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Parma_Polyhedra_Library::Polyhedron Class Reference

The base class for convex polyhedra. More...

Inherited by Parma_Polyhedra_Library::C_Polyhedron, and Parma_Polyhedra_Library::NNC_Polyhedron.

List of all members.

Public Types

enum  Degenerate_Kind { UNIVERSE, EMPTY }
 Kinds of degenerate polyhedra. More...

Public Member Functions

Member Functions that Do Not Modify the Polyhedron
dimension_type space_dimension () const
 Returns the dimension of the vector space enclosing *this.
const ConSys & constraints () const
 Returns the system of constraints.
const ConSys & minimized_constraints () const
 Returns the system of constraints, with no redundant constraint.
const GenSys & generators () const
 Returns the system of generators.
const GenSys & minimized_generators () const
 Returns the system of generators, with no redundant generator.
Poly_Con_Relation relation_with (const Constraint &c) const
 Returns the relations holding between the polyhedron *this and the constraint c.
Poly_Gen_Relation relation_with (const Generator &g) const
 Returns the relations holding between the polyhedron *this and the generator g.
bool is_empty () const
 Returns true if and only if *this is an empty polyhedron.
bool is_universe () const
 Returns true if and only if *this is a universe polyhedron.
bool is_topologically_closed () const
 Returns true if and only if *this is a topologically closed subset of the vector space.
bool is_disjoint_from (const Polyhedron &y) const
 Returns true if and only if *this and y are disjoint.
bool is_bounded () const
 Returns true if and only if *this is a bounded polyhedron.
bool bounds_from_above (const LinExpression &expr) const
 Returns true if and only if expr is bounded from above in *this.
bool bounds_from_below (const LinExpression &expr) const
 Returns true if and only if expr is bounded from below in *this.
bool maximize (const LinExpression &expr, Integer &sup_n, Integer &sup_d, bool &maximum) const
 Returns true if and only if *this is not empty and expr is bounded from above in *this, in which case the supremum value is computed.
bool maximize (const LinExpression &expr, Integer &sup_n, Integer &sup_d, bool &maximum, const Generator **const pppoint) const
 Returns true if and only if *this is not empty and expr is bounded from above in *this, in which case the supremum value and a point where expr reaches it are computed.
bool minimize (const LinExpression &expr, Integer &inf_n, Integer &inf_d, bool &minimum) const
 Returns true if and only if *this is not empty and expr is bounded from below in *this, in which case the infimum value is computed.
bool minimize (const LinExpression &expr, Integer &inf_n, Integer &inf_d, bool &minimum, const Generator **const pppoint) const
 Returns true if and only if *this is not empty and expr is bounded from below in *this, in which case the infimum value and a point where expr reaches it are computed.
bool contains (const Polyhedron &y) const
 Returns true if and only if *this contains y.
bool strictly_contains (const Polyhedron &y) const
 Returns true if and only if *this strictly contains y.
template<typename Box> void shrink_bounding_box (Box &box, Complexity_Class complexity=ANY) const
 Uses *this to shrink a generic, interval-based bounding box.
bool OK (bool check_not_empty=false) const
 Checks if all the invariants are satisfied.
Space-Dimension Preserving Member Functions that May Modify the Polyhedron
void add_constraint (const Constraint &c)
 Adds a copy of constraint c to the system of constraints of *this (without minimizing the result).
bool add_constraint_and_minimize (const Constraint &c)
 Adds a copy of constraint c to the system of constraints of *this, minimizing the result.
void add_generator (const Generator &g)
 Adds a copy of generator g to the system of generators of *this (without minimizing the result).
bool add_generator_and_minimize (const Generator &g)
 Adds a copy of generator g to the system of generators of *this, minimizing the result.
void add_constraints (const ConSys &cs)
 Adds a copy of the constraints in cs to the system of constraints of *this (without minimizing the result).
void add_recycled_constraints (ConSys &cs)
 Adds the constraints in cs to the system of constraints of *this (without minimizing the result).
bool add_constraints_and_minimize (const ConSys &cs)
 Adds a copy of the constraints in cs to the system of constraints of *this, minimizing the result.
bool add_recycled_constraints_and_minimize (ConSys &cs)
 Adds the constraints in cs to the system of constraints of *this, minimizing the result.
void add_generators (const GenSys &gs)
 Adds a copy of the generators in gs to the system of generators of *this (without minimizing the result).
void add_recycled_generators (GenSys &gs)
 Adds the generators in gs to the system of generators of *this (without minimizing the result).
bool add_generators_and_minimize (const GenSys &gs)
 Adds a copy of the generators in gs to the system of generators of *this, minimizing the result.
bool add_recycled_generators_and_minimize (GenSys &gs)
 Adds the generators in gs to the system of generators of *this, minimizing the result.
void intersection_assign (const Polyhedron &y)
 Assigns to *this the intersection of *this and y. The result is not guaranteed to be minimized.
bool intersection_assign_and_minimize (const Polyhedron &y)
 Assigns to *this the intersection of *this and y, minimizing the result.
void poly_hull_assign (const Polyhedron &y)
 Assigns to *this the poly-hull of *this and y. The result is not guaranteed to be minimized.
bool poly_hull_assign_and_minimize (const Polyhedron &y)
 Assigns to *this the poly-hull of *this and y, minimizing the result.
void poly_difference_assign (const Polyhedron &y)
 Assigns to *this the poly-difference of *this and y. The result is not guaranteed to be minimized.
void affine_image (Variable var, const LinExpression &expr, const Integer &denominator=Integer_one())
 Assigns to *this the affine image of *this under the function mapping variable var to the affine expression specified by expr and denominator.
void affine_preimage (Variable var, const LinExpression &expr, const Integer &denominator=Integer_one())
 Assigns to *this the affine preimage of *this under the function mapping variable var to the affine expression specified by expr and denominator.
void generalized_affine_image (Variable var, const Relation_Symbol relsym, const LinExpression &expr, const Integer &denominator=Integer_one())
 Assigns to *this the image of *this with respect to the generalized affine transfer function $\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$, where $\mathord{\relsym}$ is the relation symbol encoded by relsym.
void generalized_affine_image (const LinExpression &lhs, const Relation_Symbol relsym, const LinExpression &rhs)
 Assigns to *this the image of *this with respect to the generalized affine transfer function $\mathrm{lhs}' \relsym \mathrm{rhs}$, where $\mathord{\relsym}$ is the relation symbol encoded by relsym.
void time_elapse_assign (const Polyhedron &y)
 Assigns to *this the result of computing the time-elapse between *this and y.
void topological_closure_assign ()
 Assigns to *this its topological closure.
void BHRZ03_widening_assign (const Polyhedron &y, unsigned *tp=0)
 Assigns to *this the result of computing the BHRZ03-widening between *this and y.
void limited_BHRZ03_extrapolation_assign (const Polyhedron &y, const ConSys &cs, unsigned *tp=0)
 Improves the result of the BHRZ03-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this.
void bounded_BHRZ03_extrapolation_assign (const Polyhedron &y, const ConSys &cs, unsigned *tp=0)
 Improves the result of the BHRZ03-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this, plus all the constraints of the form $\pm x \leq r$ and $\pm x < r$, with $r \in \Qset$, that are satisfied by all the points of *this.
void H79_widening_assign (const Polyhedron &y, unsigned *tp=0)
 Assigns to *this the result of computing the H79-widening between *this and y.
void limited_H79_extrapolation_assign (const Polyhedron &y, const ConSys &cs, unsigned *tp=0)
 Improves the result of the H79-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this.
void bounded_H79_extrapolation_assign (const Polyhedron &y, const ConSys &cs, unsigned *tp=0)
 Improves the result of the H79-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this, plus all the constraints of the form $\pm x \leq r$ and $\pm x < r$, with $r \in \Qset$, that are satisfied by all the points of *this.
Member Functions that May Modify the Dimension of the Vector Space
void add_dimensions_and_embed (dimension_type m)
 Adds m new dimensions and embeds the old polyhedron in the new space.
void add_dimensions_and_project (dimension_type m)
 Adds m new dimensions to the polyhedron and does not embed it in the new space.
void concatenate_assign (const Polyhedron &y)
 Assigns to *this the concatenation of *this and y, taken in this order.
void remove_dimensions (const Variables_Set &to_be_removed)
 Removes all the specified dimensions.
void remove_higher_dimensions (dimension_type new_dimension)
 Removes the higher dimensions so that the resulting space will have dimension new_dimension.
template<typename PartialFunction> void map_dimensions (const PartialFunction &pfunc)
 Remaps the dimensions of the vector space according to a partial function.
void expand_dimension (Variable var, dimension_type m)
 Creates m copies of the dimension corresponding to var.
void fold_dimensions (const Variables_Set &to_be_folded, Variable var)
 Folds the dimensions in to_be_folded into var.
Miscellaneous Member Functions
 ~Polyhedron ()
 Destructor.
void swap (Polyhedron &y)
 Swaps *this with polyhedron y. (*this and y can be dimension-incompatible.).

