef CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
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};
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/// @}
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protected:
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CapacityScaling() {}
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public:
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/// \brief Constructor.
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///
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/// The constructor of the class.
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///
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/// \param graph The digraph the algorithm runs on.
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CapacityScaling(const GR& graph) :
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_graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
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INF(std::numeric_limits<Value>::has_infinity ?
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std::numeric_limits<Value>::infinity() :
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std::numeric_limits<Value>::max())
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{
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// Check the number types
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LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
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"The flow type of CapacityScaling must be signed");
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LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
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"The cost type of CapacityScaling must be signed");
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// Reset data structures
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reset();
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}
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/// \name Parameters
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/// The parameters of the algorithm can be specified using these
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/// functions.
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/// @{
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/// \brief Set the lower bounds on the arcs.
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///
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/// This function sets the lower bounds on the arcs.
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/// If it is not used before calling \ref run(), the lower bounds
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/// will be set to zero on all arcs.
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///
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/// \param map An arc map storing the lower bounds.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template <typename LowerMap>
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CapacityScaling& lowerMap(const LowerMap& map) {
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_has_lower = true;
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_lower[_arc_idf[a]] = map[a];
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}
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return *this;
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}
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/// \brief Set the upper bounds (capacities) on the arcs.
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///
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/// This function sets the upper bounds (capacities) on the arcs.
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/// If it is not used before calling \ref run(), the upper bounds
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/// will be set to \ref INF on all arcs (i.e. the flow value will be
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/// unbounded from above).
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///
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/// \param map An arc map storing the upper bounds.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename UpperMap>
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CapacityScaling& upperMap(const UpperMap& map) {
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_upper[_arc_idf[a]] = map[a];
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}
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return *this;
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}
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/// \brief Set the costs of the arcs.
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///
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/// This function sets the costs of the arcs.
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/// If it is not used before calling \ref run(), the costs
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/// will be set to \c 1 on all arcs.
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///
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/// \param map An arc map storing the costs.
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/// Its \c Value type must be convertible to the \c Cost type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename CostMap>
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CapacityScaling& costMap(const CostMap& map) {
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_cost[_arc_idf[a]] = map[a];
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_cost[_arc_idb[a]] = -map[a];
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}
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return *this;
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}
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/// \brief Set the supply values of the nodes.
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///
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/// This function sets the supply values of the nodes.
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/// If neither this function nor \ref stSupply() is used before
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/// calling \ref run(), the supply of each node will be set to zero.
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///
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/// \param map A node map storing the supply values.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename SupplyMap>
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CapacityScaling& supplyMap(const SupplyMap& map) {
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for (NodeIt n(_graph); n != INVALID; ++n) {
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_supply[_node_id[n]] = map[n];
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}
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return *this;
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}
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/// \brief Set single source and target nodes and a supply value.
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///
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/// This function sets a single source node and a single target node
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/// and the required flow value.
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/// If neither this function nor \ref supplyMap() is used before
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/// calling \ref run(), the supply of each node will be set to zero.
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///
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/// Using this function has the same effect as using \ref supplyMap()
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/// with a map in which \c k is assigned to \c s, \c -k is
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/// assigned to \c t and all other nodes have zero supply value.
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///
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/// \param s The source node.
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/// \param t The target node.
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/// \param k The required amount of flow from node \c s to node \c t
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/// (i.e. the supply of \c s and the demand of \c t).
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///
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/// \return <tt>(*this)</tt>
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CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
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for (int i = 0; i != _node_num; ++i) {
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_supply[i] = 0;
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}
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_supply[_node_id[s]] = k;
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_supply[_node_id[t]] = -k;
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return *this;
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}
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/// @}
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/// \name Execution control
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/// The algorithm can be executed using \ref run().
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/// @{
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/// \brief Run the algorithm.
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///
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/// This function runs the algorithm.
