[899] | 1 | // -*- c++ -*- |
---|
[901] | 2 | #ifndef HUGO_MIN_COST_FLOW_H |
---|
| 3 | #define HUGO_MIN_COST_FLOW_H |
---|
[899] | 4 | |
---|
| 5 | ///\ingroup flowalgs |
---|
| 6 | ///\file |
---|
| 7 | ///\brief An algorithm for finding a flow of value \c k (for small values of \c k) having minimal total cost |
---|
| 8 | |
---|
| 9 | |
---|
| 10 | #include <hugo/dijkstra.h> |
---|
| 11 | #include <hugo/graph_wrapper.h> |
---|
| 12 | #include <hugo/maps.h> |
---|
| 13 | #include <vector> |
---|
| 14 | |
---|
| 15 | namespace hugo { |
---|
| 16 | |
---|
| 17 | /// \addtogroup flowalgs |
---|
| 18 | /// @{ |
---|
| 19 | |
---|
| 20 | ///\brief Implementation of an algorithm for finding a flow of value \c k |
---|
| 21 | ///(for small values of \c k) having minimal total cost between 2 nodes |
---|
| 22 | /// |
---|
| 23 | /// |
---|
| 24 | /// The class \ref hugo::MinCostFlow "MinCostFlow" implements |
---|
| 25 | /// an algorithm for finding a flow of value \c k |
---|
| 26 | /// having minimal total cost |
---|
| 27 | /// from a given source node to a given target node in an |
---|
| 28 | /// edge-weighted directed graph. To this end, |
---|
| 29 | /// the edge-capacities and edge-weitghs have to be nonnegative. |
---|
| 30 | /// The edge-capacities should be integers, but the edge-weights can be |
---|
| 31 | /// integers, reals or of other comparable numeric type. |
---|
| 32 | /// This algorithm is intended to use only for small values of \c k, |
---|
| 33 | /// since it is only polynomial in k, |
---|
| 34 | /// not in the length of k (which is log k). |
---|
| 35 | /// In order to find the minimum cost flow of value \c k it |
---|
| 36 | /// finds the minimum cost flow of value \c i for every |
---|
| 37 | /// \c i between 0 and \c k. |
---|
| 38 | /// |
---|
| 39 | ///\param Graph The directed graph type the algorithm runs on. |
---|
| 40 | ///\param LengthMap The type of the length map. |
---|
| 41 | ///\param CapacityMap The capacity map type. |
---|
| 42 | /// |
---|
| 43 | ///\author Attila Bernath |
---|
| 44 | template <typename Graph, typename LengthMap, typename CapacityMap> |
---|
| 45 | class MinCostFlow { |
---|
| 46 | |
---|
| 47 | typedef typename LengthMap::ValueType Length; |
---|
| 48 | |
---|
| 49 | //Warning: this should be integer type |
---|
| 50 | typedef typename CapacityMap::ValueType Capacity; |
---|
| 51 | |
---|
| 52 | typedef typename Graph::Node Node; |
---|
| 53 | typedef typename Graph::NodeIt NodeIt; |
---|
| 54 | typedef typename Graph::Edge Edge; |
---|
| 55 | typedef typename Graph::OutEdgeIt OutEdgeIt; |
---|
| 56 | typedef typename Graph::template EdgeMap<int> EdgeIntMap; |
---|
| 57 | |
---|
| 58 | |
---|
| 59 | typedef ResGraphWrapper<const Graph,int,CapacityMap,EdgeIntMap> ResGraphType; |
---|
| 60 | typedef typename ResGraphType::Edge ResGraphEdge; |
---|
| 61 | |
---|
| 62 | class ModLengthMap { |
---|
| 63 | typedef typename Graph::template NodeMap<Length> NodeMap; |
---|
| 64 | const ResGraphType& G; |
---|
| 65 | const LengthMap &ol; |
---|
| 66 | const NodeMap &pot; |
---|
| 67 | public : |
---|
| 68 | typedef typename LengthMap::KeyType KeyType; |
---|
| 69 | typedef typename LengthMap::ValueType ValueType; |
---|
| 70 | |
---|
