Location: LEMON/LEMON-official/lemon/cycle_canceling.h

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/* -*- C++ -*-
*
* This file is a part of LEMON, a generic C++ optimization library
*
* Copyright (C) 2003-2008
* Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
* (Egervary Research Group on Combinatorial Optimization, EGRES).
*
* Permission to use, modify and distribute this software is granted
* provided that this copyright notice appears in all copies. For
* precise terms see the accompanying LICENSE file.
*
* This software is provided "AS IS" with no warranty of any kind,
* express or implied, and with no claim as to its suitability for any
* purpose.
*
*/
#ifndef LEMON_CYCLE_CANCELING_H
#define LEMON_CYCLE_CANCELING_H
/// \ingroup min_cost_flow_algs
/// \file
/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
#include <vector>
#include <limits>
#include <lemon/core.h>
#include <lemon/maps.h>
#include <lemon/path.h>
#include <lemon/math.h>
#include <lemon/static_graph.h>
#include <lemon/adaptors.h>
#include <lemon/circulation.h>
#include <lemon/bellman_ford.h>
#include <lemon/howard.h>
namespace lemon {
/// \addtogroup min_cost_flow_algs
/// @{
/// \brief Implementation of cycle-canceling algorithms for
/// finding a \ref min_cost_flow "minimum cost flow".
///
/// \ref CycleCanceling implements three different cycle-canceling
/// algorithms for finding a \ref min_cost_flow "minimum cost flow"
/// \ref amo93networkflows, \ref klein67primal,
/// \ref goldberg89cyclecanceling.
/// The most efficent one (both theoretically and practically)
/// is the \ref CANCEL_AND_TIGHTEN "Cancel and Tighten" algorithm,
/// thus it is the default method.
/// It is strongly polynomial, but in practice, it is typically much
/// slower than the scaling algorithms and NetworkSimplex.
///
/// Most of the parameters of the problem (except for the digraph)
/// can be given using separate functions, and the algorithm can be
/// executed using the \ref run() function. If some parameters are not
/// specified, then default values will be used.
///
/// \tparam GR The digraph type the algorithm runs on.
/// \tparam V The number type used for flow amounts, capacity bounds
/// and supply values in the algorithm. By default, it is \c int.
/// \tparam C The number type used for costs and potentials in the
/// algorithm. By default, it is the same as \c V.
///
/// \warning Both number types must be signed and all input data must
/// be integer.
/// \warning This algorithm does not support negative costs for such
/// arcs that have infinite upper bound.
///
/// \note For more information about the three available methods,
/// see \ref Method.
#ifdef DOXYGEN
template <typename GR, typename V, typename C>
#else
template <typename GR, typename V = int, typename C = V>
#endif
class CycleCanceling
{
public:
/// The type of the digraph
typedef GR Digraph;
/// The type of the flow amounts, capacity bounds and supply values
typedef V Value;
/// The type of the arc costs
typedef C Cost;
public:
/// \brief Problem type constants for the \c run() function.
///
/// Enum type containing the problem type constants that can be
/// returned by the \ref run() function of the algorithm.
enum ProblemType {
/// The problem has no feasible solution (flow).
INFEASIBLE,
/// The problem has optimal solution (i.e. it is feasible and
/// bounded), and the algorithm has found optimal flow and node
/// potentials (primal and dual solutions).
OPTIMAL,
/// The digraph contains an arc of negative cost and infinite
/// upper bound. It means that the objective function is unbounded
/// on that arc, however, note that it could actually be bounded
/// over the feasible flows, but this algroithm cannot handle
/// these cases.
UNBOUNDED
};
/// \brief Constants for selecting the used method.
///
/// Enum type containing constants for selecting the used method
/// for the \ref run() function.
///
/// \ref CycleCanceling provides three different cycle-canceling
/// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"
/// is used, which proved to be the most efficient and the most robust
/// on various test inputs.
/// However, the other methods can be selected using the \ref run()
/// function with the proper parameter.
enum Method {
/// A simple cycle-canceling method, which uses the
/// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration
/// number for detecting negative cycles in the residual network.