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/// The paramters can be specified using functions \ref lowerMap(),
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/// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
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/// For example,
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/// \code
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/// CapacityScaling<ListDigraph> cs(graph);
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/// cs.lowerMap(lower).upperMap(upper).costMap(cost)
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/// .supplyMap(sup).run();
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/// \endcode
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///
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/// This function can be called more than once. All the given parameters
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/// are kept for the next call, unless \ref resetParams() or \ref reset()
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/// is used, thus only the modified parameters have to be set again.
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/// If the underlying digraph was also modified after the construction
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/// of the class (or the last \ref reset() call), then the \ref reset()
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/// function must be called.
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///
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/// \param factor The capacity scaling factor. It must be larger than
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/// one to use scaling. If it is less or equal to one, then scaling
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/// will be disabled.
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///
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/// \return \c INFEASIBLE if no feasible flow exists,
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/// \n \c OPTIMAL if the problem has optimal solution
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/// (i.e. it is feasible and bounded), and the algorithm has found
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/// optimal flow and node potentials (primal and dual solutions),
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/// \n \c UNBOUNDED if the digraph contains an arc of negative cost
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/// and infinite upper bound. It means that the objective function
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/// is unbounded on that arc, however, note that it could actually be
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/// bounded over the feasible flows, but this algroithm cannot handle
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/// these cases.
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///
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/// \see ProblemType
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/// \see resetParams(), reset()
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ProblemType run(int factor = 4) {
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_factor = factor;
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ProblemType pt = init();
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if (pt != OPTIMAL) return pt;
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return start();
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}
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/// \brief Reset all the parameters that have been given before.
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///
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/// This function resets all the paramaters that have been given
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/// before using functions \ref lowerMap(), \ref upperMap(),
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/// \ref costMap(), \ref supplyMap(), \ref stSupply().
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///
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/// It is useful for multiple \ref run() calls. Basically, all the given
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/// parameters are kept for the next \ref run() call, unless
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/// \ref resetParams() or \ref reset() is used.
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/// If the underlying digraph was also modified after the construction
|
|
/// of the class or the last \ref reset() call, then the \ref reset()
|
|
/// function must be used, otherwise \ref resetParams() is sufficient.
|
|
///
|
|
/// For example,
|
|
/// \code
|
|
/// CapacityScaling<ListDigraph> cs(graph);
|
|
///
|
|
/// // First run
|
|
/// cs.lowerMap(lower).upperMap(upper).costMap(cost)
|
|
/// .supplyMap(sup).run();
|
|
///
|
|
/// // Run again with modified cost map (resetParams() is not called,
|
|
/// // so only the cost map have to be set again)
|
|
/// cost[e] += 100;
|
|
/// cs.costMap(cost).run();
|
|
///
|
|
/// // Run again from scratch using resetParams()
|
|
/// // (the lower bounds will be set to zero on all arcs)
|
|
/// cs.resetParams();
|
|
/// cs.upperMap(capacity).costMap(cost)
|
|
/// .supplyMap(sup).run();
|
|
/// \endcode
|
|
///
|
|
/// \return <tt>(*this)</tt>
|
|
///
|
|
/// \see reset(), run()
|
|
CapacityScaling& resetParams() {
|
|
for (int i = 0; i != _node_num; ++i) {
|
|
_supply[i] = 0;
|
|
}
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
_lower[j] = 0;
|
|
_upper[j] = INF;
|
|
_cost[j] = _forward[j] ? 1 : -1;
|
|
}
|
|
_has_lower = false;
|
|
return *this;
|
|
}
|
|
|
|
/// \brief Reset the internal data structures and all the parameters
|
|
/// that have been given before.
|
|
///
|
|
/// This function resets the internal data structures and all the
|
|
/// paramaters that have been given before using functions \ref lowerMap(),
|
|
/// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
|
|
///
|
|
/// It is useful for multiple \ref run() calls. Basically, all the given
|
|
/// parameters are kept for the next \ref run() call, unless
|
|
/// \ref resetParams() or \ref reset() is used.