Static Public Member Functions

dimension_type max_space_dimension ()
 Returns the maximum space dimension all kinds of Polyhedron can handle.

Protected Member Functions

 Polyhedron (Topology topol, dimension_type num_dimensions, Degenerate_Kind kind)
 Builds a polyhedron having the specified properties.
 Polyhedron (const Polyhedron &y)
 Ordinary copy-constructor.
 Polyhedron (Topology topol, const ConSys &cs)
 Builds a polyhedron from a system of constraints.
 Polyhedron (Topology topol, ConSys &cs)
 Builds a polyhedron recycling a system of constraints.
 Polyhedron (Topology topol, const GenSys &gs)
 Builds a polyhedron from a system of generators.
 Polyhedron (Topology topol, GenSys &gs)
 Builds a polyhedron recycling a system of generators.
template<typename Box>  Polyhedron (Topology topol, const Box &box)
 Builds a polyhedron out of a generic, interval-based bounding box.
Polyhedronoperator= (const Polyhedron &y)
 The assignment operator. (*this and y can be dimension-incompatible.).

Related Functions

(Note that these are not member functions.)

std::ostream & operator<< (std::ostream &s, const Polyhedron &ph)
 Output operator.
bool operator== (const Polyhedron &x, const Polyhedron &y)
 Returns true if and only if x and y are the same polyhedron.
bool operator!= (const Polyhedron &x, const Polyhedron &y)
 Returns true if and only if x and y are different polyhedra.
void swap (Parma_Polyhedra_Library::Polyhedron &x, Parma_Polyhedra_Library::Polyhedron &y)
 Specializes std::swap.
template<typename PH> bool poly_hull_assign_if_exact (PH &p, const PH &q)
 If the poly-hull between p and q is exact it is assigned to p.