SIMPLE_CYCLE_CANCELING,
/// The "Minimum Mean Cycle-Canceling" algorithm, which is a
/// well-known strongly polynomial method
/// \ref goldberg89cyclecanceling. It improves along a
/// \ref min_mean_cycle "minimum mean cycle" in each iteration.
/// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).
MINIMUM_MEAN_CYCLE_CANCELING,
/// The "Cancel And Tighten" algorithm, which can be viewed as an
/// improved version of the previous method
/// \ref goldberg89cyclecanceling.
/// It is faster both in theory and in practice, its running time
/// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).
CANCEL_AND_TIGHTEN
};
private:
TEMPLATE_DIGRAPH_TYPEDEFS(GR);
typedef std::vector<int> IntVector;
typedef std::vector<char> CharVector;
typedef std::vector<double> DoubleVector;
typedef std::vector<Value> ValueVector;
typedef std::vector<Cost> CostVector;
private:
template <typename KT, typename VT>
class StaticVectorMap {
public:
typedef KT Key;
typedef VT Value;
StaticVectorMap(std::vector<Value>& v) : _v(v) {}
const Value& operator[](const Key& key) const {
return _v[StaticDigraph::id(key)];
}
Value& operator[](const Key& key) {
return _v[StaticDigraph::id(key)];
}
void set(const Key& key, const Value& val) {
_v[StaticDigraph::id(key)] = val;
}
private:
std::vector<Value>& _v;
};
typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
private:
// Data related to the underlying digraph
const GR &_graph;
int _node_num;
int _arc_num;
int _res_node_num;
int _res_arc_num;
int _root;
// Parameters of the problem
bool _have_lower;
Value _sum_supply;
// Data structures for storing the digraph
IntNodeMap _node_id;
IntArcMap _arc_idf;
IntArcMap _arc_idb;
IntVector _first_out;
CharVector _forward;
IntVector _source;
IntVector _target;
IntVector _reverse;
// Node and arc data
ValueVector _lower;
ValueVector _upper;
CostVector _cost;
ValueVector _supply;
ValueVector _res_cap;
CostVector _pi;
// Data for a StaticDigraph structure
typedef std::pair<int, int> IntPair;
StaticDigraph _sgr;
std::vector<IntPair> _arc_vec;
std::vector<Cost> _cost_vec;
IntVector _id_vec;
CostArcMap _cost_map;
CostNodeMap _pi_map;
public:
/// \brief Constant for infinite upper bounds (capacities).
///
/// Constant for infinite upper bounds (capacities).
/// It is \c std::numeric_limits<Value>::infinity() if available,
/// \c std::numeric_limits<Value>::max() otherwise.
const Value INF;
public:
/// \brief Constructor.
///
/// The constructor of the class.
///
/// \param graph The digraph the algorithm runs on.
CycleCanceling(const GR& graph) :
_graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
_cost_map(_cost_vec), _pi_map(_pi),
INF(std::numeric_limits<Value>::has_infinity ?
std::numeric_limits<Value>::infinity() :
std::numeric_limits<Value>::max())
{
// Check the number types
LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
"The flow type of CycleCanceling must be signed");
LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
"The cost type of CycleCanceling must be signed");
// Resize vectors
_node_num = countNodes(_graph);
_arc_num = countArcs(_graph);
_res_node_num = _node_num + 1;
_res_arc_num = 2 * (_arc_num + _node_num);
_root = _node_num;
_first_out.resize(_res_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(_res_node_num);
_res_cap.resize(_res_arc_num);
_pi.resize(_res_node_num);
_arc_vec.reserve(_res_arc_num);
_cost_vec.reserve(_res_arc_num);
_id_vec.reserve(_res_arc_num);
// Copy the graph
int i = 0, j = 0, k = 2 * _arc_num + _node_num;
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[_res_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
reset();
}
/// \name Parameters
/// The parameters of the algorithm can be specified using these
/// functions.
/// @{
/// \brief Set the lower bounds on the arcs.
///
/// This function sets the lower bounds on the arcs.
/// If it is not used before calling \ref run(), the lower bounds
/// will be set to zero on all arcs.
///
/// \param map An arc map storing the lower bounds.
/// Its \c Value type must be convertible to the \c Value type
/// of the algorithm.