|
|
/// If the underlying digraph was also modified after the construction
|
|
/// of the class or the last \ref reset() call, then the \ref reset()
|
|
/// function must be used, otherwise \ref resetParams() is sufficient.
|
|
///
|
|
/// See \ref resetParams() for examples.
|
|
///
|
|
/// \return <tt>(*this)</tt>
|
|
///
|
|
/// \see resetParams(), run()
|
|
CapacityScaling& reset() {
|
|
// Resize vectors
|
|
_node_num = countNodes(_graph);
|
|
_arc_num = countArcs(_graph);
|
|
_res_arc_num = 2 * (_arc_num + _node_num);
|
|
_root = _node_num;
|
|
++_node_num;
|
|
|
|
_first_out.resize(_node_num + 1);
|
|
_forward.resize(_res_arc_num);
|
|
_source.resize(_res_arc_num);
|
|
_target.resize(_res_arc_num);
|
|
_reverse.resize(_res_arc_num);
|
|
|
|
_lower.resize(_res_arc_num);
|
|
_upper.resize(_res_arc_num);
|
|
_cost.resize(_res_arc_num);
|
|
_supply.resize(_node_num);
|
|
|
|
_res_cap.resize(_res_arc_num);
|
|
_pi.resize(_node_num);
|
|
_excess.resize(_node_num);
|
|
_pred.resize(_node_num);
|
|
|
|
// Copy the graph
|
|
int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
|
|
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
|
|
_node_id[n] = i;
|
|
}
|
|
i = 0;
|
|
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
|
|
_first_out[i] = j;
|
|
for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
|
|
_arc_idf[a] = j;
|
|
_forward[j] = true;
|
|
_source[j] = i;
|
|
_target[j] = _node_id[_graph.runningNode(a)];
|
|
}
|
|
for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
|
|
_arc_idb[a] = j;
|
|
_forward[j] = false;
|
|
_source[j] = i;
|
|
_target[j] = _node_id[_graph.runningNode(a)];
|
|
}
|
|
_forward[j] = false;
|
|
_source[j] = i;
|
|
_target[j] = _root;
|
|
_reverse[j] = k;
|
|
_forward[k] = true;
|
|
_source[k] = _root;
|
|
_target[k] = i;
|
|
_reverse[k] = j;
|
|
++j; ++k;
|
|
}
|
|
_first_out[i] = j;
|
|
_first_out[_node_num] = k;
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
int fi = _arc_idf[a];
|
|
int bi = _arc_idb[a];
|
|
_reverse[fi] = bi;
|
|
_reverse[bi] = fi;
|
|
}
|
|
|
|
// Reset parameters
|
|
resetParams();
|
|
return *this;
|
|
}
|
|
|
|
/// @}
|
|
|
|
/// \name Query Functions
|
|
/// The results of the algorithm can be obtained using these
|
|
/// functions.\n
|
|
/// The \ref run() function must be called before using them.
|
|
|
|
/// @{
|
|
|
|
/// \brief Return the total cost of the found flow.
|
|
///
|
|
/// This function returns the total cost of the found flow.
|
|
/// Its complexity is O(m).
|
|
///
|
|
/// \note The return type of the function can be specified as a
|
|
/// template parameter. For example,
|
|
/// \code
|
|
/// cs.totalCost<double>();
|
|
/// \endcode
|
|
/// It is useful if the total cost cannot be stored in the \c Cost
|
|
/// type of the algorithm, which is the default return type of the
|
|
/// function.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename Number>
|
|
Number totalCost() const {
|
|
Number c = 0;
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
int i = _arc_idb[a];
|
|
c += static_cast<Number>(_res_cap[i]) *
|
|
(-static_cast<Number>(_cost[i]));
|
|
}
|
|
return c;
|
|
}
|
|
|
|
#ifndef DOXYGEN
|
|
Cost totalCost() const {
|
|
return totalCost<Cost>();
|
|
}
|
|
#endif
|
|
|
|
/// \brief Return the flow on the given arc.
