1 | /* -*- C++ -*- |
---|
2 | * |
---|
3 | * This file is a part of LEMON, a generic C++ optimization library |
---|
4 | * |
---|
5 | * Copyright (C) 2003-2007 |
---|
6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
---|
7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
---|
8 | * |
---|
9 | * Permission to use, modify and distribute this software is granted |
---|
10 | * provided that this copyright notice appears in all copies. For |
---|
11 | * precise terms see the accompanying LICENSE file. |
---|
12 | * |
---|
13 | * This software is provided "AS IS" with no warranty of any kind, |
---|
14 | * express or implied, and with no claim as to its suitability for any |
---|
15 | * purpose. |
---|
16 | * |
---|
17 | */ |
---|
18 | |
---|
19 | #ifndef LEMON_SSP_MIN_COST_FLOW_H |
---|
20 | #define LEMON_SSP_MIN_COST_FLOW_H |
---|
21 | |
---|
22 | ///\ingroup min_cost_flow |
---|
23 | /// |
---|
24 | ///\file |
---|
25 | ///\brief An algorithm for finding a flow of value \c k (for |
---|
26 | ///small values of \c k) having minimal total cost |
---|
27 | |
---|
28 | |
---|
29 | #include <lemon/dijkstra.h> |
---|
30 | #include <lemon/graph_adaptor.h> |
---|
31 | #include <lemon/maps.h> |
---|
32 | #include <vector> |
---|
33 | |
---|
34 | namespace lemon { |
---|
35 | |
---|
36 | /// \addtogroup min_cost_flow |
---|
37 | /// @{ |
---|
38 | |
---|
39 | /// \brief Implementation of an algorithm for finding a flow of |
---|
40 | /// value \c k (for small values of \c k) having minimal total cost |
---|
41 | /// between two nodes |
---|
42 | /// |
---|
43 | /// |
---|
44 | /// The \ref lemon::SspMinCostFlow "Successive Shortest Path Minimum |
---|
45 | /// Cost Flow" implements an algorithm for finding a flow of value |
---|
46 | /// \c k having minimal total cost from a given source node to a |
---|
47 | /// given target node in a directed graph with a cost function on |
---|
48 | /// the edges. To this end, the edge-capacities and edge-costs have |
---|
49 | /// to be nonnegative. The edge-capacities should be integers, but |
---|
50 | /// the edge-costs can be integers, reals or of other comparable |
---|
51 | /// numeric type. This algorithm is intended to be used only for |
---|
52 | /// small values of \c k, since it is only polynomial in k, not in |
---|
53 | /// the length of k (which is log k): in order to find the minimum |
---|
54 | /// cost flow of value \c k it finds the minimum cost flow of value |
---|
55 | /// \c i for every \c i between 0 and \c k. |
---|
56 | /// |
---|
57 | ///\param Graph The directed graph type the algorithm runs on. |
---|
58 | ///\param LengthMap The type of the length map. |
---|
59 | ///\param CapacityMap The capacity map type. |
---|
60 | /// |
---|
61 | ///\author Attila Bernath |
---|
62 | template <typename Graph, typename LengthMap, typename CapacityMap> |
---|
63 | class SspMinCostFlow { |
---|
64 | |
---|
65 | typedef typename LengthMap::Value Length; |
---|
66 | |
---|
67 | //Warning: this should be integer type |
---|
68 | typedef typename CapacityMap::Value Capacity; |
---|
69 | |
---|
70 | typedef typename Graph::Node Node; |
---|
71 | typedef typename Graph::NodeIt NodeIt; |
---|
72 | typedef typename Graph::Edge Edge; |
---|
73 | typedef typename Graph::OutEdgeIt OutEdgeIt; |
---|
74 | typedef typename Graph::template EdgeMap<int> EdgeIntMap; |
---|
75 | |
---|
76 | typedef ResGraphAdaptor<const Graph,int,CapacityMap,EdgeIntMap> ResGW; |
---|
77 | typedef typename ResGW::Edge ResGraphEdge; |