| 71 | ValueType operator[](typename ResGraphType::Edge e) const { |
---|
| 72 | if (G.forward(e)) |
---|
| 73 | return ol[e]-(pot[G.head(e)]-pot[G.tail(e)]); |
---|
| 74 | else |
---|
| 75 | return -ol[e]-(pot[G.head(e)]-pot[G.tail(e)]); |
---|
| 76 | } |
---|
| 77 | |
---|
| 78 | ModLengthMap(const ResGraphType& _G, |
---|
| 79 | const LengthMap &o, const NodeMap &p) : |
---|
| 80 | G(_G), /*rev(_rev),*/ ol(o), pot(p){}; |
---|
| 81 | };//ModLengthMap |
---|
| 82 | |
---|
| 83 | |
---|
| 84 | protected: |
---|
| 85 | |
---|
| 86 | //Input |
---|
| 87 | const Graph& G; |
---|
| 88 | const LengthMap& length; |
---|
| 89 | const CapacityMap& capacity; |
---|
| 90 | |
---|
| 91 | |
---|
| 92 | //auxiliary variables |
---|
| 93 | |
---|
| 94 | //To store the flow |
---|
| 95 | EdgeIntMap flow; |
---|
| 96 | //To store the potential (dual variables) |
---|
| 97 | typedef typename Graph::template NodeMap<Length> PotentialMap; |
---|
| 98 | PotentialMap potential; |
---|
| 99 | |
---|
| 100 | |
---|
| 101 | Length total_length; |
---|
| 102 | |
---|
| 103 | |
---|
| 104 | public : |
---|
| 105 | |
---|
| 106 | /// The constructor of the class. |
---|
| 107 | |
---|
| 108 | ///\param _G The directed graph the algorithm runs on. |
---|
| 109 | ///\param _length The length (weight or cost) of the edges. |
---|
| 110 | ///\param _cap The capacity of the edges. |
---|
| 111 | MinCostFlow(Graph& _G, LengthMap& _length, CapacityMap& _cap) : G(_G), |
---|
| 112 | length(_length), capacity(_cap), flow(_G), potential(_G){ } |
---|
| 113 | |
---|
| 114 | |
---|
| 115 | ///Runs the algorithm. |
---|
| 116 | |
---|
| 117 | ///Runs the algorithm. |
---|
| 118 | ///Returns k if there is a flow of value at least k edge-disjoint |
---|
| 119 | ///from s to t. |
---|
| 120 | ///Otherwise it returns the maximum value of a flow from s to t. |
---|
| 121 | /// |
---|
| 122 | ///\param s The source node. |
---|
| 123 | ///\param t The target node. |
---|
| 124 | ///\param k The value of the flow we are looking for. |
---|
| 125 | /// |
---|
| 126 | ///\todo May be it does make sense to be able to start with a nonzero |
---|
| 127 | /// feasible primal-dual solution pair as well. |
---|
| 128 | int run(Node s, Node t, int k) { |
---|
| 129 | |
---|
| 130 | //Resetting variables from previous runs |
---|
| 131 | total_length = 0; |
---|
| 132 | |
---|
| 133 | for (typename Graph::EdgeIt e(G); e!=INVALID; ++e) flow.set(e, 0); |
---|
| 134 | |
---|
| 135 | //Initialize the potential to zero |
---|
| 136 | for (typename Graph::NodeIt n(G); n!=INVALID; ++n) potential.set(n, 0); |
---|
| 137 | |
---|
| 138 | |
---|
| 139 | //We need a residual graph |
---|
| 140 | ResGraphType res_graph(G, capacity, flow); |
---|
| 141 | |
---|
| 142 | |
---|
| 143 | ModLengthMap mod_length(res_graph, length, potential); |
---|
| 144 | |
---|
| 145 | Dijkstra<ResGraphType, ModLengthMap> dijkstra(res_graph, mod_length); |
---|
| 146 | |
---|
| 147 | int i; |
---|
| 148 | for (i=0; i<k; ++i){ |
---|
| 149 | dijkstra.run(s); |
---|
| 150 | if (!dijkstra.reached(t)){ |
---|
| 151 | //There are no flow of value k from s to t |
---|
| 152 | break; |
---|
| 153 | }; |
---|
| 154 | |
---|
| 155 | //We have to change the potential |
---|
| 156 | for(typename ResGraphType::NodeIt n(res_graph); n!=INVALID; ++n) |
