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...  
class  regina::LPInitialTableaux< LPConstraint > 
Stores an adjusted matrix of homogeneous linear matching equations based on a given triangulation, in sparse form. More...  
struct  regina::LPInitialTableaux< LPConstraint >::Col 
Stores a single column of the adjusted matching equation matrix 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...  
Typedefs  
typedef NDoubleDescription  regina::NDoubleDescriptor 
A legacy typedef provided for backward compatibility only. 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 (LPInitialTableaux< LPConstraintBase >::Col *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 (LPInitialTableaux< regina::LPConstraintNone >::Col *, 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 (LPInitialTableaux< regina::LPConstraintEuler >::Col *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 (LPInitialTableaux< regina::LPConstraintNonSpun >::Col *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) 
regina::LPMatrix< Integer >::LPMatrix ()  
Creates an uninitialised matrix with no memory storage. More...  
regina::LPMatrix< Integer >::LPMatrix (unsigned rows, unsigned cols)  
Creates a fully initialised rows by cols matrix with all elements set to zero. More...  
regina::LPMatrix< Integer >::~LPMatrix ()  
Destroys this matrix and all of the data it contains. More...  
void  regina::LPMatrix< Integer >::reserve (unsigned maxRows, unsigned maxCols) 
Reserves enough space to store the elements of a maxRows by maxCols matrix. More...  
void  regina::LPMatrix< Integer >::initClone (const LPMatrix &clone) 
Initialises this matrix to a copy of the given matrix. More...  
void  regina::LPMatrix< Integer >::initIdentity (unsigned size) 
Initialises this matrix to the identity matrix of the given size. More...  
Integer &  regina::LPMatrix< Integer >::entry (unsigned row, unsigned col) 
Returns a readwrite reference to the given element of this matrix. More...  
const Integer &  regina::LPMatrix< Integer >::entry (unsigned row, unsigned col) const 
Returns a readonly reference to the given element of this matrix. More...  
unsigned  regina::LPMatrix< Integer >::rows () const 
Returns the number of rows in this matrix. More...  
unsigned  regina::LPMatrix< Integer >::columns () const 
Returns the number of columns in this matrix. More...  
void  regina::LPMatrix< Integer >::swapRows (unsigned r1, unsigned r2) 
Swaps the two given rows of this matrix. More...  
void  regina::LPMatrix< Integer >::combRow (const Integer &destCoeff, unsigned dest, const Integer &srcCoeff, unsigned src, const Integer &div) 
Applies a particular row operation to this matrix. More...  
Integer  regina::LPMatrix< Integer >::combRowAndNorm (const Integer &destCoeff, unsigned dest, const Integer &srcCoeff, unsigned src) 
Applies a particular row operation to this matrix, and then normalises. More...  
void  regina::LPMatrix< Integer >::negateRow (unsigned row) 
Negates all elements in the given row of this matrix. More...  
void  regina::LPMatrix< Integer >::dump (std::ostream &out) const 
Writes this matrix to the given output stream. More...  
regina::LPInitialTableaux< LPConstraint >::Col::Col ()  
Initialises an empty column. More...  
void  regina::LPInitialTableaux< LPConstraint >::Col::push (unsigned row, int val) 
Adds the given entry in the given row to this column. More...  
regina::LPInitialTableaux< LPConstraint >::LPInitialTableaux (const NTriangulation *tri, NormalCoords coords, bool enumeration)  
Construts this adjusted sparse matrix of matching equations. More...  
regina::LPInitialTableaux< LPConstraint >::~LPInitialTableaux ()  
Destroys this matrix. More...  
const NTriangulation *  regina::LPInitialTableaux< LPConstraint >::tri () const 
Returns the underlying 3manifold triangulation from which the matching equations were derived. More...  
NormalCoords  regina::LPInitialTableaux< LPConstraint >::coords () const 
Returns the coordinate system that is used for the matrix of matching equations. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::rank () const 
Returns the rank of this matrix. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::columns () const 
Returns the number of columns in this matrix. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::coordinateColumns () const 
Returns the number of columns that correspond to normal coordinates or angle structure coordinates. More...  
bool  regina::LPInitialTableaux< LPConstraint >::constraintsBroken () const 
Indicates whether or not the extra constraints from the template parameter LPConstraints were added successfully. More...  
const int *  regina::LPInitialTableaux< LPConstraint >::columnPerm () const 
Returns the permutation that describes how the columns of the matching equation matrix were reordered. More...  
template<typename Integer >  
Integer  regina::LPInitialTableaux< LPConstraint >::multColByRow (const LPMatrix< Integer > &m, unsigned mRow, unsigned thisCol) const 
Computes the inner product of (i) the given row of the given matrix with (ii) the given column of this matrix. More...  