Detailed Description

The base class for convex polyhedra.

An object of the class Polyhedron represents a convex polyhedron in the vector space $\Rset^n$.

A polyhedron can be specified as either a finite system of constraints or a finite system of generators (see Section Representations of Convex Polyhedra) and it is always possible to obtain either representation. That is, if we know the system of constraints, we can obtain from this the system of generators that define the same polyhedron and vice versa. These systems can contain redundant members: in this case we say that they are not in the minimal form. Most operators on polyhedra are provided with two implementations: one of these, denoted <operator-name>_and_minimize, also enforces the minimization of the representations, and returns the Boolean value false whenever the resulting polyhedron turns out to be empty.

Two key attributes of any polyhedron are its topological kind (recording whether it is a C_Polyhedron or an NNC_Polyhedron object) and its space dimension (the dimension $n \in \Nset$ of the enclosing vector space):

Note that four different polyhedra can be defined on the zero-dimension space: the empty polyhedron, either closed or NNC, and the universe polyhedron $R^0$, again either closed or NNC.

In all the examples it is assumed that variables x and y are defined (where they are used) as follows:
  Variable x(0);
  Variable y(1);
Example 1
The following code builds a polyhedron corresponding to a square in $\Rset^2$, given as a system of constraints:
  ConSys cs;
  cs.insert(x >= 0);
  cs.insert(x <= 3);
  cs.insert(y >= 0);
  cs.insert(y <= 3);
  C_Polyhedron ph(cs);
The following code builds the same polyhedron as above, but starting from a system of generators specifying the four vertices of the square:
  GenSys gs;
  gs.insert(point(0*x + 0*y));
  gs.insert(point(0*x + 3*y));
  gs.insert(point(3*x + 0*y));
  gs.insert(point(3*x + 3*y));
  C_Polyhedron ph(gs);
Example 2
The following code builds an unbounded polyhedron corresponding to a half-strip in $\Rset^2$, given as a system of constraints:
  ConSys cs;
  cs.insert(x >= 0);
  cs.insert(x - y <= 0);
  cs.insert(x - y + 1 >= 0);
  C_Polyhedron ph(cs);
The following code builds the same polyhedron as above, but starting from the system of generators specifying the two vertices of the polyhedron and one ray:
  GenSys gs;
  gs.insert(point(0*x + 0*y));
  gs.insert(point(0*x + y));
  gs.insert(ray(x - y));
  C_Polyhedron ph(gs);
Example 3
The following code builds the polyhedron corresponding to an half-plane by adding a single constraint to the universe polyhedron in $\Rset^2$:
  C_Polyhedron ph(2);
  ph.add_constraint(y >= 0);
The following code builds the same polyhedron as above, but starting from the empty polyhedron in the space $\Rset^2$ and inserting the appropriate generators (a point, a ray and a line).
  C_Polyhedron ph(2, Polyhedron::EMPTY);
  ph.add_generator(point(0*x + 0*y));
  ph.add_generator(ray(y));
  ph.add_generator(line(x));
Note that, although the above polyhedron has no vertices, we must add one point, because otherwise the result of the Minkowsky's sum would be an empty polyhedron. To avoid subtle errors related to the minimization process, it is required that the first generator inserted in an empty polyhedron is a point (otherwise, an exception is thrown).
Example 4
The following code shows the use of the function add_dimensions_and_embed:
  C_Polyhedron ph(1);
  ph.add_constraint(x == 2);
  ph.add_dimensions_and_embed(1);
We build the universe polyhedron in the 1-dimension space $\Rset$. Then we add a single equality constraint, thus obtaining the polyhedron corresponding to the singleton set $\{ 2 \} \sseq \Rset$. After the last line of code, the resulting polyhedron is

\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \]

Example 5
The following code shows the use of the function add_dimensions_and_project:
  C_Polyhedron ph(1);
  ph.add_constraint(x == 2);
  ph.add_dimensions_and_project(1);
The first two lines of code are the same as in Example 4 for add_dimensions_and_embed. After the last line of code, the resulting polyhedron is the singleton set $\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$.
Example 6
The following code shows the use of the function affine_image:
  C_Polyhedron ph(2, Polyhedron::EMPTY);
  ph.add_generator(point(0*x + 0*y));
  ph.add_generator(point(0*x + 3*y));
  ph.add_generator(point(3*x + 0*y));
  ph.add_generator(point(3*x + 3*y));
  LinExpression coeff = x + 4;
  ph.affine_image(x, coeff);
In this example the starting polyhedron is a square in $\Rset^2$, the considered variable is $x$ and the affine expression is $x+4$. The resulting polyhedron is the same square translated to the right. Moreover, if the affine transformation for the same variable x is $x+y$:
  LinExpression coeff = x + y;
the resulting polyhedron is a parallelogram with the height equal to the side of the square and the oblique sides parallel to the line $x-y$. Instead, if we do not use an invertible transformation for the same variable; for example, the affine expression $y$:
  LinExpression coeff = y;
the resulting polyhedron is a diagonal of the square.
Example 7
The following code shows the use of the function affine_preimage:
  C_Polyhedron ph(2);
  ph.add_constraint(x >= 0);
  ph.add_constraint(x <= 3);
  ph.add_constraint(y >= 0);
  ph.add_constraint(y <= 3);
  LinExpression coeff = x + 4;
  ph.affine_preimage(x, coeff);
In this example the starting polyhedron, var and the affine expression and the denominator are the same as in Example 6, while the resulting polyhedron is again the same square, but translated to the left. Moreover, if the affine transformation for x is $x+y$
  LinExpression coeff = x + y;
the resulting polyhedron is a parallelogram with the height equal to the side of the square and the oblique sides parallel to the line $x+y$. Instead, if we do not use an invertible transformation for the same variable x, for example, the affine expression $y$:
  LinExpression coeff = y;
the resulting polyhedron is a line that corresponds to the $y$ axis.
Example 8
For this example we use also the variables:
  Variable z(2);
  Variable w(3);
The following code shows the use of the function remove_dimensions:
  GenSys gs;
  gs.insert(point(3*x + y +0*z + 2*w));
  C_Polyhedron ph(gs);
  set<Variable> to_be_removed;
  to_be_removed.insert(y);
  to_be_removed.insert(z);
  ph.remove_dimensions(to_be_removed);
The starting polyhedron is the singleton set $\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4$, while the resulting polyhedron is $\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$. Be careful when removing dimensions incrementally: since dimensions are automatically renamed after each application of the remove_dimensions operator, unexpected results can be obtained. For instance, by using the following code we would obtain a different result:
  set<Variable> to_be_removed1;
  to_be_removed1.insert(y);
  ph.remove_dimensions(to_be_removed1);
  set<Variable> to_be_removed2;
  to_be_removed2.insert(z);
  ph.remove_dimensions(to_be_removed2);
In this case, the result is the polyhedron $\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$: when removing the set of dimensions to_be_removed2 we are actually removing variable $w$ of the original polyhedron. For the same reason, the operator remove_dimensions is not idempotent: removing twice the same set of dimensions is never a no-op.


