Regina Calculation Engine

Polytope vertex enumeration algorithms. More...
Classes  
class  regina::NDoubleDescription 
Implements a modified double description method for polytope vertex enumeration. More...  
class  regina::NEnumConstraintList 
Represents an individual validity constraint for use with polytope vertex enumeration. More...  
class  regina::NHilbertCD 
Implements a modified ContejeanDevie algorithm for enumerating Hilbert bases. More...  
class  regina::NHilbertDual 
Implements a modified dual algorithm for enumerating Hilbert bases. More...  
class  regina::NHilbertPrimal 
Implements a modified primal algorithm for enumerating Hilbert bases. More...  
class  regina::NMaxAdmissible 
Used to enumerate all maximal admissible faces of a polyhedral cone under a given set of admissibility constraints. More...  
class  regina::LPConstraintBase 
A base class for additional linear constraints that we can add to the tableaux of normal surface or angle structure matching equations. More...  
struct  regina::LPConstraintBase::Coefficients 
Stores the extra coefficients in a single column for the nConstraints additional rows that we add to the tableaux to describe the nConstraints additional linear equations or inequalities. More...  
class  regina::LPConstraintSubspace 
A subclass of LPConstraintBase used for constraints defined entirely by homogeneous linear equations. More...  
class  regina::LPConstraintNone 
A donothing class that imposes no additional linear constraints on the tableaux of normal surface or angle structure matching equations. More...  
class  regina::LPConstraintEuler 
A class that constraints the tableaux of normal surface matching equations to ensure that Euler characteristic is strictly positive. More...  
class  regina::LPConstraintNonSpun 
A class that constraints the tableaux of normal surface matching equations to ensure that normal surfaces in an ideal triangulation are compact (thereby avoiding spun normal surfaces with infinitely many triangles). More...  
class  regina::BanConstraintBase 
A base class for additional banning and marking constraints that we can place on tree traversal algorithms. More...  
class  regina::BanNone 
A donothing class that bans no coordinates and marks no coordinates. More...  
class  regina::BanBoundary 
A class that bans normal disc types that meet the boundary of the underlying triangulation. More...  
class  regina::BanTorusBoundary 
A class that bans and marks disc types associated with torus boundary components. More...  
class  regina::LPMatrix< Integer > 
A matrix class for use with linear programming. More...  
struct  regina::LPCol< LPConstraint > 
Used by LPInitialTableaux<LPConstraint> to store a single column of the adjusted matching equation matrix in sparse form. More...  
class  regina::LPInitialTableaux< LPConstraint > 
Stores an adjusted matrix of homogeneous linear matching equations based on a given triangulation, in sparse form. More...  
class  regina::LPData< LPConstraint, Integer > 
Stores an intermediate tableaux for the dual simplex method, and contains all of the core machinery for using the dual simplex method. More...  
class  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer > 
A base class for searches that employ the tree traversal algorithm for enumerating and locating vertex normal surfaces and taut angle structures. More...  
class  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer > 
The main entry point for the tree traversal algorithm to enumerate all vertex normal or almost normal surfaces in a 3manifold triangulation. More...  
class  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer > 
The main entry point for the tree traversal algorithm to enumerate all taut angle structures in a 3manifold triangulation. More...  
class  regina::NTreeSingleSoln< LPConstraint, BanConstraint, Integer > 
The main entry point for the tree traversal / branching algorithm to locate a single nontrivial normal surface satisfying given constraints within a 3manifold triangulation. More...  
class  regina::NTypeTrie< nTypes > 
A trie that stores a set of type vectors of a fixed length. More...  
class  regina::NPosOrder 
A comparison object that sorts hyperplanes by position vectors. More...  
Enumerations  
enum  { regina::LPConstraintBase::nConstraints } 
enum  { nConstraints = 0 } 
enum  { nConstraints = 1 } 
enum  { nConstraints = 2 } 
Functions  
regina::LPConstraintBase::Coefficients::Coefficients ()  
Creates an uninitialised set of coefficients for a single column. More...  
template<typename Integer >  
void  regina::LPConstraintBase::Coefficients::fillFinalRows (LPMatrix< Integer > &m, unsigned col) const 
Explicitly fills the final row(s) of the given tableaux matrix with the coefficients stored in this Coefficients structure. More...  
template<typename Integer >  
Integer  regina::LPConstraintBase::Coefficients::innerProduct (const LPMatrix< Integer > &m, unsigned mRow) const 
Computes the inner product of (i) the final nConstraints entries in the given row of the given matrix with (ii) the nConstraints column coefficients stored in this data structure. More...  
template<typename Integer >  
Integer  regina::LPConstraintBase::Coefficients::innerProductOct (const LPMatrix< Integer > &m, unsigned mRow) const 
A variant of innerProduct() that takes into account any adjustments to these linear constraint(s) that are required when this is a quadrilateral column being used to represent an octagon type. More...  
static bool  regina::LPConstraintBase::addRows (LPCol< LPConstraintBase > *col, const int *columnPerm, const NTriangulation *tri) 
Explicitly constructs equations for the linear function(s) constrained by this class. More...  
template<typename Integer >  
static void  regina::LPConstraintBase::constrain (LPData< LPConstraintNone, Integer > &lp, unsigned numCols) 
Explicitly constraints each of these linear functions to an equality or inequality in the underlying tableaux. More...  
static bool  regina::LPConstraintBase::verify (const NNormalSurface *s) 
Ensures that the given normal surface satisfies the extra constraints described by this class. More...  
static bool  regina::LPConstraintBase::verify (const NAngleStructure *s) 