template<typename Integer >  
Integer  regina::LPInitialTableaux< LPConstraint >::multColByRowOct (const LPMatrix< Integer > &m, unsigned mRow, unsigned thisCol) const 
A variant of multColByRow() that takes into account any adjustments to the tableaux that are required when this is a quadrilateral column being used to represent an octagon type. More...  
template<typename Integer >  
void  regina::LPInitialTableaux< LPConstraint >::fillInitialTableaux (LPMatrix< Integer > &m) const 
Fills the given matrix with the contents of this matrix. More...  
regina::LPData< LPConstraint, Integer >::LPData ()  
Constructs a new tableaux. More...  
regina::LPData< LPConstraint, Integer >::~LPData ()  
Destroys this tableaux. More...  
void  regina::LPData< LPConstraint, Integer >::reserve (const LPInitialTableaux< LPConstraint > *origTableaux) 
Reserves enough memory for this tableaux to work with. More...  
void  regina::LPData< LPConstraint, Integer >::initStart () 
Initialises this tableaux by beginning at the original starting tableaux and working our way to any feasible basis. More...  
void  regina::LPData< LPConstraint, Integer >::initClone (const LPData &parent) 
Initialises this tableaux to be a clone of the given tableaux. More...  
unsigned  regina::LPData< LPConstraint, Integer >::columns () const 
Returns the number of columns in this tableaux. More...  
unsigned  regina::LPData< LPConstraint, Integer >::coordinateColumns () const 
Returns the number of columns in this tableaux that correspond to normal coordinates or angle structure coordinates. More...  
bool  regina::LPData< LPConstraint, Integer >::isFeasible () const 
Returns whether or not this system is feasible. More...  
bool  regina::LPData< LPConstraint, Integer >::isActive (unsigned pos) const 
Determines whether the given variable is currently active. More...  
int  regina::LPData< LPConstraint, Integer >::sign (unsigned pos) const 
Returns the sign of the given variable under the current basis. More...  
void  regina::LPData< LPConstraint, Integer >::constrainZero (unsigned pos) 
Constrains this system further by setting the given variable to zero and deactivating it. More...  
void  regina::LPData< LPConstraint, Integer >::constrainPositive (unsigned pos) 
Constrains this system further by constraining the given variable to be strictly positive. More...  
void  regina::LPData< LPConstraint, Integer >::constrainOct (unsigned quad1, unsigned quad2) 
Declares that two quadrilateral coordinates within a tetrahedron are to be combined into a single octagon coordinate, for use with almost normal surfaces, and constrains the system accordingly. More...  
void  regina::LPData< LPConstraint, Integer >::dump (std::ostream &out) const 
Writes details of this tableaux to the given output stream. More...  
void  regina::LPData< LPConstraint, Integer >::extractSolution (NRay &v, const char *type) const 
Extracts the values of the individual variables from the current basis, with some modifications (as described below). More...  
static bool  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::supported (NormalCoords coords) 
Indicates whether the given coordinate system is supported by this tree traversal infrastructure. More...  
bool  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::constraintsBroken () const 
Indicates whether or not the extra constraints from the template parameter LPConstraints were added successfully to the infrastructure for the search tree. More...  
unsigned long  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nVisited () const 
Returns the total number of nodes in the search tree that we have visited thus far in the tree traversal. More...  
void  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::dumpTypes (std::ostream &out) const 
Writes the current type vector to the given output stream. More...  
NNormalSurface *  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::buildSurface () const 
Reconstructs the full normal surface that is represented by the type vector at the current stage of the search. More...  
NAngleStructure *  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::buildStructure () const 
Reconstructs the full taut angle structure that is represented by the type vector at the current stage of the search. More...  
bool  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::verify (const NNormalSurface *s, const NMatrixInt *matchingEqns=0) const 
Ensures that the given normal or almost normal surface satisfies the matching equations, as well as any additional constraints from the template parameter LPConstraint. More...  
bool  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::verify (const NAngleStructure *s, const NMatrixInt *angleEqns=0) const 
Ensures that the given angle structure satisfies the angle equations, as well as any additional constraints from the template parameter LPConstraint. More...  
regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::NTreeTraversal (const NTriangulation *tri, NormalCoords coords, int branchesPerQuad, int branchesPerTri, bool enumeration)  