Member Enumeration Documentation

enum Parma_Polyhedra_Library::Polyhedron::Degenerate_Kind
 

Kinds of degenerate polyhedra.

Enumeration values:
UNIVERSE  The universe polyhedron, i.e., the whole vector space.
EMPTY  The empty polyhedron, i.e., the empty set.


Constructor & Destructor Documentation

Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
dimension_type  num_dimensions,
Degenerate_Kind  kind
[protected]
 

Builds a polyhedron having the specified properties.

Parameters:
topol The topology of the polyhedron;
num_dimensions The number of dimensions of the vector space enclosing the polyhedron;
kind Specifies whether the universe or the empty polyhedron has to be built.

Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
const ConSys &  cs
[protected]
 

Builds a polyhedron from a system of constraints.

The polyhedron inherits the space dimension of the constraint system.

Parameters:
topol The topology of the polyhedron;
cs The system of constraints defining the polyhedron.
Exceptions:
std::invalid_argument Thrown if the topology of cs is incompatible with topol.

Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
ConSys &  cs
[protected]
 

Builds a polyhedron recycling a system of constraints.

The polyhedron inherits the space dimension of the constraint system.

Parameters:
topol The topology of the polyhedron;
cs The system of constraints defining the polyhedron. It is not declared const because its data-structures will be recycled to build the polyhedron.
Exceptions:
std::invalid_argument Thrown if the topology of cs is incompatible with topol.

Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
const GenSys &  gs
[protected]
 

Builds a polyhedron from a system of generators.

The polyhedron inherits the space dimension of the generator system.

Parameters:
topol The topology of the polyhedron;
gs The system of generators defining the polyhedron.
Exceptions:
std::invalid_argument Thrown if if the topology of gs is incompatible with topol, or if the system of generators is not empty but has no points.

Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
GenSys &  gs
[protected]
 

Builds a polyhedron recycling a system of generators.

The polyhedron inherits the space dimension of the generator system.

Parameters:
topol The topology of the polyhedron;
gs The system of generators defining the polyhedron. It is not declared const because its data-structures will be recycled to build the polyhedron.
Exceptions:
std::invalid_argument Thrown if if the topology of gs is incompatible with topol, or if the system of generators is not empty but has no points.

template<typename Box>
Parma_Polyhedra_Library::Polyhedron::Polyhedron Topology  topol,
const Box &  box
[protected]
 

Builds a polyhedron out of a generic, interval-based bounding box.

Parameters:
topol The topology of the polyhedron;
box The bounding box representing the polyhedron to be built.
Exceptions:
std::invalid_argument Thrown if box has intervals that are incompatible with topol.
The template class Box must provide the following methods.
      dimension_type space_dimension() const
returns the dimension of the vector space enclosing the polyhedron represented by the bounding box.
      bool is_empty() const
returns true if and only if the bounding box describes the empty set. The is_empty() method will always be called before the methods below. However, if is_empty() returns true, none of the functions below will be called.
      bool get_lower_bound(dimension_type k, bool closed,
                           Integer& n, Integer& d) const
Let $I$ the interval corresponding to the k-th dimension. If $I$ is not bounded from below, simply return false. Otherwise, set closed, n and d as follows: closed is set to true if the the lower boundary of $I$ is closed and is set to false otherwise; n and d are assigned the integers $n$ and $d$ such that the canonical fraction $n/d$ corresponds to the greatest lower bound of $I$. The fraction $n/d$ is in canonical form if and only if $n$ and $d$ have no common factors and $d$ is positive, $0/1$ being the unique representation for zero.
      bool get_upper_bound(dimension_type k, bool closed,
                           Integer& n, Integer& d) const
Let $I$ the interval corresponding to the k-th dimension. If $I$ is not bounded from above, simply return false. Otherwise, set closed, n and d as follows: closed is set to true if the the upper boundary of $I$ is closed and is set to false otherwise; n and d are assigned the integers $n$ and $d$ such that the canonical fraction $n/d$ corresponds to the least upper bound of $I$.