Ensures that the given angle structure satisfies the extra constraints described by this class. More...  
static bool  regina::LPConstraintBase::supported (NormalCoords coords) 
Indicates whether the given coordinate system is supported by this constraint class. More...  
template<typename Integer >  
void  regina::LPConstraintNone::Coefficients::fillFinalRows (LPMatrix< Integer > &m, unsigned col) const 
template<typename Integer >  
Integer  regina::LPConstraintNone::Coefficients::innerProduct (const LPMatrix< Integer > &, unsigned) const 
template<typename Integer >  
Integer  regina::LPConstraintNone::Coefficients::innerProductOct (const LPMatrix< Integer > &, unsigned) const 
static bool  regina::LPConstraintNone::addRows (LPCol< regina::LPConstraintNone > *, const int *, const NTriangulation *) 
template<typename Integer >  
static void  regina::LPConstraintNone::constrain (LPData< regina::LPConstraintNone, Integer > &, unsigned) 
static bool  regina::LPConstraintNone::verify (const NNormalSurface *) 
static bool  regina::LPConstraintNone::verify (const NAngleStructure *) 
static bool  regina::LPConstraintNone::supported (NormalCoords coords) 
template<typename Integer >  
void  regina::LPConstraintEuler::Coefficients::fillFinalRows (LPMatrix< Integer > &m, unsigned col) const 
template<typename Integer >  
Integer  regina::LPConstraintEuler::Coefficients::innerProduct (const LPMatrix< Integer > &m, unsigned mRow) const 
template<typename Integer >  
Integer  regina::LPConstraintEuler::Coefficients::innerProductOct (const LPMatrix< Integer > &m, unsigned mRow) const 
static bool  regina::LPConstraintEuler::addRows (LPCol< regina::LPConstraintEuler > *col, const int *columnPerm, const NTriangulation *tri) 
template<typename Integer >  
static void  regina::LPConstraintEuler::constrain (LPData< regina::LPConstraintEuler, Integer > &lp, unsigned numCols) 
static bool  regina::LPConstraintEuler::verify (const NNormalSurface *s) 
static bool  regina::LPConstraintEuler::verify (const NAngleStructure *) 
static bool  regina::LPConstraintEuler::supported (NormalCoords coords) 
template<typename Integer >  
void  regina::LPConstraintNonSpun::Coefficients::fillFinalRows (LPMatrix< Integer > &m, unsigned col) const 
template<typename Integer >  
Integer  regina::LPConstraintNonSpun::Coefficients::innerProduct (const LPMatrix< Integer > &m, unsigned mRow) const 
template<typename Integer >  
Integer  regina::LPConstraintNonSpun::Coefficients::innerProductOct (const LPMatrix< Integer > &m, unsigned mRow) const 
static bool  regina::LPConstraintNonSpun::addRows (LPCol< regina::LPConstraintNonSpun > *col, const int *columnPerm, const NTriangulation *tri) 
template<typename Integer >  
static void  regina::LPConstraintNonSpun::constrain (LPData< regina::LPConstraintNonSpun, Integer > &lp, unsigned numCols) 
static bool  regina::LPConstraintNonSpun::verify (const NNormalSurface *s) 
static bool  regina::LPConstraintNonSpun::verify (const NAngleStructure *) 
static bool  regina::LPConstraintNonSpun::supported (NormalCoords coords) 
regina::BanConstraintBase::BanConstraintBase (const NTriangulation *tri, int coords)  
Constructs and initialises the banned_ and marked_ arrays to be entirely false . More...  
regina::BanConstraintBase::~BanConstraintBase ()  
Destroys this object and all associated data. More...  
template<class LPConstraint , typename Integer >  
void  regina::BanConstraintBase::enforceBans (LPData< LPConstraint, Integer > &lp) const 
Enforces all bans described by this class in the given tableaux. More...  
void  regina::BanConstraintBase::init (const int *columnPerm) 
Identifies which coordinates to ban and mark, and records the corresponding tableaux columns in the banned_ and marked_ arrays respectively. More...  
static bool  regina::BanConstraintBase::supported (NormalCoords coords) 
Indicates whether the given coordinate system is supported by this constraint class. More...  
regina::BanNone::BanNone (const NTriangulation *tri, int coords)  
Constructs and initialises the banned_ and marked_ arrays to be entirely false , as described in the BanConstraintBase superclass constructor. More...  
void  regina::BanNone::init (const int *) 
static bool  regina::BanNone::supported (NormalCoords coords) 
regina::BanBoundary::BanBoundary (const NTriangulation *tri, int coords)  
Constructs and initialises the banned_ and marked_ arrays to be entirely false , as described in the BanConstraintBase superclass constructor. More...  
void  regina::BanBoundary::init (const int *columnPerm) 
static bool  regina::BanBoundary::supported (NormalCoords coords) 
regina::BanTorusBoundary::BanTorusBoundary (const NTriangulation *tri, int coords)  
Constructs and initialises the banned_ and marked_ arrays to be entirely false , as described in the BanConstraintBase superclass constructor. More...  
void  regina::BanTorusBoundary::init (const int *columnPerm) 
static bool  regina::BanTorusBoundary::supported (NormalCoords coords) 
Variables  
REGINA_DEPRECATED typedef NDoubleDescription  regina::NDoubleDescriptor 
A legacy typedef provided for backward compatibility only. More...  
int  regina::LPConstraintEuler::Coefficients::euler 
The coefficient of the Euler characteristic function for the corresponding column of the matching equation matrix. More...  
int  regina::LPConstraintNonSpun::Coefficients::meridian 
The coefficient of the meridian equation for the corresponding column of the matching equation matrix. More...  
int  regina::LPConstraintNonSpun::Coefficients::longitude 
The coefficient of the longitude equation for the corresponding column of the matching equation matrix. More...  
const NTriangulation *  regina::BanConstraintBase::tri_ 
The triangulation with which we are working. More...  
int  regina::BanConstraintBase::coords_ 
The normal or almost normal coordinate system in which we are working. More...  
bool *  regina::BanConstraintBase::banned_ 
Indicates which columns of a tableaux correspond to banned coordinates (e.g., banned normal disc types). More...  
bool *  regina::BanConstraintBase::marked_ 
Indicates which columns of a tableaux correspond to marked coordinates (e.g., marked normal disc types). More...  
Polytope vertex enumeration algorithms.
anonymous enum 