|
|
///
|
|
/// This function returns the flow on the given arc.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
Value flow(const Arc& a) const {
|
|
return _res_cap[_arc_idb[a]];
|
|
}
|
|
|
|
/// \brief Copy the flow values (the primal solution) into the
|
|
/// given map.
|
|
///
|
|
/// This function copies the flow value on each arc into the given
|
|
/// map. The \c Value type of the algorithm must be convertible to
|
|
/// the \c Value type of the map.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename FlowMap>
|
|
void flowMap(FlowMap &map) const {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
map.set(a, _res_cap[_arc_idb[a]]);
|
|
}
|
|
}
|
|
|
|
/// \brief Return the potential (dual value) of the given node.
|
|
///
|
|
/// This function returns the potential (dual value) of the
|
|
/// given node.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
Cost potential(const Node& n) const {
|
|
return _pi[_node_id[n]];
|
|
}
|
|
|
|
/// \brief Copy the potential values (the dual solution) into the
|
|
/// given map.
|
|
///
|
|
/// This function copies the potential (dual value) of each node
|
|
/// into the given map.
|
|
/// The \c Cost type of the algorithm must be convertible to the
|
|
/// \c Value type of the map.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename PotentialMap>
|
|
void potentialMap(PotentialMap &map) const {
|
|
for (NodeIt n(_graph); n != INVALID; ++n) {
|
|
map.set(n, _pi[_node_id[n]]);
|
|
}
|
|
}
|
|
|
|
/// @}
|
|
|
|
private:
|
|
|
|
// Initialize the algorithm
|
|
ProblemType init() {
|
|
if (_node_num <= 1) return INFEASIBLE;
|
|
|
|
// Check the sum of supply values
|
|
_sum_supply = 0;
|
|
for (int i = 0; i != _root; ++i) {
|
|
_sum_supply += _supply[i];
|
|
}
|
|
if (_sum_supply > 0) return INFEASIBLE;
|
|
|
|
// Check lower and upper bounds
|
|
LEMON_DEBUG(checkBoundMaps(),
|
|
"Upper bounds must be greater or equal to the lower bounds");
|
|
|
|
|
|
// Initialize vectors
|
|
for (int i = 0; i != _root; ++i) {
|
|
_pi[i] = 0;
|
|
_excess[i] = _supply[i];
|
|
}
|
|
|
|
// Remove non-zero lower bounds
|
|
const Value MAX = std::numeric_limits<Value>::max();
|
|
int last_out;
|
|
if (_has_lower) {
|
|
for (int i = 0; i != _root; ++i) {
|
|
last_out = _first_out[i+1];
|
|
for (int j = _first_out[i]; j != last_out; ++j) {
|
|
if (_forward[j]) {
|
|
Value c = _lower[j];
|
|
if (c >= 0) {
|
|
_res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
|
|
} else {
|
|
_res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
|
|
}
|
|
_excess[i] -= c;
|
|
_excess[_target[j]] += c;
|
|
} else {
|
|
_res_cap[j] = 0;
|
|
}
|
|
}
|
|
}
|
|
} else {
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
_res_cap[j] = _forward[j] ? _upper[j] : 0;
|
|
}
|
|
}
|
|
|