///
/// \return <tt>(*this)</tt>
template <typename LowerMap>
CycleCanceling& lowerMap(const LowerMap& map) {
_have_lower = true;
for (ArcIt a(_graph); a != INVALID; ++a) {
_lower[_arc_idf[a]] = map[a];
_lower[_arc_idb[a]] = map[a];
}
return *this;
}
/// \brief Set the upper bounds (capacities) on the arcs.
///
/// This function sets the upper bounds (capacities) on the arcs.
/// If it is not used before calling \ref run(), the upper bounds
/// will be set to \ref INF on all arcs (i.e. the flow value will be
/// unbounded from above).
///
/// \param map An arc map storing the upper bounds.
/// Its \c Value type must be convertible to the \c Value type
/// of the algorithm.
///
/// \return <tt>(*this)</tt>
template<typename UpperMap>
CycleCanceling& upperMap(const UpperMap& map) {
for (ArcIt a(_graph); a != INVALID; ++a) {
_upper[_arc_idf[a]] = map[a];
}
return *this;
}
/// \brief Set the costs of the arcs.
///
/// This function sets the costs of the arcs.
/// If it is not used before calling \ref run(), the costs
/// will be set to \c 1 on all arcs.
///
/// \param map An arc map storing the costs.
/// Its \c Value type must be convertible to the \c Cost type
/// of the algorithm.
///
/// \return <tt>(*this)</tt>
template<typename CostMap>
CycleCanceling& costMap(const CostMap& map) {
for (ArcIt a(_graph); a != INVALID; ++a) {
_cost[_arc_idf[a]] = map[a];
_cost[_arc_idb[a]] = -map[a];
}
return *this;
}
/// \brief Set the supply values of the nodes.
///
/// This function sets the supply values of the nodes.
/// If neither this function nor \ref stSupply() is used before
/// calling \ref run(), the supply of each node will be set to zero.
///
/// \param map A node map storing the supply values.
/// Its \c Value type must be convertible to the \c Value type
/// of the algorithm.
///
/// \return <tt>(*this)</tt>
template<typename SupplyMap>
CycleCanceling& supplyMap(const SupplyMap& map) {
for (NodeIt n(_graph); n != INVALID; ++n) {
_supply[_node_id[n]] = map[n];
}
return *this;
}
/// \brief Set single source and target nodes and a supply value.
///
/// This function sets a single source node and a single target node
/// and the required flow value.
/// If neither this function nor \ref supplyMap() is used before
/// calling \ref run(), the supply of each node will be set to zero.
///
/// Using this function has the same effect as using \ref supplyMap()
/// with such a map in which \c k is assigned to \c s, \c -k is
/// assigned to \c t and all other nodes have zero supply value.
///
/// \param s The source node.
/// \param t The target node.
/// \param k The required amount of flow from node \c s to node \c t
/// (i.e. the supply of \c s and the demand of \c t).
///
/// \return <tt>(*this)</tt>
CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
for (int i = 0; i != _res_node_num; ++i) {
_supply[i] = 0;
}
_supply[_node_id[s]] = k;
_supply[_node_id[t]] = -k;
return *this;
}
/// @}
/// \name Execution control
/// The algorithm can be executed using \ref run().
/// @{
/// \brief Run the algorithm.
///
/// This function runs the algorithm.
/// The paramters can be specified using functions \ref lowerMap(),
/// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
/// For example,
/// \code
/// CycleCanceling<ListDigraph> cc(graph);
/// cc.lowerMap(lower).upperMap(upper).costMap(cost)
/// .supplyMap(sup).run();
/// \endcode
///
/// This function can be called more than once. All the parameters
/// that have been given are kept for the next call, unless
/// \ref reset() is called, thus only the modified parameters
/// have to be set again. See \ref reset() for examples.
/// However, the underlying digraph must not be modified after this
/// class have been constructed, since it copies and extends the graph.
///
/// \param method The cycle-canceling method that will be used.
/// For more information, see \ref Method.
