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

## 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 be a digraph, denote the lower and upper bounds on the arcs, for which holds for all , and denotes the signed supply values of the nodes. If , then is a supply node with supply, if , then is a demand node with demand. A feasible circulation is an solution of the following problem.

The sum of the supply values, i.e. 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 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

  GR | The type of the digraph the algorithm runs on.
  LM | The type of the lower bound map. The default map type is GR::ArcMap<int>.
  UM | The type of the upper bound (capacity) map. The default map type is `LM`.
  SM | The type of the supply map. The default map type is GR::NodeMap<UM::Value>.
  TR | The 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. More...
| ~Circulation ()
| Destructor.
Circulation & | lowerMap (const LowerMap &map)
| Sets the lower bound map. More...
Circulation & | upperMap (const UpperMap &map)
| Sets the upper bound (capacity) map. More...
Circulation & | supplyMap (const SupplyMap &map)
| Sets the supply map. More...
Circulation & | flowMap (FlowMap &map)
| Sets the flow map. More...
Circulation & | elevator (Elevator &elevator)
| Sets the elevator used by algorithm. More...
const Elevator & | elevator () const
| Returns a const reference to the elevator. More...
Circulation & | tolerance (const Tolerance &tolerance)
| Sets the tolerance used by the algorithm. More...
const Tolerance & | tolerance () const
| Returns a const reference to the tolerance. More...
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. More...
void | greedyInit ()
| Initializes the internal data structures using a greedy approach. More...
bool | start ()
| Executes the algorithm. More...
bool | run ()
| Runs the algorithm. More...
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. More...
const FlowMap & | flowMap () const
| Returns a const reference to the flow map. More...
bool | barrier (const Node &node) const
| Returns `true` if the given node is in a barrier. More...
template<class BarrierMap >
void | barrierMap (BarrierMap &bar) const
| Gives back a barrier. More...
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. More...
bool | checkBarrier () const
| Check whether or not the last execution provides a barrier. More...

## Constructor & Destructor Documentation

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

The constructor of the class.

Parameters

  graph | The digraph the algorithm runs on.
  lower | The lower bounds for the flow values on the arcs.
  upper | The upper bounds (capacities) for the flow values on the arcs.
  supply | The 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

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()