static 
Explicitly constructs equations for the linear function(s) constrained by this class.
Specifically, this routine takes an array of Coefficients objects (one for each column of the initial tableaux) and fills in the necessary coefficient data.
The precise form of the linear function(s) will typically depend upon the underlying triangulation. For this reason, the triangulation is explicitly passed, along with the permutation that indicates which columns of the initial tableaux correspond to which normal or angle structure coordinates.
More precisely: recall that, for each linear function, the initial tableaux acquires one new variable x_i that evaluates this linear function f(x). This routine must create the corresponding row that sets f(x)  x_i = 0
. Thus it must construct the coefficients of f(x) in the columns corresponding to normal coordinates, and it must also set a coefficient of 1 in the column for the corresponding new variable.
For each subclass S of LPConstraintBase, the array col must be an array of objects of type LPCol<S>. The class LPCol<S> is itself a larger subclass of the Coefficients class. This exact type must be used because the compiler must know how large each column object is in order to correct access each element of the given array.
As described in the LPInitialTableaux class notes, it might not be possible to construct the linear functions (since the triangulation might not satisfy the necessary requirements). In this case, this routine should ensure that the linear functions are in fact the zero functions, and should return false
(but it must still set 1 coefficients for the new variables as described above). Otherwise (if the linear function were successfully constructed) this routine should return true
.
If you are implementing this routine in a subclass that works with angle structure coordinates, remember that your linear constraints must not interact with the scaling coordinate (the final angle structure coordinate that is used to projectivise the angle structure polytope into a polyhedral cone). Your implementation of this routine must ensure that your linear constraints all have coefficient zero in this column.
col  the array of columns as stored in the initial tableaux (i.e., the data member LPInitialTableaux::col_). 
columnPerm  the corresponding permutation of columns that describes how columns of the tableaux correspond to normal or angle structure coordinates in the underlying triangulation (i.e., the data member LPInitialTableaux::columnPerm_). 
tri  the underlying triangulation. 
true
if the linear functions were successfully constructed, or false
if not (in which case they will be replaced with the zero functions instead).