Member Function Documentation

Poly_Con_Relation Parma_Polyhedra_Library::Polyhedron::relation_with const Constraint c  )  const
 

Returns the relations holding between the polyhedron *this and the constraint c.

Exceptions:
std::invalid_argument Thrown if *this and constraint c are dimension-incompatible.

Poly_Gen_Relation Parma_Polyhedra_Library::Polyhedron::relation_with const Generator g  )  const
 

Returns the relations holding between the polyhedron *this and the generator g.

Exceptions:
std::invalid_argument Thrown if *this and generator g are dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::is_disjoint_from const Polyhedron y  )  const
 

Returns true if and only if *this and y are disjoint.

Exceptions:
std::invalid_argument Thrown if x and y are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::bounds_from_above const LinExpression expr  )  const
 

Returns true if and only if expr is bounded from above in *this.

Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::bounds_from_below const LinExpression expr  )  const
 

Returns true if and only if expr is bounded from below in *this.

Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::maximize const LinExpression expr,
Integer sup_n,
Integer sup_d,
bool &  maximum
const
 

Returns true if and only if *this is not empty and expr is bounded from above in *this, in which case the supremum value is computed.

Parameters:
expr The linear expression to be maximized subject to *this;
sup_n The numerator of the supremum value;
sup_d The denominator of the supremum value;
maximum true if and only if the supremum is also the maximum value.
Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.
If *this is empty or expr is not bounded from above, false is returned and sup_n, sup_d and maximum are left untouched.

bool Parma_Polyhedra_Library::Polyhedron::maximize const LinExpression expr,
Integer sup_n,
Integer sup_d,
bool &  maximum,
const Generator **const  pppoint
const
 

Returns true if and only if *this is not empty and expr is bounded from above in *this, in which case the supremum value and a point where expr reaches it are computed.

Parameters:
expr The linear expression to be maximized subject to *this;
sup_n The numerator of the supremum value;
sup_d The denominator of the supremum value;
maximum true if and only if the supremum is also the maximum value;
pppoint When nonzero and maximization succeeds, a pointer to a point or closure point where expr reaches its supremum value will be written at this address.
Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.
If *this is empty or expr is not bounded from above, false is returned and sup_n, sup_d, maximum and pppoint are left untouched.

bool Parma_Polyhedra_Library::Polyhedron::minimize const LinExpression expr,
Integer inf_n,
Integer inf_d,
bool &  minimum
const
 

Returns true if and only if *this is not empty and expr is bounded from below in *this, in which case the infimum value is computed.

Parameters:
expr The linear expression to be minimized subject to *this;
inf_n The numerator of the infimum value;
inf_d The denominator of the infimum value;
minimum true if and only if the infimum is also the minimum value.
Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.
If *this is empty or expr is not bounded from below, false is returned and inf_n, inf_d and minimum are left untouched.

bool Parma_Polyhedra_Library::Polyhedron::minimize const LinExpression expr,
Integer inf_n,
Integer inf_d,
bool &  minimum,
const Generator **const  pppoint
const
 

Returns true if and only if *this is not empty and expr is bounded from below in *this, in which case the infimum value and a point where expr reaches it are computed.

Parameters:
expr The linear expression to be minimized subject to *this;
inf_n The numerator of the infimum value;
inf_d The denominator of the infimum value;
minimum true if and only if the infimum is also the minimum value;
pppoint When nonzero and minimization succeeds, a pointer to a point or closure point where expr reaches its infimum value will be written at this address.
Exceptions:
std::invalid_argument Thrown if expr and *this are dimension-incompatible.
If *this is empty or expr is not bounded from below, false is returned and inf_n, inf_d, minimum and pppoint are left untouched.

bool Parma_Polyhedra_Library::Polyhedron::contains const Polyhedron y  )  const
 

Returns true if and only if *this contains y.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::strictly_contains const Polyhedron y  )  const
 

Returns true if and only if *this strictly contains y.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

template<typename Box>
void Parma_Polyhedra_Library::Polyhedron::shrink_bounding_box Box &  box,
Complexity_Class  complexity = ANY
const
 

Uses *this to shrink a generic, interval-based bounding box.

Parameters:
box The bounding box to be shrunk;
complexity The complexity class of the algorithm to be used.
The template class Box must provide the following methods, whose return value, if any, is simply ignored.
      set_empty()
causes the box to become empty, i.e., to represent the empty set.
      raise_lower_bound(dimension_type k, bool closed,
                        const Integer& n, const Integer& d)
intersects the interval corresponding to the k-th dimension with $[n/d, +\infty)$ if closed is true, with $(n/d, +\infty)$ if closed is false.
      lower_upper_bound(dimension_type k, bool closed,
                        const Integer& n, const Integer& d)
intersects the interval corresponding to the k-th dimension with $(-\infty, n/d]$ if closed is true, with $(-\infty, n/d)$ if closed is false.

The function raise_lower_bound(k, closed, n, d) will be called at most once for each possible value for k and for all such calls the fraction $n/d$ will be in canonical form, that is, $n$ and $d$ have no common factors and $d$ is positive, $0/1$ being the unique representation for zero. The same guarantee is offered for the function lower_upper_bound(k, closed, n, d).

bool Parma_Polyhedra_Library::Polyhedron::OK bool  check_not_empty = false  )  const
 

Checks if all the invariants are satisfied.