Initialises a new base object for running the tree traversal algorithm. More...  
regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::~NTreeTraversal ()  
Destroys this object. More...  
void  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::setNext (int nextType) 
Rearranges the search tree so that nextType becomes the next type that we process. More...  
int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nextUnmarkedTriangleType (int startFrom) 
Returns the next unmarked triangle type from a given starting point. More...  
int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::feasibleBranches (int quadType) 
Determines how many different values we could assign to the given quadrilateral or angle type and still obtain a feasible system. More...  
double  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::percent () const 
Gives a rough estimate as to what percentage of the way the current type vector is through a full enumeration of the search tree. More...  
regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::NTreeEnumeration (const NTriangulation *tri, NormalCoords coords)  
Creates a new object for running the tree traversal algorithm. More...  
unsigned long  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::nSolns () const 
Returns the total number of vertex normal or almost normal surfaces found thus far in the tree traversal search. More...  
void  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::run (bool(*useSoln)(const NTreeEnumeration &, void *), void *arg=0) 
Runs the complete tree traversal algorithm to enumerate vertex normal or almost normal surfaces. More...  
bool  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::next (NProgressTracker *tracker=0) 
An incremental step in the tree traversal algorithm that runs forward until it finds the next solution. More...  
static bool  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::writeTypes (const NTreeEnumeration &tree, void *) 
A callback function that writes to standard output the type vector at the current point in the given tree traversal search. More...  
static bool  regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::writeSurface (const NTreeEnumeration &tree, void *) 
A callback function that writes to standard output the full trianglequadrilateral coordinates of the vertex normal or almost normal surface at the current point in the given tree traversal search. More...  
regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::NTautEnumeration (const NTriangulation *tri)  
Creates a new object for running the tree traversal algorithm. More...  
unsigned long  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::nSolns () const 
Returns the total number of taut angle structures found thus far in the tree traversal search. More...  
void  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::run (bool(*useSoln)(const NTautEnumeration &, void *), void *arg=0) 
Runs the complete tree traversal algorithm to enumerate all taut angle structures. More...  
bool  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::next (NProgressTracker *tracker=0) 
An incremental step in the enumeration algorithm that runs forward until it finds the next solution. More...  
static bool  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::writeTypes (const NTautEnumeration &tree, void *) 
A callback function that writes to standard output the type vector at the current point in the given tree traversal search. More...  
static bool  regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::writeStructure (const NTautEnumeration &tree, void *) 
A callback function that writes to standard output the full angle structure coordinates of the taut angle structure at the current point in the given tree traversal search. More...  
regina::NTreeSingleSoln< LPConstraint, BanConstraint, Integer >::NTreeSingleSoln (const NTriangulation *tri, NormalCoords coords)  
Creates a new object for running the tree traversal / branching algorithm to locate a nontrivial surface that satisfies the chosen constraints. More...  
bool  regina::NTreeSingleSoln< LPConstraint, BanConstraint, Integer >::find () 
Runs the tree traversal algorithm until it finds some nontrivial surface that satisfies the chosen constraints, or else proves that no such solution exists. More...  
void  regina::NTreeSingleSoln< LPConstraint, BanConstraint, Integer >::cancel () 
Cancels the current find() operation. More...  
regina::NTypeTrie< nTypes >::NTypeTrie ()  
Initialises an empty trie. More...  
regina::NTypeTrie< nTypes >::~NTypeTrie ()  
Destroys this trie. More...  
void  regina::NTypeTrie< nTypes >::clear () 
Resets this to the empty trie. More...  
void  regina::NTypeTrie< nTypes >::insert (const char *entry, unsigned len) 
Inserts the given type vector into this trie. More...  
bool  regina::NTypeTrie< nTypes >::dominates (const char *vec, unsigned len) const 
Determines whether the given type vector dominates any vector in this trie. More...  
Variables  