|
// Handle negative costs
|
|
for (int i = 0; i != _root; ++i) {
|
|
last_out = _first_out[i+1] - 1;
|
|
for (int j = _first_out[i]; j != last_out; ++j) {
|
|
Value rc = _res_cap[j];
|
|
if (_cost[j] < 0 && rc > 0) {
|
|
if (rc >= MAX) return UNBOUNDED;
|
|
_excess[i] -= rc;
|
|
_excess[_target[j]] += rc;
|
|
_res_cap[j] = 0;
|
|
_res_cap[_reverse[j]] += rc;
|
|
}
|
|
}
|
|
}
|
|
|
|
// Handle GEQ supply type
|
|
if (_sum_supply < 0) {
|
|
_pi[_root] = 0;
|
|
_excess[_root] = -_sum_supply;
|
|
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
|
|
int ra = _reverse[a];
|
|
_res_cap[a] = -_sum_supply + 1;
|
|
_res_cap[ra] = 0;
|
|
_cost[a] = 0;
|
|
_cost[ra] = 0;
|
|
}
|
|
} else {
|
|
_pi[_root] = 0;
|
|
_excess[_root] = 0;
|
|
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
|
|
int ra = _reverse[a];
|
|
_res_cap[a] = 1;
|
|
_res_cap[ra] = 0;
|
|
_cost[a] = 0;
|
|
_cost[ra] = 0;
|
|
}
|
|
}
|
|
|
|
// Initialize delta value
|
|
if (_factor > 1) {
|
|
// With scaling
|
|
Value max_sup = 0, max_dem = 0, max_cap = 0;
|
|
for (int i = 0; i != _root; ++i) {
|
|
Value ex = _excess[i];
|
|
if ( ex > max_sup) max_sup = ex;
|
|
if (-ex > max_dem) max_dem = -ex;
|
|
int last_out = _first_out[i+1] - 1;
|
|
for (int j = _first_out[i]; j != last_out; ++j) {
|
|
if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
|
|
}
|
|
}
|
|
max_sup = std::min(std::min(max_sup, max_dem), max_cap);
|
|
for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
|
|
} else {
|
|
// Without scaling
|
|
_delta = 1;
|
|
}
|
|
|
|
return OPTIMAL;
|
|
}
|
|
|
|
// Check if the upper bound is greater than or equal to the lower bound
|
|
// on each forward arc.
|
|
bool checkBoundMaps() {
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
if (_forward[j] && _upper[j] < _lower[j]) return false;
|
|
}
|
|
return true;
|
|
}
|
|
|
|
ProblemType start() {
|
|
// Execute the algorithm
|
|
ProblemType pt;
|
|
if (_delta > 1)
|
|
pt = startWithScaling();
|
|
else
|
|
pt = startWithoutScaling();
|
|
|
|
// Handle non-zero lower bounds
|
|
if (_has_lower) {
|
|
int limit = _first_out[_root];
|
|
for (int j = 0; j != limit; ++j) {
|
|
if (_forward[j]) _res_cap[_reverse[j]] += _lower[j];
|
|
}
|
|
}
|
|
|
|
// Shift potentials if necessary
|
|
Cost pr = _pi[_root];
|
|
if (_sum_supply < 0 || pr > 0) {
|
|
for (int i = 0; i != _node_num; ++i) {
|
|
_pi[i] -= pr;
|
|
}
|
|
}
|
|
|
|
return pt;
|
|
}
|
|
|
|
// Execute the capacity scaling algorithm
|
|
ProblemType startWithScaling() {
|
|
// Perform capacity scaling phases
|
|
int s, t;
|
|
ResidualDijkstra _dijkstra(*this);
|
|
while (true) {
|
|
// Saturate all arcs not satisfying the optimality condition
|
|
int last_out;
|
|
for (int u = 0; u != _node_num; ++u) {
|
|
last_out = _sum_supply < 0 ?