Returns:
true if and only if *this satisfies all the invariants and either check_not_empty is false or *this is not empty.
Parameters:
check_not_empty true if and only if, in addition to checking the invariants, *this must be checked to be not empty.
The check is performed so as to intrude as little as possible. If the library has been compiled with run-time assertions enabled, error messages are written on std::cerr in case invariants are violated. This is useful for the purpose of debugging the library.

void Parma_Polyhedra_Library::Polyhedron::add_constraint const Constraint c  ) 
 

Adds a copy of constraint c to the system of constraints of *this (without minimizing the result).

Exceptions:
std::invalid_argument Thrown if *this and constraint c are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::add_constraint_and_minimize const Constraint c  ) 
 

Adds a copy of constraint c to the system of constraints of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Exceptions:
std::invalid_argument Thrown if *this and constraint c are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::add_generator const Generator g  ) 
 

Adds a copy of generator g to the system of generators of *this (without minimizing the result).

Exceptions:
std::invalid_argument Thrown if *this and generator g are topology-incompatible or dimension-incompatible, or if *this is an empty polyhedron and g is not a point.

bool Parma_Polyhedra_Library::Polyhedron::add_generator_and_minimize const Generator g  ) 
 

Adds a copy of generator g to the system of generators of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Exceptions:
std::invalid_argument Thrown if *this and generator g are topology-incompatible or dimension-incompatible, or if *this is an empty polyhedron and g is not a point.

void Parma_Polyhedra_Library::Polyhedron::add_constraints const ConSys &  cs  ) 
 

Adds a copy of the constraints in cs to the system of constraints of *this (without minimizing the result).

Parameters:
cs Contains the constraints that will be added to the system of constraints of *this.
Exceptions:
std::invalid_argument Thrown if *this and cs are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::add_recycled_constraints ConSys &  cs  ) 
 

Adds the constraints in cs to the system of constraints of *this (without minimizing the result).

Parameters:
cs The constraint system that will be recycled, adding its constraints to the system of constraints of *this.
Exceptions:
std::invalid_argument Thrown if *this and cs are topology-incompatible or dimension-incompatible.
Warning:
The only assumption that can be made on cs upon successful or exceptional return is that it can be safely destroyed.

bool Parma_Polyhedra_Library::Polyhedron::add_constraints_and_minimize const ConSys &  cs  ) 
 

Adds a copy of the constraints in cs to the system of constraints of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Parameters:
cs Contains the constraints that will be added to the system of constraints of *this.
Exceptions:
std::invalid_argument Thrown if *this and cs are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::add_recycled_constraints_and_minimize ConSys &  cs  ) 
 

Adds the constraints in cs to the system of constraints of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Parameters:
cs The constraint system that will be recycled, adding its constraints to the system of constraints of *this.
Exceptions:
std::invalid_argument Thrown if *this and cs are topology-incompatible or dimension-incompatible.
Warning:
The only assumption that can be made on cs upon successful or exceptional return is that it can be safely destroyed.

void Parma_Polyhedra_Library::Polyhedron::add_generators const GenSys &  gs  ) 
 

Adds a copy of the generators in gs to the system of generators of *this (without minimizing the result).

Parameters:
gs Contains the generators that will be added to the system of generators of *this.
Exceptions:
std::invalid_argument Thrown if *this and gs are topology-incompatible or dimension-incompatible, or if *this is empty and the system of generators gs is not empty, but has no points.

void Parma_Polyhedra_Library::Polyhedron::add_recycled_generators GenSys &  gs  ) 
 

Adds the generators in gs to the system of generators of *this (without minimizing the result).

Parameters:
gs The generator system that will be recycled, adding its generators to the system of generators of *this.
Exceptions:
std::invalid_argument Thrown if *this and gs are topology-incompatible or dimension-incompatible, or if *this is empty and the system of generators gs is not empty, but has no points.
Warning:
The only assumption that can be made on gs upon successful or exceptional return is that it can be safely destroyed.

bool Parma_Polyhedra_Library::Polyhedron::add_generators_and_minimize const GenSys &  gs  ) 
 

Adds a copy of the generators in gs to the system of generators of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Parameters:
gs Contains the generators that will be added to the system of generators of *this.
Exceptions:
std::invalid_argument Thrown if *this and gs are topology-incompatible or dimension-incompatible, or if *this is empty and the the system of generators gs is not empty, but has no points.

bool Parma_Polyhedra_Library::Polyhedron::add_recycled_generators_and_minimize GenSys &  gs  ) 
 

Adds the generators in gs to the system of generators of *this, minimizing the result.

Returns:
false if and only if the result is empty.
Parameters:
gs The generator system that will be recycled, adding its generators to the system of generators of *this.
Exceptions:
std::invalid_argument Thrown if *this and gs are topology-incompatible or dimension-incompatible, or if *this is empty and the the system of generators gs is not empty, but has no points.
Warning:
The only assumption that can be made on gs upon successful or exceptional return is that it can be safely destroyed.

void Parma_Polyhedra_Library::Polyhedron::intersection_assign const Polyhedron y  ) 
 

Assigns to *this the intersection of *this and y. The result is not guaranteed to be minimized.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::intersection_assign_and_minimize const Polyhedron y  ) 
 

Assigns to *this the intersection of *this and y, minimizing the result.