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...  
unsigned  regina::LPInitialTableaux< LPConstraint >::Col::nPlus 
The total number of +1 entries in this column. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::Col::plus [4] 
The rows containing these +1 entries, in any order. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::Col::nMinus 
The total number of 1 entries in this column. More...  
unsigned  regina::LPInitialTableaux< LPConstraint >::Col::minus [4] 
The rows containing these 1 entries, in any order. More...  
const LPInitialTableaux < LPConstraint >  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::origTableaux_ 
The original starting tableaux that holds the adjusted matrix of matching equations, before the tree traversal algorithm begins. More...  
const NormalCoords  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::coords_ 
The coordinate system in which we are enumerating or searching for normal surfaces, almost normal surfaces, or taut angle structures. More...  
const int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nTets_ 
The number of tetrahedra in the underlying triangulation. More...  
const int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nTypes_ 
The total length of a type vector. More...  
const int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nTableaux_ 
The maximum number of tableaux that we need to keep in memory at any given time during the backtracking search. More...  
char *  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::type_ 
The current working type vector. More...  
int *  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::typeOrder_ 
A permutation of 0,...,nTypes_1 that indicates in which order we select types: the first type we select (at the root of the tree) is type_[typeOrder_[0]], and the last type we select (at the leaves of the tree) is type_[typeOrder_[nTypes_1]]. More...  
int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::level_ 
The current level in the search tree. More...  
int  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::octLevel_ 
The level at which we are enforcing an octagon type (with a strictly positive number of octagons). More...  
LPData< LPConstraint, Integer > *  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::lp_ 
Stores tableaux for linear programming at various nodes in the search tree. More...  
LPData< LPConstraint, Integer > **  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::lpSlot_ 
Recall from above that the array lp_ stores tableaux for the current node in the search tree and all of its ancestors. More...  
LPData< LPConstraint, Integer > **  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nextSlot_ 
Points to the next available tableaux in lp_ that is free to use at each level of the search tree. More...  
unsigned long  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::nVisited_ 
Counts the total number of nodes in the search tree that we have visited thus far. More...  
LPData< LPConstraint, Integer >  regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::tmpLP_ [4] 
Temporary tableaux used by the function feasibleBranches() to determine which quadrilateral types or angle types have good potential for pruning the search tree. More...  
Polytope vertex enumeration algorithms.
typedef NDoubleDescription regina::NDoubleDescriptor 
A legacy typedef provided for backward compatibility only.
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 LPInitialTableaux<S>::Col. The class LPInitialTableaux<S>::Col 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. 
NAngleStructure* regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::buildStructure  (  )  const 
Reconstructs the full taut angle structure that is represented by the type vector at the current stage of the search.
This routine is for use only with taut angle structures, not normal or almost normal surfaces.
The angle structure that is returned will be newly constructed, and it is the caller's responsibility to destroy it when it is no longer required.
There will always be a unique taut angle structure corresponding to this type vector (this follows from the preconditions below).
true
, or any time that NTautEnumeration::run() calls its callback function.NNormalSurface* regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::buildSurface  (  )  const 
Reconstructs the full normal surface that is represented by the type vector at the current stage of the search.
This routine is for use only with normal (or almost normal) surfaces, not taut angle structures.
The surface that is returned will be newly constructed, and it is the caller's responsibility to destroy it when it is no longer required.
If the current type vector does not represent a vertex normal surface (which may be the case when calling NTreeSingleSoln::find()), then there may be many normal surfaces all represented by the same type vector; in this case there are no further guarantees about which of these normal surfaces you will get.
true
, or any time that NTreeEnumeration::run() calls its callback function.