inlineprotected 
Constructs and initialises the banned_ and marked_ arrays to be entirely false
, as described in the BanConstraintBase superclass constructor.
tri  the triangulation with which we are working. 
coords  the normal or almost normal coordinate system in which we are working. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, or NS_AN_STANDARD. 

protected 
Constructs and initialises the banned_ and marked_ arrays to be entirely false
.
The only purpose of passing the triangulation and coordinate system is to determine how many normal or angle structure coordinates we are dealing with.
tri  the triangulation with which we are working. 
coords  the coordinate system in which we are working. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, NS_AN_STANDARD, or NS_ANGLE. 

inlineprotected 
Constructs and initialises the banned_ and marked_ arrays to be entirely false
, as described in the BanConstraintBase superclass constructor.
Although one should normally call the routine init() before using this object, for BanNone this is not strictly necessary since there are no coordinates to ban or mark.
tri  the triangulation with which we are working. 
coords  the coordinate system in which we are working. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, NS_AN_STANDARD, or NS_ANGLE. 

inlineprotected 
Constructs and initialises the banned_ and marked_ arrays to be entirely false
, as described in the BanConstraintBase superclass constructor.
tri  the triangulation with which we are working. 
coords  the normal or almost normal coordinate system in which we are working. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, or NS_AN_STANDARD. 
regina::LPConstraintBase::Coefficients::Coefficients  (  ) 
Creates an uninitialised set of coefficients for a single column.
These cofficients must be initialised through a call to addRows() before they can be used.

static 
Explicitly constraints each of these linear functions to an equality or inequality in the underlying tableaux.
This will typically consist of a series of calls to LPData::constrainZero() and/or LPData::constrainPositive().
The variables for these extra linear functions are stored in columns numCols  nConstraints
, ..., numCols  1
of the given tableaux, and so your calls to LPData::constrainZero() and/or LPData::constrainPositive() should operate on these (and only these) columns.
lp  the tableaux in which to constrain these linear functions. 
numCols  the number of columns in the given tableaux. 

inlineprotected 
Enforces all bans described by this class in the given tableaux.
Specifically, for each banned coordinate, this routine calls LPData::constrainZero() on the corresponding coordinate column.
lp  the tableaux in which to enforce the bans. 
void regina::LPConstraintBase::Coefficients::fillFinalRows  (  LPMatrix< Integer > &  m, 
unsigned  col  
)  const 
Explicitly fills the final row(s) of the given tableaux matrix with the coefficients stored in this Coefficients structure.
In essence, this routine simply copies this sparse and/or specialised representation of the final row(s) into a more standard dense matrix representation.
This routine should only affect the final nConstraints entries in the given column of the matrix. It may assume that these final row(s) have already been initialised to zero.
m  the matrix in which to place these column coefficients. 
col  the column of the given matrix in which to place these coefficients. 

