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Circulation< GR, LM, UM, SM, TR > Class Template Reference

Detailed Description

template<typename GR, typename LM, typename UM, typename SM, typename TR>
class lemon::Circulation< GR, LM, UM, SM, TR >

This class implements a push-relabel algorithm for the network circulation problem. It is to find a feasible circulation when lower and upper bounds are given for the flow values on the arcs and lower bounds are given for the difference between the outgoing and incoming flow at the nodes.

The exact formulation of this problem is the following. Let \(G=(V,A)\) be a digraph, \(lower: A\rightarrow\mathbf{R}\) \(upper: A\rightarrow\mathbf{R}\cup\{\infty\}\) denote the lower and upper bounds on the arcs, for which \(lower(uv) \leq upper(uv)\) holds for all \(uv\in A\), and \(sup: V\rightarrow\mathbf{R}\) denotes the signed supply values of the nodes. If \(sup(u)>0\), then \(u\) is a supply node with \(sup(u)\) supply, if \(sup(u)<0\), then \(u\) is a demand node with \(-sup(u)\) demand. A feasible circulation is an \(f: A\rightarrow\mathbf{R}\) solution of the following problem.

\[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq sup(u) \quad \forall u\in V, \]

\[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A. \]

The sum of the supply values, i.e. \(\sum_{u\in V} sup(u)\) must be zero or negative in order to have a feasible solution (since the sum of the expressions on the left-hand side of the inequalities is zero). It means that the total demand must be greater or equal to the total supply and all the supplies have to be carried out from the supply nodes, but there could be demands that are not satisfied. If \(\sum_{u\in V} sup(u)\) is zero, then all the supply/demand constraints have to be satisfied with equality, i.e. all demands have to be satisfied and all supplies have to be used.

If you need the opposite inequalities in the supply/demand constraints (i.e. the total demand is less than the total supply and all the demands have to be satisfied while there could be supplies that are not used), then you could easily transform the problem to the above form by reversing the direction of the arcs and taking the negative of the supply values (e.g. using ReverseDigraph and NegMap adaptors).

This algorithm either calculates a feasible circulation, or provides a barrier, which prooves that a feasible soultion cannot exist.

Note that this algorithm also provides a feasible solution for the minimum cost flow problem.

Template Parameters
GRThe type of the digraph the algorithm runs on.
LMThe type of the lower bound map. The default map type is GR::ArcMap<int>.
UMThe type of the upper bound (capacity) map. The default map type is LM.
SMThe type of the supply map. The default map type is GR::NodeMap<UM::Value>.
TRThe traits class that defines various types used by the algorithm. By default, it is CirculationDefaultTraits<GR, LM, UM, SM>. In most cases, this parameter should not be set directly, consider to use the named template parameters instead.

#include <lemon/circulation.h>

Classes

struct  SetElevator
 Named parameter for setting Elevator type More...
 
struct  SetFlowMap
 Named parameter for setting FlowMap type More...
 
struct  SetStandardElevator
 Named parameter for setting Elevator type with automatic allocation More...
 

Public Types

typedef TR Traits
 The traits class of the algorithm.
 
typedef Traits::Digraph Digraph
 The type of the digraph the algorithm runs on.
 
typedef Traits::Value Value
 The type of the flow and supply values.
 
typedef Traits::LowerMap LowerMap
 The type of the lower bound map.
 
typedef Traits::UpperMap UpperMap
 The type of the upper bound (capacity) map.
 
typedef Traits::SupplyMap SupplyMap
 The type of the supply map.
 
typedef Traits::FlowMap FlowMap
 The type of the flow map.
 
typedef Traits::Elevator Elevator
 The type of the elevator.
 
typedef Traits::Tolerance Tolerance
 The type of the tolerance.
 

Public Member Functions

 Circulation (const Digraph &graph, const LowerMap &lower, const UpperMap &upper, const SupplyMap &supply)
 Constructor.
 
 ~Circulation ()
 Destructor.
 
CirculationlowerMap (const LowerMap &map)
 Sets the lower bound map.
 
CirculationupperMap (const UpperMap &map)
 Sets the upper bound (capacity) map.
 
CirculationsupplyMap (const SupplyMap &map)
 Sets the supply map.
 
CirculationflowMap (FlowMap &map)
 Sets the flow map.
 
Circulationelevator (Elevator &elevator)
 Sets the elevator used by algorithm.
 
const Elevatorelevator () const
 Returns a const reference to the elevator.
 
Circulationtolerance (const Tolerance &tolerance)
 Sets the tolerance used by the algorithm.
 
const Tolerancetolerance () const
 Returns a const reference to the tolerance.
 