---|
78 | |
---|
79 | protected: |
---|
80 | |
---|
81 | const Graph& g; |
---|
82 | const LengthMap& length; |
---|
83 | const CapacityMap& capacity; |
---|
84 | |
---|
85 | EdgeIntMap flow; |
---|
86 | typedef typename Graph::template NodeMap<Length> PotentialMap; |
---|
87 | PotentialMap potential; |
---|
88 | |
---|
89 | Node s; |
---|
90 | Node t; |
---|
91 | |
---|
92 | Length total_length; |
---|
93 | |
---|
94 | class ModLengthMap { |
---|
95 | typedef typename Graph::template NodeMap<Length> NodeMap; |
---|
96 | const ResGW& g; |
---|
97 | const LengthMap &length; |
---|
98 | const NodeMap &pot; |
---|
99 | public : |
---|
100 | typedef typename LengthMap::Key Key; |
---|
101 | typedef typename LengthMap::Value Value; |
---|
102 | |
---|
103 | ModLengthMap(const ResGW& _g, |
---|
104 | const LengthMap &_length, const NodeMap &_pot) : |
---|
105 | g(_g), /*rev(_rev),*/ length(_length), pot(_pot) { } |
---|
106 | |
---|
107 | Value operator[](typename ResGW::Edge e) const { |
---|
108 | if (g.forward(e)) |
---|
109 | return length[e]-(pot[g.target(e)]-pot[g.source(e)]); |
---|
110 | else |
---|
111 | return -length[e]-(pot[g.target(e)]-pot[g.source(e)]); |
---|
112 | } |
---|
113 | |
---|
114 | }; //ModLengthMap |
---|
115 | |
---|
116 | ResGW res_graph; |
---|
117 | ModLengthMap mod_length; |
---|
118 | Dijkstra<ResGW, ModLengthMap> dijkstra; |
---|
119 | |
---|
120 | public : |
---|
121 | |
---|
122 | /// \brief The constructor of the class. |
---|
123 | /// |
---|
124 | /// \param _g The directed graph the algorithm runs on. |
---|
125 | /// \param _length The length (cost) of the edges. |
---|
126 | /// \param _cap The capacity of the edges. |
---|
127 | /// \param _s Source node. |
---|
128 | /// \param _t Target node. |
---|
129 | SspMinCostFlow(Graph& _g, LengthMap& _length, CapacityMap& _cap, |
---|
130 | Node _s, Node _t) : |
---|
131 | g(_g), length(_length), capacity(_cap), flow(_g), potential(_g), |
---|
132 | s(_s), t(_t), |
---|
133 | res_graph(g, capacity, flow), |
---|
134 | mod_length(res_graph, length, potential), |
---|
135 | dijkstra(res_graph, mod_length) { |
---|
136 | reset(); |
---|
137 | } |
---|
138 | |
---|
139 | /// \brief Tries to augment the flow between s and t by 1. The |
---|
140 | /// return value shows if the augmentation is successful. |
---|
141 | bool augment() { |
---|
142 | dijkstra.run(s); |
---|
143 | if (!dijkstra.reached(t)) { |
---|
144 | |
---|
145 | //Unsuccessful augmentation. |
---|
146 | return false; |
---|
147 | } else { |
---|
148 | |
---|
149 | //We have to change the potential |
---|
150 | for(typename ResGW::NodeIt n(res_graph); n!=INVALID; ++n) |
---|
151 | potential.set(n, potential[n]+dijkstra.distMap()[n]); |
---|
152 | |
---|
153 | //Augmenting on the shortest path |
---|
154 | Node n=t; |
---|
155 | ResGraphEdge e; |
---|
156 | while (n!=s){ |
---|
157 | e = dijkstra.predEdge(n); |
---|
158 | n = dijkstra.predNode(n); |
---|
159 | res_graph.augment(e,1); |
---|
160 | //Let's update the total length |
---|
161 | if (res_graph.forward(e)) |
---|
162 | total_length += length[e]; |
---|
163 | else |
---|
164 | total_length -= length[e]; |
---|
165 | } |
---|
166 | |
---|
167 | return true; |
---|
168 | } |
---|
169 | } |
---|
170 | |
---|
171 | /// \brief Runs the algorithm. |
---|
172 | /// |
---|
173 | /// Runs the algorithm. |
---|
174 | /// Returns k if there is a flow of value at least k from s to t. |
---|
175 | /// Otherwise it returns the maximum value of a flow from s to t. |