---|
| 157 | potential[n] += dijkstra.distMap()[n]; |
---|
| 158 | |
---|
| 159 | |
---|
| 160 | //Augmenting on the sortest path |
---|
| 161 | Node n=t; |
---|
| 162 | ResGraphEdge e; |
---|
| 163 | while (n!=s){ |
---|
| 164 | e = dijkstra.pred(n); |
---|
| 165 | n = dijkstra.predNode(n); |
---|
| 166 | res_graph.augment(e,1); |
---|
| 167 | //Let's update the total length |
---|
| 168 | if (res_graph.forward(e)) |
---|
| 169 | total_length += length[e]; |
---|
| 170 | else |
---|
| 171 | total_length -= length[e]; |
---|
| 172 | } |
---|
| 173 | |
---|
| 174 | |
---|
| 175 | } |
---|
| 176 | |
---|
| 177 | |
---|
| 178 | return i; |
---|
| 179 | } |
---|
| 180 | |
---|
| 181 | |
---|
| 182 | |
---|
| 183 | /// Gives back the total weight of the found flow. |
---|
| 184 | |
---|
| 185 | ///This function gives back the total weight of the found flow. |
---|
| 186 | ///Assumes that \c run() has been run and nothing changed since then. |
---|
| 187 | Length totalLength(){ |
---|
| 188 | return total_length; |
---|
| 189 | } |
---|
| 190 | |
---|
| 191 | ///Returns a const reference to the EdgeMap \c flow. |
---|
| 192 | |
---|
| 193 | ///Returns a const reference to the EdgeMap \c flow. |
---|
| 194 | ///\pre \ref run() must |
---|
| 195 | ///be called before using this function. |
---|
| 196 | const EdgeIntMap &getFlow() const { return flow;} |
---|
| 197 | |
---|
| 198 | ///Returns a const reference to the NodeMap \c potential (the dual solution). |
---|
| 199 | |
---|
| 200 | ///Returns a const reference to the NodeMap \c potential (the dual solution). |
---|
| 201 | /// \pre \ref run() must be called before using this function. |
---|
| 202 | const PotentialMap &getPotential() const { return potential;} |
---|
| 203 | |
---|
| 204 | /// Checking the complementary slackness optimality criteria |
---|
| 205 | |
---|
| 206 | ///This function checks, whether the given solution is optimal |
---|
| 207 | ///If executed after the call of \c run() then it should return with true. |
---|
| 208 | ///This function only checks optimality, doesn't bother with feasibility. |
---|
| 209 | ///It is meant for testing purposes. |
---|
| 210 | /// |
---|
| 211 | bool checkComplementarySlackness(){ |
---|
| 212 | Length mod_pot; |
---|
| 213 | Length fl_e; |
---|
| 214 | for(typename Graph::EdgeIt e(G); e!=INVALID; ++e) { |
---|
| 215 | //C^{\Pi}_{i,j} |
---|
| 216 | mod_pot = length[e]-potential[G.head(e)]+potential[G.tail(e)]; |
---|
| 217 | fl_e = flow[e]; |
---|
| 218 | if (0<fl_e && fl_e<capacity[e]) { |
---|
| 219 | /// \todo better comparison is needed for real types, moreover, |
---|
| 220 | /// this comparison here is superfluous. |
---|
| 221 | if (mod_pot != 0) |
---|
| 222 | return false; |
---|
| 223 | } |
---|
| 224 | else { |
---|
| 225 | if (mod_pot > 0 && fl_e != 0) |
---|
| 226 | return false; |
---|
| 227 | if (mod_pot < 0 && fl_e != capacity[e]) |
---|
| 228 | return false; |
---|
| 229 | } |
---|
| 230 | } |
---|
| 231 | return true; |
---|
| 232 | } |
---|
| 233 | |
---|
| 234 | |
---|
| 235 | }; //class MinCostFlow |
---|
| 236 | |
---|
| 237 | ///@} |
---|
| 238 | |
---|
| 239 | } //namespace hugo |
---|
| 240 | |
---|
[901] | 241 | #endif //HUGO_MIN_COST_FLOW_H |
---|