///
/// \return \c INFEASIBLE if no feasible flow exists,
/// \n \c OPTIMAL if the problem has optimal solution
/// (i.e. it is feasible and bounded), and the algorithm has found
/// optimal flow and node potentials (primal and dual solutions),
/// \n \c UNBOUNDED if the digraph contains an arc of negative cost
/// and infinite upper bound. It means that the objective function
/// is unbounded on that arc, however, note that it could actually be
/// bounded over the feasible flows, but this algroithm cannot handle
/// these cases.
///
/// \see ProblemType, Method
ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
ProblemType pt = init();
if (pt != OPTIMAL) return pt;
start(method);
return OPTIMAL;
}
/// \brief Reset all the parameters that have been given before.
///
/// This function resets 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 run() calls. If this function is not
/// used, all the parameters given before are kept for the next
/// \ref run() call.
/// However, the underlying digraph must not be modified after this
/// class have been constructed, since it copies and extends the graph.
///
/// For example,
/// \code
/// CycleCanceling<ListDigraph> cs(graph);
///
/// // First run
/// cc.lowerMap(lower).upperMap(upper).costMap(cost)
/// .supplyMap(sup).run();
///
/// // Run again with modified cost map (reset() is not called,
/// // so only the cost map have to be set again)
/// cost[e] += 100;
/// cc.costMap(cost).run();
///
/// // Run again from scratch using reset()
/// // (the lower bounds will be set to zero on all arcs)
/// cc.reset();
/// cc.upperMap(capacity).costMap(cost)
/// .supplyMap(sup).run();
/// \endcode
///
/// \return <tt>(*this)</tt>
CycleCanceling& reset() {
for (int i = 0; i != _res_node_num; ++i) {
_supply[i] = 0;
}
int limit = _first_out[_root];
for (int j = 0; j != limit; ++j) {
_lower[j] = 0;
_upper[j] = INF;
_cost[j] = _forward[j] ? 1 : -1;
}
for (int j = limit; j != _res_arc_num; ++j) {
_lower[j] = 0;
_upper[j] = INF;
_cost[j] = 0;
_cost[_reverse[j]] = 0;
}
_have_lower = false;
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(e).
///
/// \note The return type of the function can be specified as a
/// template parameter. For example,
/// \code
/// cc.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 Return the flow map (the primal solution).
///
/// 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 static_cast<Cost>(_pi[_node_id[n]]);
}
/// \brief Return the potential map (the dual solution).
///
/// 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, static_cast<Cost>(_pi[_node_id[n]]));
}
}
/// @}
private:
// Initialize the algorithm
ProblemType init() {
if (_res_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;
// Initialize vectors
for (int i = 0; i != _res_node_num; ++i) {
_pi[i] = 0;
}
ValueVector excess(_supply);
// Remove infinite upper bounds and check negative arcs
const Value MAX = std::numeric_limits<Value>::max();
int last_out;
if (_have_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 = _cost[j] < 0 ? _upper[j] : _lower[j];
if (c >= MAX) return UNBOUNDED;
excess[i] -= c;
excess[_target[j]] += c;
}
}
}
} else {
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] && _cost[j] < 0) {
Value c = _upper[j];
if (c >= MAX) return UNBOUNDED;
excess[i] -= c;
excess[_target[j]] += c;
}
}
}
}
Value ex, max_cap = 0;
for (int i = 0; i != _res_node_num; ++i) {
ex = excess[i];
if (ex < 0) max_cap -= ex;
}
for (int j = 0; j != _res_arc_num; ++j) {
if (_upper[j] >= MAX) _upper[j] = max_cap;
}
// Initialize maps for Circulation and remove non-zero lower bounds
ConstMap<Arc, Value> low(0);
typedef typename Digraph::template ArcMap<Value> ValueArcMap;
typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
ValueArcMap cap(_graph), flow(_graph);
ValueNodeMap sup(_graph);
for (NodeIt n(_graph); n != INVALID; ++n) {
sup[n] = _supply[_node_id[n]];
}
if (_have_lower) {
for (ArcIt a(_graph); a != INVALID; ++a) {
int j = _arc_idf[a];
Value c = _lower[j];
cap[a] = _upper[j] - c;
sup[_graph.source(a)] -= c;
sup[_graph.target(a)] += c;
}
} else {
for (ArcIt a(_graph); a != INVALID; ++a) {
cap[a] = _upper[_arc_idf[a]];
}
}
// Find a feasible flow using Circulation
Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
circ(_graph, low, cap, sup);
if (!circ.flowMap(flow).run()) return INFEASIBLE;
// Set residual capacities and handle GEQ supply type
if (_sum_supply < 0) {
for (ArcIt a(_graph); a != INVALID; ++a) {
Value fa = flow[a];
_res_cap[_arc_idf[a]] = cap[a] - fa;
_res_cap[_arc_idb[a]] = fa;
sup[_graph.source(a)] -= fa;
sup[_graph.target(a)] += fa;
}
for (NodeIt n(_graph); n != INVALID; ++n) {
excess[_node_id[n]] = sup[n];
}
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
int u = _target[a];
int ra = _reverse[a];
_res_cap[a] = -_sum_supply + 1;
_res_cap[ra] = -excess[u];
_cost[a] = 0;
_cost[ra] = 0;
}
} else {
for (ArcIt a(_graph); a != INVALID; ++a) {
Value fa = flow[a];
_res_cap[_arc_idf[a]] = cap[a] - fa;
_res_cap[_arc_idb[a]] = fa;
}
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;
}
}
return OPTIMAL;
}
// Build a StaticDigraph structure containing the current
// residual network
void buildResidualNetwork() {
_arc_vec.clear();
_cost_vec.clear();
_id_vec.clear();
for (int j = 0; j != _res_arc_num; ++j) {
if (_res_cap[j] > 0) {
_arc_vec.push_back(IntPair(_source[j], _target[j]));
_cost_vec.push_back(_cost[j]);
_id_vec.push_back(j);
}
}
_sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
}
// Execute the algorithm and transform the results
void start(Method method) {
// Execute the algorithm
switch (method) {
case SIMPLE_CYCLE_CANCELING:
startSimpleCycleCanceling();
break;
case MINIMUM_MEAN_CYCLE_CANCELING:
startMinMeanCycleCanceling();
break;
case CANCEL_AND_TIGHTEN:
startCancelAndTighten();
break;
}
// Compute node potentials
if (method != SIMPLE_CYCLE_CANCELING) {
buildResidualNetwork();
typename BellmanFord<StaticDigraph, CostArcMap>
::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
bf.distMap(_pi_map);
bf.init(0);
bf.start();
}
// Handle non-zero lower bounds
if (_have_lower) {
int limit = _first_out[_root];
for (int j = 0; j != limit; ++j) {
if (!_forward[j]) _res_cap[j] += _lower[j];
}
}
}
// Execute the "Simple Cycle Canceling" method
void startSimpleCycleCanceling() {
// Constants for computing the iteration limits
const int BF_FIRST_LIMIT = 2;
const double BF_LIMIT_FACTOR = 1.5;
typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
typedef typename BellmanFord<ResDigraph, CostArcMap>
::template SetDistMap<CostNodeMap>
::template SetPredMap<PredMap>::Create BF;
// Build the residual network
_arc_vec.clear();
_cost_vec.clear();
for (int j = 0; j != _res_arc_num; ++j) {
_arc_vec.push_back(IntPair(_source[j], _target[j]));
_cost_vec.push_back(_cost[j]);
}
_sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
FilterMap filter_map(_res_cap);
ResDigraph rgr(_sgr, filter_map);
std::vector<int> cycle;
std::vector<StaticDigraph::Arc> pred(_res_arc_num);
PredMap pred_map(pred);
BF bf(rgr, _cost_map);
bf.distMap(_pi_map).predMap(pred_map);
int length_bound = BF_FIRST_LIMIT;
bool optimal = false;
while (!optimal) {
bf.init(0);
int iter_num = 0;
bool cycle_found = false;
while (!cycle_found) {
// Perform some iterations of the Bellman-Ford algorithm
int curr_iter_num = iter_num + length_bound <= _node_num ?