|
|
_first_out[u+1] : _first_out[u+1] - 1;
|
|
for (int a = _first_out[u]; a != last_out; ++a) {
|
|
int v = _target[a];
|
|
Cost c = _cost[a] + _pi[u] - _pi[v];
|
|
Value rc = _res_cap[a];
|
|
if (c < 0 && rc >= _delta) {
|
|
_excess[u] -= rc;
|
|
_excess[v] += rc;
|
|
_res_cap[a] = 0;
|
|
_res_cap[_reverse[a]] += rc;
|
|
}
|
|
}
|
|
}
|
|
|
|
// Find excess nodes and deficit nodes
|
|
_excess_nodes.clear();
|
|
_deficit_nodes.clear();
|
|
for (int u = 0; u != _node_num; ++u) {
|
|
Value ex = _excess[u];
|
|
if (ex >= _delta) _excess_nodes.push_back(u);
|
|
if (ex <= -_delta) _deficit_nodes.push_back(u);
|
|
}
|
|
int next_node = 0, next_def_node = 0;
|
|
|
|
// Find augmenting shortest paths
|
|
while (next_node < int(_excess_nodes.size())) {
|
|
// Check deficit nodes
|
|
if (_delta > 1) {
|
|
bool delta_deficit = false;
|
|
for ( ; next_def_node < int(_deficit_nodes.size());
|
|
++next_def_node ) {
|
|
if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
|
|
delta_deficit = true;
|
|
break;
|
|
}
|
|
}
|
|
if (!delta_deficit) break;
|
|
}
|
|
|
|
// Run Dijkstra in the residual network
|
|
s = _excess_nodes[next_node];
|
|
if ((t = _dijkstra.run(s, _delta)) == -1) {
|
|
if (_delta > 1) {
|
|
++next_node;
|
|
continue;
|
|
}
|
|
return INFEASIBLE;
|
|
}
|
|
|
|
// Augment along a shortest path from s to t
|
|
Value d = std::min(_excess[s], -_excess[t]);
|
|
int u = t;
|
|
int a;
|
|
if (d > _delta) {
|
|
while ((a = _pred[u]) != -1) {
|
|
if (_res_cap[a] < d) d = _res_cap[a];
|
|
u = _source[a];
|
|
}
|
|
}
|
|
u = t;
|
|
while ((a = _pred[u]) != -1) {
|
|
_res_cap[a] -= d;
|
|
_res_cap[_reverse[a]] += d;
|
|
u = _source[a];
|
|
}
|
|
_excess[s] -= d;
|
|
_excess[t] += d;
|
|
|
|
if (_excess[s] < _delta) ++next_node;
|
|
}
|
|
|
|
if (_delta == 1) break;
|
|
_delta = _delta <= _factor ? 1 : _delta / _factor;
|
|
}
|
|
|
|
return OPTIMAL;
|
|
}
|
|
|
|
// Execute the successive shortest path algorithm
|
|
ProblemType startWithoutScaling() {
|
|
// Find excess nodes
|
|
_excess_nodes.clear();
|
|
for (int i = 0; i != _node_num; ++i) {
|
|
if (_excess[i] > 0) _excess_nodes.push_back(i);
|
|
}
|
|
if (_excess_nodes.size() == 0) return OPTIMAL;
|
|
int next_node = 0;
|
|
|
|
// Find shortest paths
|
|
int s, t;
|
|
ResidualDijkstra _dijkstra(*this);
|
|
while ( _excess[_excess_nodes[next_node]] > 0 ||
|
|
++next_node < int(_excess_nodes.size()) )
|
|
{
|
|
// Run Dijkstra in the residual network
|
|
s = _excess_nodes[next_node];
|
|
if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
|
|
|
|
// Augment along a shortest path from s to t
|
|
Value d = std::min(_excess[s], -_excess[t]);
|
|
int u = t;
|
|
int a;
|
|
if (d > 1) {
|
|
while ((a = _pred[u]) != -1) {
|
|
if (_res_cap[a] < d) d = _res_cap[a];
|
|
u = _source[a];
|
|
}
|
|
}
|
|
u = t;
|
|
while ((a = _pred[u]) != -1) {
|
|
_res_cap[a] -= d;
|
|
_res_cap[_reverse[a]] += d;
|
|
u = _source[a];
|
|
}
|
|
_excess[s] -= d;
|
|
_excess[t] += d;
|
|
}
|
|
|
|
return OPTIMAL;
|
|
}
|
|
|
|
}; //class CapacityScaling
|
|
|
|
///@}
|
|
|
|
} //namespace lemon
|
|
|
|
#endif //LEMON_CAPACITY_SCALING_H
|