Returns:
false if and only if the result is empty.
Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::poly_hull_assign const Polyhedron y  ) 
 

Assigns to *this the poly-hull of *this and y. The result is not guaranteed to be minimized.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

bool Parma_Polyhedra_Library::Polyhedron::poly_hull_assign_and_minimize const Polyhedron y  ) 
 

Assigns to *this the poly-hull of *this and y, minimizing the result.

Returns:
false if and only if the result is empty.
Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::poly_difference_assign const Polyhedron y  ) 
 

Assigns to *this the poly-difference of *this and y. The result is not guaranteed to be minimized.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::affine_image Variable  var,
const LinExpression expr,
const Integer denominator = Integer_one()
 

Assigns to *this the affine image of *this under the function mapping variable var to the affine expression specified by expr and denominator.

Parameters:
var The variable to which the affine expression is assigned;
expr The numerator of the affine expression;
denominator The denominator of the affine expression (optional argument with default value 1.)
Exceptions:
std::invalid_argument Thrown if denominator is zero or if expr and *this are dimension-incompatible or if var is not a dimension of this.

void Parma_Polyhedra_Library::Polyhedron::affine_preimage Variable  var,
const LinExpression expr,
const Integer denominator = Integer_one()
 

Assigns to *this the affine preimage of *this under the function mapping variable var to the affine expression specified by expr and denominator.

Parameters:
var The variable to which the affine expression is substituted;
expr The numerator of the affine expression;
denominator The denominator of the affine expression (optional argument with default value 1.)
Exceptions:
std::invalid_argument Thrown if denominator is zero or if expr and *this are dimension-incompatible or if var is not a dimension of *this.

void Parma_Polyhedra_Library::Polyhedron::generalized_affine_image Variable  var,
const Relation_Symbol  relsym,
const LinExpression expr,
const Integer denominator = Integer_one()
 

Assigns to *this the image of *this with respect to the generalized affine transfer function $\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$, where $\mathord{\relsym}$ is the relation symbol encoded by relsym.

Parameters:
var The left hand side variable of the generalized affine transfer function;
relsym The relation symbol;
expr The numerator of the right hand side affine expression;
denominator The denominator of the right hand side affine expression (optional argument with default value 1.)
Exceptions:
std::invalid_argument Thrown if denominator is zero or if expr and *this are dimension-incompatible or if var is not a dimension of *this or if *this is a C_Polyhedron and relsym is a strict relation symbol.

void Parma_Polyhedra_Library::Polyhedron::generalized_affine_image const LinExpression lhs,
const Relation_Symbol  relsym,
const LinExpression rhs
 

Assigns to *this the image of *this with respect to the generalized affine transfer function $\mathrm{lhs}' \relsym \mathrm{rhs}$, where $\mathord{\relsym}$ is the relation symbol encoded by relsym.

Parameters:
lhs The left hand side affine expression;
relsym The relation symbol;
rhs The right hand side affine expression.
Exceptions:
std::invalid_argument Thrown if *this is dimension-incompatible with lhs or rhs or if *this is a C_Polyhedron and relsym is a strict relation symbol.

void Parma_Polyhedra_Library::Polyhedron::time_elapse_assign const Polyhedron y  ) 
 

Assigns to *this the result of computing the time-elapse between *this and y.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::BHRZ03_widening_assign const Polyhedron y,
unsigned *  tp = 0
 

Assigns to *this the result of computing the BHRZ03-widening between *this and y.

Parameters:
y A polyhedron that must be contained in *this;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::limited_BHRZ03_extrapolation_assign const Polyhedron y,
const ConSys &  cs,
unsigned *  tp = 0
 

Improves the result of the BHRZ03-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this.

Parameters:
y A polyhedron that must be contained in *this;
cs The system of constraints used to improve the widened polyhedron;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this, y and cs are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::bounded_BHRZ03_extrapolation_assign const Polyhedron y,
const ConSys &  cs,
unsigned *  tp = 0
 

Improves the result of the BHRZ03-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this, plus all the constraints of the form $\pm x \leq r$ and $\pm x < r$, with $r \in \Qset$, that are satisfied by all the points of *this.

Parameters:
y A polyhedron that must be contained in *this;
cs The system of constraints used to improve the widened polyhedron;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this, y and cs are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::H79_widening_assign const Polyhedron y,
unsigned *  tp = 0
 

Assigns to *this the result of computing the H79-widening between *this and y.

Parameters:
y A polyhedron that must be contained in *this;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::limited_H79_extrapolation_assign const Polyhedron y,
const ConSys &  cs,
unsigned *  tp = 0
 

Improves the result of the H79-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this.

Parameters:
y A polyhedron that must be contained in *this;
cs The system of constraints used to improve the widened polyhedron;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this, y and cs are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::bounded_H79_extrapolation_assign const Polyhedron y,
const ConSys &  cs,
unsigned *  tp = 0
 

Improves the result of the H79-widening computation by also enforcing those constraints in cs that are satisfied by all the points of *this, plus all the constraints of the form $\pm x \leq r$ and $\pm x < r$, with $r \in \Qset$, that are satisfied by all the points of *this.

Parameters:
y A polyhedron that must be contained in *this;
cs The system of constraints used to improve the widened polyhedron;
tp An optional pointer to an unsigned variable storing the number of available tokens (to be used when applying the widening with tokens delay technique).
Exceptions:
std::invalid_argument Thrown if *this, y and cs are topology-incompatible or dimension-incompatible.

void Parma_Polyhedra_Library::Polyhedron::add_dimensions_and_embed dimension_type  m  ) 
 

Adds m new dimensions and embeds the old polyhedron in the new space.