Execution Control

The simplest way to execute the algorithm is to call run().
If you need better control on the initial solution or the execution, you have to call one of the init() functions first, then the start() function.

void init ()
 Initializes the internal data structures.
 
void greedyInit ()
 Initializes the internal data structures using a greedy approach.
 
bool start ()
 Executes the algorithm.
 
bool run ()
 Runs the algorithm.
 
Query Functions

The results of the circulation algorithm can be obtained using these functions.
Either run() or start() should be called before using them.

Value flow (const Arc &arc) const
 Returns the flow value on the given arc.
 
const FlowMapflowMap () const
 Returns a const reference to the flow map.
 
bool barrier (const Node &node) const
 Returns true if the given node is in a barrier.
 
template<class BarrierMap >
void barrierMap (BarrierMap &bar) const
 Gives back a barrier.
 
Checker Functions

The feasibility of the results can be checked using these functions.
Either run() or start() should be called before using them.

bool checkFlow () const
 Check if the found flow is a feasible circulation.
 
bool checkBarrier () const
 Check whether or not the last execution provides a barrier.
 

Constructor & Destructor Documentation

Circulation ( const Digraph graph,
const LowerMap lower,
const UpperMap upper,
const SupplyMap supply 
)
inline

The constructor of the class.

Parameters
graphThe digraph the algorithm runs on.
lowerThe lower bounds for the flow values on the arcs.
upperThe upper bounds (capacities) for the flow values on the arcs.
supplyThe signed supply values of the nodes.

Member Function Documentation

Circulation& lowerMap ( const LowerMap map)
inline

Sets the lower bound map.

Returns
(*this)
Circulation& upperMap ( const UpperMap map)
inline

Sets the upper bound (capacity) map.

Returns
(*this)
Circulation& supplyMap ( const SupplyMap map)
inline

Sets the supply map.

Returns
(*this)
Circulation& flowMap ( FlowMap map)
inline

Sets the flow map. If you don't use this function before calling run() or init(), an instance will be allocated automatically. The destructor deallocates this automatically allocated map, of course.

Returns
(*this)
Circulation& elevator ( Elevator elevator)
inline

Sets the elevator used by algorithm. If you don't use this function before calling run() or init(), an instance will be allocated automatically. The destructor deallocates this automatically allocated elevator, of course.

Returns
(*this)
const Elevator& elevator ( ) const
inline

Returns a const reference to the elevator.

Precondition
Either run() or init() must be called before using this function.
Circulation& tolerance ( const Tolerance tolerance)
inline

Sets the tolerance object used by the algorithm.

Returns
(*this)
const Tolerance& tolerance ( ) const
inline

Returns a const reference to the tolerance object used by the algorithm.

void init ( )
inline

Initializes the internal data structures and sets all flow values to the lower bound.

void greedyInit ( )
inline

Initializes the internal data structures using a greedy approach to construct the initial solution.

bool start ( )
inline

This function executes the algorithm.

Returns
true if a feasible circulation is found.
See Also
barrier()
barrierMap()
bool run ( )
inline

This function runs the algorithm.

Returns
true if a feasible circulation is found.
Note
Apart from the return value, c.run() is just a shortcut of the following code.
c.greedyInit();
c.start();
Value flow ( const Arc &  arc) const
inline

Returns the flow value on the given arc.

Precondition
Either run() or init() must be called before using this function.
const FlowMap& flowMap ( ) const
inline

Returns a const reference to the arc map storing the found flow.

Precondition
Either run() or init() must be called before using this function.
bool barrier ( const Node &  node) const
inline

Barrier is a set B of nodes for which

\[ \sum_{uv\in A: u\in B} upper(uv) - \sum_{uv\in A: v\in B} lower(uv) < \sum_{v\in B} sup(v) \]

holds. The existence of a set with this property prooves that a feasible circualtion cannot exist.

This function returns true if the given node is in the found barrier. If a feasible circulation is found, the function gives back false for every node.

Precondition
Either run() or init() must be called before using this function.
See Also
barrierMap()
checkBarrier()
void barrierMap ( BarrierMap &  bar) const
inline

This function sets bar to the characteristic vector of the found barrier. bar should be a writable node map with bool (or convertible) value type.

If a feasible circulation is found, the function gives back an empty set, so bar[v] will be false for all nodes v.

Note
This function calls barrier() for each node, so it runs in O(n) time.
Precondition
Either run() or init() must be called before using this function.
See Also
barrier()
checkBarrier()
bool checkFlow ( ) const
inline

Check if the found flow is a feasible circulation,

bool checkBarrier ( ) const
inline

Check whether or not the last execution provides a barrier.

See Also
barrier()
barrierMap()