inline 

inline 
Resets this to the empty trie.
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.

inline 
Initialises an empty column.

inline 
Returns the permutation that describes how the columns of the matching equation matrix were reordered.
This permutation maps column numbers in this adjusted matching equation matrix to column numbers in the original (unmodified) matching equation matrix that was originally derived from the triangulation.
The permutation is returned as an array of columns() integers, such that column i of this adjusted matrix corresponds to column columnPerm()[i]
of the original matrix.
If you are imposing additional constraints through the template parameter LPConstraint, then the corresponding extra variables will be included in the permutation; however, these are never moved and will always remain the rightmost variables in this system (i.e., the columns of highest index).
As well as the requirement that this is a genuine permutation of 0,...,columns()1, this array will also adhere to the following constraints. In the following discussion, n refers to the number of tetrahedra in the underlying triangulation.

inline 
Returns the number of columns in this matrix.
This relates to the currently assigned matrix size, not the total amount of memory that was originally reserved.

inline 
Returns the number of columns in this matrix.
Note that, if we are imposing extra constraints through the template parameter LPConstraint, then there will be extra variables to enforce these, and so the number of columns will be larger than in the original matching equation matrix.

inline 
Returns the number of columns in this tableaux.
Note that, if we are imposing extra constraints through the template parameter LPConstraint, then there will be extra variables to enforce these, and so the number of columns will be larger than in the original matching equation matrix.

inline 
Applies a particular row operation to this matrix.
Specifically, row dest will be replaced with the linear combination: (destCoeff * row dest  srcCoeff * row src) / div.
destCoeff  the coefficient applied to row dest in the linear combination. 
dest  the index of the row to replace. This must be between 0 and rows()1 inclusive. 
srcCoeff  the coefficient applied to row src in the linear combination. 
src  the index of the other row used in this linear combination. This must be between 0 and rows()1 inclusive. 
div  the integer to divide the final row by. This must be nonzero. 

inline 
Applies a particular row operation to this matrix, and then normalises.
Specifically, row dest will be replaced with the linear combination: (destCoeff * row dest  srcCoeff * row src); then, if row dest is nonzero, it will be normalised by dividing through by the gcd of its elements. Note that this gcd is always taken to be positive (i.e., the final normalisation will never change the signs of the elements in the row).
destCoeff  the coefficient applied to row dest in the linear combination. 
dest  the index of the row to replace. This must be between 0 and rows()1 inclusive. 
srcCoeff  the coefficient applied to row src in the linear combination. 
src  the index of the other row used in this linear combination. This must be between 0 and rows()1 inclusive. 

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. 
void regina::LPData< LPConstraint, Integer >::constrainOct  (  unsigned  quad1, 
unsigned  quad2  
) 
Declares that two quadrilateral coordinates within a tetrahedron are to be combined into a single octagon coordinate, for use with almost normal surfaces, and constrains the system accordingly.
This constrains the system in several ways, as discussed in detail in the LPData class notes. In theory, we set the two quadrilateral coordinates to be equal, and also insist that the number of octagons be strictly positive. In practice, we do this through several changes of variable; see the LPData class notes for a detailed discussion of precisely how the variables and tableaux will change.
This routine will work even if one of the given quadrilateral variables has already been deactivated, but in this case the routine will immediately set the system to infeasible and return.
This routine is not used with angle structure coordinates.
quad1  one of the two quadrilateral types that we combine to form the new octagon type. 
quad2  the other of the two quadrilateral types that we combine to form the new octagon type. 
void regina::LPData< LPConstraint, Integer >::constrainPositive  (  unsigned  pos  ) 
Constrains this system further by constraining the given variable to be strictly positive.
We do this using a change of variable that effectively replaces x_pos with the new variable x'_pos = x_pos  1 (which we simply constrain to be nonnegative as usual). See the LPData class notes for details.
This routine will work even if the given variable has already been deactivated, but in this case the routine will immediately set the system to infeasible and return.
pos  the index of the variable that is to be constrained as positive. This must be between 0 and origTableaux_>columns()1 inclusive. 