Parameters:
m The number of dimensions to add.
The new dimensions will be those having the highest indexes in the new polyhedron, which is characterized by a system of constraints in which the variables running through the new dimensions are not constrained. For instance, when starting from the polyhedron $\cP \sseq \Rset^2$ and adding a third dimension, the result will be the polyhedron

\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]

void Parma_Polyhedra_Library::Polyhedron::add_dimensions_and_project dimension_type  m  ) 
 

Adds m new dimensions to the polyhedron and does not embed it in the new space.

Parameters:
m The number of dimensions to add.
The new dimensions will be those having the highest indexes in the new polyhedron, which is characterized by a system of constraints in which the variables running through the new dimensions are all constrained to be equal to 0. For instance, when starting from the polyhedron $\cP \sseq \Rset^2$ and adding a third dimension, the result will be the polyhedron

\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]

void Parma_Polyhedra_Library::Polyhedron::concatenate_assign const Polyhedron y  ) 
 

Assigns to *this the concatenation of *this and y, taken in this order.

Exceptions:
std::invalid_argument Thrown if *this and y are topology-incompatible.

void Parma_Polyhedra_Library::Polyhedron::remove_dimensions const Variables_Set to_be_removed  ) 
 

Removes all the specified dimensions.

Parameters:
to_be_removed The set of Variable objects corresponding to the dimensions to be removed.
Exceptions:
std::invalid_argument Thrown if *this is dimension-incompatible with one of the Variable objects contained in to_be_removed.

void Parma_Polyhedra_Library::Polyhedron::remove_higher_dimensions dimension_type  new_dimension  ) 
 

Removes the higher dimensions so that the resulting space will have dimension new_dimension.

Exceptions:
std::invalid_argument Thrown if new_dimensions is greater than the space dimension of *this.

template<typename PartialFunction>
void Parma_Polyhedra_Library::Polyhedron::map_dimensions const PartialFunction &  pfunc  ) 
 

Remaps the dimensions of the vector space according to a partial function.

Parameters:
pfunc The partial function specifying the destiny of each dimension.
The template class PartialFunction must provide the following methods.
      bool has_empty_codomain() const
returns true if and only if the represented partial function has an empty codomain (i.e., it is always undefined). The has_empty_codomain() method will always be called before the methods below. However, if has_empty_codomain() returns true, none of the functions below will be called.
      dimension_type max_in_codomain() const
returns the maximum value that belongs to the codomain of the partial function. The max_in_codomain() method is called at most once.
      bool maps(dimension_type i, dimension_type& j) const
Let $f$ be the represented function and $k$ be the value of i. If $f$ is defined in $k$, then $f(k)$ is assigned to j and true is returned. If $f$ is undefined in $k$, then false is returned. This method is called at most $n$ times, where $n$ is the dimension of the vector space enclosing the polyhedron.

The result is undefined if pfunc does not encode a partial function with the properties described in the specification of the mapping operator.

void Parma_Polyhedra_Library::Polyhedron::expand_dimension Variable  var,
dimension_type  m
 

Creates m copies of the dimension corresponding to var.

Parameters:
var The variable corresponding to the dimension to be replicated;
m The number of replica to be created.
Exceptions:
std::invalid_argument Thrown if var does not correspond to a dimension of the polyhedron.
If *this is $n$-dimensional, with $n > 0$, and $i < n$ is var.id(), then the $i$-th dimension is expanded to m new dimensions $n$, $n+1$, $\dots$, $n+m-1$.

void Parma_Polyhedra_Library::Polyhedron::fold_dimensions const Variables_Set to_be_folded,
Variable  var
 

Folds the dimensions in to_be_folded into var.

Parameters:
to_be_folded The set of Variable objects corresponding to the dimensions to be folded;
var The variable corresponding to the dimension that is the destination of the folding operation.
Exceptions:
std::invalid_argument Thrown if *this is dimension-incompatible with var or with one of the Variable objects contained in to_be_folded. Also thrown if var is contained in to_be_folded.
If *this is $n$-dimensional, with $n > 0$, $i < n$ is var.id(), to_be_folded is a set of variables whose id() is also less than $n$, and var is not a member of to_be_folded, then the dimensions corresponding to variables in to_be_folded are folded into dimension $i$.

void Parma_Polyhedra_Library::Polyhedron::swap Polyhedron y  ) 
 

Swaps *this with polyhedron y. (*this and y can be dimension-incompatible.).

Exceptions:
std::invalid_argument Thrown if x and y are topology-incompatible.


Friends And Related Function Documentation

std::ostream & operator<< std::ostream &  s,
const Polyhedron ph
[related]
 

Output operator.

Writes a textual representation of ph on s: false is written if ph is an empty polyhedron; true is written if ph is a universe polyhedron; a minimized system of constraints defining ph is written otherwise, all constraints in one row separated by ", ".

bool operator== const Polyhedron x,
const Polyhedron y
[related]
 

Returns true if and only if x and y are the same polyhedron.

Note that x and y may be topology- and/or dimension-incompatible polyhedra: in those cases, the value false is returned.

bool operator!= const Polyhedron x,
const Polyhedron y
[related]
 

Returns true if and only if x and y are different polyhedra.

Note that x and y may be topology- and/or dimension-incompatible polyhedra: in those cases, the value true is returned.


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