protected 
Identifies which coordinates to ban and mark, and records the corresponding tableaux columns in the banned_ and marked_ arrays respectively.
columnPerm  the permutation of columns that describes how columns of the tableaux correspond to normal or angle strutcure coordinates in the underlying triangulation. Specifically, this permutation must be the same permutation returned by LPInitialTableaux::columnPerm(). 
Integer regina::LPConstraintBase::Coefficients::innerProduct  (  const LPMatrix< Integer > &  m, 
unsigned  mRow  
)  const 
Computes the inner product of (i) the final nConstraints entries in the given row of the given matrix with (ii) the nConstraints column coefficients stored in this data structure.
m  the matrix whose row we will use in the inner product. 
mRow  the row of the matrix m to use in the inner product. 
Integer regina::LPConstraintBase::Coefficients::innerProductOct  (  const LPMatrix< Integer > &  m, 
unsigned  mRow  
)  const 
A variant of innerProduct() that takes into account any adjustments to these linear constraint(s) that are required when this is a quadrilateral column being used to represent an octagon type.
The LPData class offers support for octagonal almost normal surfaces, in which exactly one tetrahedron is allowed to have exactly one octagon type. We represent such an octagon as a pair of incompatible quadrilaterals within the same tetrahedron. See the LPData class notes for details on how this works.
In some settings, our extra linear constraints must behave differently in the presence of octagons (i.e., the coefficient of the octagon type is not just the sum of coefficients of the two constituent quadrilateral types). This routine effectively allows us to adjust the tableaux accordingly.
Specifically: this routine computes the inner product of (i) the final nConstraints entries in the given row of the given matrix with (ii) the nConstraints column coefficients stored in this data structure. We assume that this column in the underlying tableaux describes one of the two quadrilateral coordinates in some tetrahedron that together form an octagon type, and if necessary we implicitly adjust the coefficients stored in this data structure accordingly.
This routine is not used with angle structure coordinates.
m  the matrix whose row we will use in the inner product. 
mRow  the row of the matrix m to use in the inner product. 

static 
Indicates whether the given coordinate system is supported by this constraint class.
This routine assumes that the given system is already known to be supported by the generic tree traversal infrastructure, and only returns false
if there are additional prerequisites imposed by this particular constraint class that the given system does not satisfy. If this constraint class does not impose any of its own additional conditions, this routine may simply return true
.
coords  the coordinate system being queried; this must be one of the coordinate systems known to be supported by the generic NTreeTraversal infrastructure. 
true
if and only if this coordinate system is also supported by this specific constraint class.

staticprotected 
Indicates whether the given coordinate system is supported by this constraint class.
This routine assumes that the given system is already known to be supported by the generic tree traversal infrastructure, and only returns false
if there are additional prerequisites imposed by this particular constraint class that the given system does not satisfy. If this constraint class does not impose any of its own additional conditions, this routine may simply return true
.
coords  the coordinate system being queried; this must be one of the coordinate systems known to be supported by the generic NTreeTraversal infrastructure. 
true
if and only if this coordinate system is also supported by this specific constraint class.

static 
Ensures that the given normal surface satisfies the extra constraints described by this class.
Ideally this test is not based on explicitly recomputing the linear function(s), but instead runs independent tests. For instance, if this class is used to constraint Euler characteristic, then ideally this routine would call s>eulerChar() and test the return value of that routine instead.
If these linear constraints work with angle structure coordinates (not normal or almost normal surfaces), then this routine should return false
.
s  the surface to test. 
true
if the given surface satisfies these linear constraints, or false
if it does not.

static 
Ensures that the given angle structure satisfies the extra constraints described by this class.
Ideally this test is not based on explicitly recomputing the linear function(s), but instead runs independent tests; see the related routine verify(const NNormalSurface*) for examples.
If these linear constraints work with normal or almost normal surfaces (not angle structure coordinates), then this routine should return false
.
s  the angle structure to test. 
true
if the given angle structure satisfies these linear constraints, or false
if it does not.

inlineprotected 
Destroys this object and all associated data.

protected 
Indicates which columns of a tableaux correspond to banned coordinates (e.g., banned normal disc types).
The size of this array is the number of normal or angle structure coordinates (so we explicitly exclude extra columns that arise from the template parameter LPConstraint.

protected 
The normal or almost normal coordinate system in which we are working.
This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, NS_AN_STANDARD, or NS_ANGLE.
int regina::LPConstraintEuler::Coefficients::euler 
The coefficient of the Euler characteristic function for the corresponding column of the matching equation matrix.
int regina::LPConstraintNonSpun::Coefficients::longitude 
The coefficient of the longitude equation for the corresponding column of the matching equation matrix.

protected 
Indicates which columns of a tableaux correspond to marked coordinates (e.g., marked normal disc types).
The size of this array is the number of normal or angle structure coordinates (so we explicitly exclude extra columns that arise from the template parameter LPConstraint.
int regina::LPConstraintNonSpun::Coefficients::meridian 
The coefficient of the meridian equation for the corresponding column of the matching equation matrix.
REGINA_DEPRECATED typedef NDoubleDescription regina::NDoubleDescriptor 
A legacy typedef provided for backward compatibility only.

protected 
The triangulation with which we are working.