inline 
Indicates whether or not the extra constraints from the template parameter LPConstraints were added successfully to the infrastructure for the search tree.
This query function is important because some constraints require additional preconditions on the underlying triangulation, and so these constraints cannot be added in some circumstances. If it is possible that the constraints might not be added successfully, this function should be tested as soon as the NTreeTraversal object has been created.
If the extra constraints were not added successfully, the search tree will be left in a consistent state but will give incorrect results (specifically, the extra constraints will be treated as zero functions).
true
if the constraints were not added successfully, or false
if the constraints were added successfully.

inline 
Indicates whether or not the extra constraints from the template parameter LPConstraints were added successfully.
This query function is important because some constraints require additional preconditions on the underlying triangulation, and cannot be added if these preconditions are not satisfied.
Even if the extra constraints were not added successfully, this tableaux will be left in a consistent state (the extra constraints will be treated as zero functions). See the LPInitialTableaux class notes for further details.
true
if the constraints were not added successfully, or false
if the constraints were added successfully. void regina::LPData< LPConstraint, Integer >::constrainZero  (  unsigned  pos  ) 
Constrains this system further by setting the given variable to zero and deactivating it.
See the LPData class notes for details.
This routine will work even if the given variable has already been deactivated (and it will do nothing in this case).
pos  the index of the variable that is to be set to zero. This must be between 0 and origTableaux_>columns()1 inclusive. 

inline 
Returns the number of columns that correspond to normal coordinates or angle structure coordinates.
This is precisely the number of columns in the original matrix of matching equations.

inline 
Returns the number of columns in this tableaux that correspond to normal coordinates or angle structure coordinates.
This is precisely the number of columns in the original matrix of matching equations.

inline 
Returns the coordinate system that is used for the matrix of matching equations.
This will be the same coordinate system that was passed to the LPInitialTableaux constructor; in particular, it will always be one of NS_QUAD, NS_STANDARD, or NS_ANGLE.
bool regina::NTypeTrie< nTypes >::dominates  (  const char *  vec, 
unsigned  len  
)  const 
Determines whether the given type vector dominates any vector in this trie.
vec  the type vector to test. 
len  the number of elements in the given type vector. 
true
if and only if vec dominates some type vector stored in this trie. void regina::LPMatrix< Integer >::dump  (  std::ostream &  out  )  const 
Writes this matrix to the given output stream.
The output is "rough" and wasteful, and is intended for debugging purposes only. The precise output format is subject to change in future versions of Regina.
out  the output stream to write to. 
void regina::LPData< LPConstraint, Integer >::dump  (  std::ostream &  out  )  const 
Writes details of this tableaux to the given output stream.
The output is "rough" and wasteful, and is intended for debugging purposes only.
The precise output is subject to change in future versions of Regina.
out  the output stream to write to. 

inline 
Writes the current type vector to the given output stream.
There will be no spaces between the types, and there will be no final newline.
out  the output stream to which to write. 

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. 

inline 

inline 
void regina::LPData< LPConstraint, Integer >::extractSolution  (  NRay &  v, 
const char *  type  
)  const 
Extracts the values of the individual variables from the current basis, with some modifications (as described below).
The values of the variables are store in the given vector v.
The modifications are as follows:
This routine is not used as an internal part of the tree traversal algorithm; instead it is offered as a helper routine for reconstructing the normal surfaces or angle structures that result.
v  the vector into which the values of the variables will be placed. 
type  the type vector corresponding to the current state of this tableaux, indicating which variables were previously fixed as positive via calls to constrainPositive(). This is necessary because LPData does not keep such historical data on its own. As a special case, when extracting a strict angle structure one may pass type = 0, in which case this routine will assume that every coordinate was constrained as positive. 

protected 
Determines how many different values we could assign to the given quadrilateral or angle type and still obtain a feasible system.
This will involve solving three or four linear programs, all based on the current state of the tableaux at the current level of the search tree. These assign 0, 1, 2 and 3 to the given quadrilateral or angle type in turn (here 0 is not used for angle types), and then enforce the corresponding constraints. For quadrilateral types, we count types 0 and 1 separately as in NTreeEnumeration, not merged together as in NTreeSingleSoln.
quadType  the quadrilateral or angle type to examine. 
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. 