length_bound : _node_num - iter_num;
iter_num += curr_iter_num;
int real_iter_num = curr_iter_num;
for (int i = 0; i < curr_iter_num; ++i) {
if (bf.processNextWeakRound()) {
real_iter_num = i;
break;
}
}
if (real_iter_num < curr_iter_num) {
// Optimal flow is found
optimal = true;
break;
} else {
// Search for node disjoint negative cycles
std::vector<int> state(_res_node_num, 0);
int id = 0;
for (int u = 0; u != _res_node_num; ++u) {
if (state[u] != 0) continue;
++id;
int v = u;
for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
-1 : rgr.id(rgr.source(pred[v]))) {
state[v] = id;
}
if (v != -1 && state[v] == id) {
// A negative cycle is found
cycle_found = true;
cycle.clear();
StaticDigraph::Arc a = pred[v];
Value d, delta = _res_cap[rgr.id(a)];
cycle.push_back(rgr.id(a));
while (rgr.id(rgr.source(a)) != v) {
a = pred_map[rgr.source(a)];
d = _res_cap[rgr.id(a)];
if (d < delta) delta = d;
cycle.push_back(rgr.id(a));
}
// Augment along the cycle
for (int i = 0; i < int(cycle.size()); ++i) {
int j = cycle[i];
_res_cap[j] -= delta;
_res_cap[_reverse[j]] += delta;
}
}
}
}
// Increase iteration limit if no cycle is found
if (!cycle_found) {
length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
}
}
}
}
// Execute the "Minimum Mean Cycle Canceling" method
void startMinMeanCycleCanceling() {
typedef SimplePath<StaticDigraph> SPath;
typedef typename SPath::ArcIt SPathArcIt;
typedef typename Howard<StaticDigraph, CostArcMap>
::template SetPath<SPath>::Create MMC;
SPath cycle;
MMC mmc(_sgr, _cost_map);
mmc.cycle(cycle);
buildResidualNetwork();
while (mmc.findMinMean() && mmc.cycleLength() < 0) {
// Find the cycle
mmc.findCycle();
// Compute delta value
Value delta = INF;
for (SPathArcIt a(cycle); a != INVALID; ++a) {
Value d = _res_cap[_id_vec[_sgr.id(a)]];
if (d < delta) delta = d;
}
// Augment along the cycle
for (SPathArcIt a(cycle); a != INVALID; ++a) {
int j = _id_vec[_sgr.id(a)];
_res_cap[j] -= delta;
_res_cap[_reverse[j]] += delta;
}
// Rebuild the residual network
buildResidualNetwork();
}
}
// Execute the "Cancel And Tighten" method
void startCancelAndTighten() {
// Constants for the min mean cycle computations
const double LIMIT_FACTOR = 1.0;
const int MIN_LIMIT = 5;
// Contruct auxiliary data vectors
DoubleVector pi(_res_node_num, 0.0);
IntVector level(_res_node_num);
CharVector reached(_res_node_num);
CharVector processed(_res_node_num);
IntVector pred_node(_res_node_num);
IntVector pred_arc(_res_node_num);
std::vector<int> stack(_res_node_num);
std::vector<int> proc_vector(_res_node_num);
// Initialize epsilon
double epsilon = 0;
for (int a = 0; a != _res_arc_num; ++a) {
if (_res_cap[a] > 0 && -_cost[a] > epsilon)
epsilon = -_cost[a];
}
// Start phases
Tolerance<double> tol;
tol.epsilon(1e-6);
int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
if (limit < MIN_LIMIT) limit = MIN_LIMIT;
int iter = limit;
while (epsilon * _res_node_num >= 1) {
// Find and cancel cycles in the admissible network using DFS
for (int u = 0; u != _res_node_num; ++u) {
reached[u] = false;
processed[u] = false;
}
int stack_head = -1;
int proc_head = -1;
for (int start = 0; start != _res_node_num; ++start) {
if (reached[start]) continue;
// New start node
reached[start] = true;
pred_arc[start] = -1;
pred_node[start] = -1;
// Find the first admissible outgoing arc
double p = pi[start];
int a = _first_out[start];
int last_out = _first_out[start+1];
for (; a != last_out && (_res_cap[a] == 0 ||
!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
if (a == last_out) {
processed[start] = true;
proc_vector[++proc_head] = start;
continue;
}