---|
176 | /// |
---|
177 | /// \param k The value of the flow we are looking for. |
---|
178 | /// |
---|
179 | /// \todo May be it does make sense to be able to start with a |
---|
180 | /// nonzero feasible primal-dual solution pair as well. |
---|
181 | /// |
---|
182 | /// \todo If the current flow value is bigger than k, then |
---|
183 | /// everything is cleared and the algorithm starts from zero |
---|
184 | /// flow. Is it a good approach? |
---|
185 | int run(int k) { |
---|
186 | if (flowValue()>k) reset(); |
---|
187 | while (flowValue()<k && augment()) { } |
---|
188 | return flowValue(); |
---|
189 | } |
---|
190 | |
---|
191 | /// \brief The class is reset to zero flow and potential. |
---|
192 | void reset() { |
---|
193 | total_length=0; |
---|
194 | for (typename Graph::EdgeIt e(g); e!=INVALID; ++e) flow.set(e, 0); |
---|
195 | for (typename Graph::NodeIt n(g); n!=INVALID; ++n) potential.set(n, 0); |
---|
196 | } |
---|
197 | |
---|
198 | /// \brief Returns the value of the current flow. |
---|
199 | int flowValue() const { |
---|
200 | int i=0; |
---|
201 | for (typename Graph::OutEdgeIt e(g, s); e!=INVALID; ++e) i+=flow[e]; |
---|
202 | for (typename Graph::InEdgeIt e(g, s); e!=INVALID; ++e) i-=flow[e]; |
---|
203 | return i; |
---|
204 | } |
---|
205 | |
---|
206 | /// \brief Total cost of the found flow. |
---|
207 | /// |
---|
208 | /// This function gives back the total cost of the found flow. |
---|
209 | Length totalLength(){ |
---|
210 | return total_length; |
---|
211 | } |
---|
212 | |
---|
213 | /// \brief Returns a const reference to the EdgeMap \c flow. |
---|
214 | /// |
---|
215 | /// Returns a const reference to the EdgeMap \c flow. |
---|
216 | const EdgeIntMap &getFlow() const { return flow;} |
---|
217 | |
---|
218 | /// \brief Returns a const reference to the NodeMap \c potential |
---|
219 | /// (the dual solution). |
---|
220 | /// |
---|
221 | /// Returns a const reference to the NodeMap \c potential (the |
---|
222 | /// dual solution). |
---|
223 | const PotentialMap &getPotential() const { return potential;} |
---|
224 | |
---|
225 | /// \brief Checking the complementary slackness optimality criteria. |
---|
226 | /// |
---|
227 | /// This function checks, whether the given flow and potential |
---|
228 | /// satisfy the complementary slackness conditions (i.e. these are optimal). |
---|
229 | /// This function only checks optimality, doesn't bother with feasibility. |
---|
230 | /// For testing purpose. |
---|
231 | bool checkComplementarySlackness(){ |
---|
232 | Length mod_pot; |
---|
233 | Length fl_e; |
---|
234 | for(typename Graph::EdgeIt e(g); e!=INVALID; ++e) { |
---|
235 | //C^{\Pi}_{i,j} |
---|
236 | mod_pot = length[e]-potential[g.target(e)]+potential[g.source(e)]; |
---|
237 | fl_e = flow[e]; |
---|
238 | if (0<fl_e && fl_e<capacity[e]) { |
---|
239 | /// \todo better comparison is needed for real types, moreover, |
---|
240 | /// this comparison here is superfluous. |
---|
241 | if (mod_pot != 0) |
---|
242 | return false; |
---|
243 | } |
---|
244 | else { |
---|
245 | if (mod_pot > 0 && fl_e != 0) |
---|
246 | return false; |
---|
247 | if (mod_pot < 0 && fl_e != capacity[e]) |
---|
248 | return false; |
---|
249 | } |
---|
250 | } |
---|
251 | return true; |
---|
252 | } |
---|
253 | |
---|
254 | }; //class SspMinCostFlow |
---|
255 | |
---|
256 | ///@} |
---|
257 | |
---|
258 | } //namespace lemon |
---|
259 | |
---|
260 | #endif //LEMON_SSP_MIN_COST_FLOW_H |
---|