stack[++stack_head] = a;
while (stack_head >= 0) {
int sa = stack[stack_head];
int u = _source[sa];
int v = _target[sa];
if (!reached[v]) {
// A new node is reached
reached[v] = true;
pred_node[v] = u;
pred_arc[v] = sa;
p = pi[v];
a = _first_out[v];
last_out = _first_out[v+1];
for (; a != last_out && (_res_cap[a] == 0 ||
!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
stack[++stack_head] = a == last_out ? -1 : a;
} else {
if (!processed[v]) {
// A cycle is found
int n, w = u;
Value d, delta = _res_cap[sa];
for (n = u; n != v; n = pred_node[n]) {
d = _res_cap[pred_arc[n]];
if (d <= delta) {
delta = d;
w = pred_node[n];
}
}
// Augment along the cycle
_res_cap[sa] -= delta;
_res_cap[_reverse[sa]] += delta;
for (n = u; n != v; n = pred_node[n]) {
int pa = pred_arc[n];
_res_cap[pa] -= delta;
_res_cap[_reverse[pa]] += delta;
}
for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
--stack_head;
reached[n] = false;
}
u = w;
}
v = u;
// Find the next admissible outgoing arc
p = pi[v];
a = stack[stack_head] + 1;
last_out = _first_out[v+1];
for (; a != last_out && (_res_cap[a] == 0 ||
!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
stack[stack_head] = a == last_out ? -1 : a;
}
while (stack_head >= 0 && stack[stack_head] == -1) {
processed[v] = true;
proc_vector[++proc_head] = v;
if (--stack_head >= 0) {
// Find the next admissible outgoing arc
v = _source[stack[stack_head]];
p = pi[v];
a = stack[stack_head] + 1;
last_out = _first_out[v+1];
for (; a != last_out && (_res_cap[a] == 0 ||
!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
stack[stack_head] = a == last_out ? -1 : a;
}
}
}
}
// Tighten potentials and epsilon
if (--iter > 0) {
for (int u = 0; u != _res_node_num; ++u) {
level[u] = 0;
}
for (int i = proc_head; i > 0; --i) {
int u = proc_vector[i];
double p = pi[u];
int l = level[u] + 1;
int last_out = _first_out[u+1];
for (int a = _first_out[u]; a != last_out; ++a) {
int v = _target[a];
if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
l > level[v]) level[v] = l;
}
}
// Modify potentials
double q = std::numeric_limits<double>::max();
for (int u = 0; u != _res_node_num; ++u) {
int lu = level[u];
double p, pu = pi[u];
int last_out = _first_out[u+1];
for (int a = _first_out[u]; a != last_out; ++a) {
if (_res_cap[a] == 0) continue;
int v = _target[a];
int ld = lu - level[v];
if (ld > 0) {
p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
if (p < q) q = p;
}
}
}
for (int u = 0; u != _res_node_num; ++u) {
pi[u] -= q * level[u];
}
// Modify epsilon
epsilon = 0;
for (int u = 0; u != _res_node_num; ++u) {
double curr, pu = pi[u];
int last_out = _first_out[u+1];
for (int a = _first_out[u]; a != last_out; ++a) {
if (_res_cap[a] == 0) continue;
curr = _cost[a] + pu - pi[_target[a]];
if (-curr > epsilon) epsilon = -curr;
}
}
} else {
typedef Howard<StaticDigraph, CostArcMap> MMC;
typedef typename BellmanFord<StaticDigraph, CostArcMap>
::template SetDistMap<CostNodeMap>::Create BF;
// Set epsilon to the minimum cycle mean
buildResidualNetwork();
MMC mmc(_sgr, _cost_map);
mmc.findMinMean();
epsilon = -mmc.cycleMean();
Cost cycle_cost = mmc.cycleLength();
int cycle_size = mmc.cycleArcNum();
// Compute feasible potentials for the current epsilon
for (int i = 0; i != int(_cost_vec.size()); ++i) {
_cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
}
BF bf(_sgr, _cost_map);
bf.distMap(_pi_map);
bf.init(0);
bf.start();
for (int u = 0; u != _res_node_num; ++u) {
pi[u] = static_cast<double>(_pi[u]) / cycle_size;
}
iter = limit;
}
}
}
}; //class CycleCanceling
///@}
} //namespace lemon
#endif //LEMON_CYCLE_CANCELING_H