inline 
Fills the given matrix with the contents of this matrix.
This effectively copies this sparse but highly specialised matrix representation into a dense but more flexible matrix representation.
m  the matrix to fill. 
bool regina::NTreeSingleSoln< LPConstraint, BanConstraint, Integer >::find  (  ) 
Runs the tree traversal algorithm until it finds some nontrivial surface that satisfies the chosen constraints, or else proves that no such solution exists.
Note that, if a solution is found, it will have a maximal (but not necessarily maximum) set of zero coordinates, which in some settings is enough to guarantee a vertex normal surface. See the NTreeSingleSoln class notes for details.
If find() does return true
, you can extract details of the corresponding surface directly from this tree enumeration object: for instance, you can dump the type vector using dumpTypes(), or you can reconstruct the full surface using buildSurface(). Be warned that this class defines the type vector in an unusual way (see the NTreeSingleSoln class notes for details). If you call buildSurface(), remember to delete the surface once you are finished with it.
true
if we found a nontrivial solution as described in the class notes, or false
if no such solution exists.

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(). 

inline 
Initialises this matrix to a copy of the given matrix.
This matrix does not yet need to be initialised, but it does need to have enough space reserved.
You may call this routine on an alreadyinitialised matrix, and you may use this routine to assign it a different size (as long as enough space was originally reserved).
clone  the matrix to copy. 
void regina::LPData< LPConstraint, Integer >::initClone  (  const LPData< LPConstraint, Integer > &  parent  ) 
Initialises this tableaux to be a clone of the given tableaux.
This is used in the tree traversal algorithm as we work our way down the search tree, and child nodes "inherit" tableaux from their parent nodes.
parent  the tableaux to clone. 

inline 
Initialises this matrix to the identity matrix of the given size.
This matrix does not yet need to be initialised, but it does need to have enough space reserved.
You may call this routine on an alreadyinitialised matrix, and you may use this routine to assign it a different size (as long as enough space was originally reserved).
size  the number of rows, and also the number of columns, that will be assigned to this matrix. This must be strictly positive. 
void regina::LPData< LPConstraint, Integer >::initStart  (  ) 
Initialises this tableaux by beginning at the original starting tableaux and working our way to any feasible basis.
This routine also explicitly enforces the additional constraints from the template parameter LPConstraint (i.e., this routine is responsible for forcing the corresponding linear function(s) to be zero or strictly positive as appropriate).
It is possible that a feasible basis cannot be found; you should test isFeasible() after running this routine to see whether this is the case.
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. 
void regina::NTypeTrie< nTypes >::insert  (  const char *  entry, 
unsigned  len  
) 
Inserts the given type vector into this trie.
entry  the type vector to insert. 
len  the number of elements in the given type vector. 

inline 

inline 
Returns whether or not this system is feasible.
A system may become infeasible when we add too many extra constraints on the variables (such as forcing them to be positive, or setting them to zero); see the LPData class notes for details on these constraints.
true
if this system is feasible, or false
if it is infeasible.

inline 
Constructs a new tableaux.
You must call reserve() before doing anything else with this tableaux.
regina::LPInitialTableaux< LPConstraint >::LPInitialTableaux  (  const NTriangulation *  tri, 
NormalCoords  coords,  
bool  enumeration  
) 
Construts this adjusted sparse matrix of matching equations.
tri  the underlying 3manifold triangulation. 
coords  the coordinate system to use for the matrix of matching equations; this must be one of NS_QUAD, NS_STANDARD, or NS_ANGLE. 
enumeration  true if we should optimise the tableaux for a full enumeration of vertex surfaces or taut angle structures, or false if we should optimise the tableaux for an existence test (such as searching for a nontrivial normal disc or sphere, or a strict angle structure). 

inline 
Creates an uninitialised matrix with no memory storage.
You must call reserve() and then either initClone() or initIdentity() before this matrix will become initialised.

inline 
Creates a fully initialised rows by cols matrix with all elements set to zero.
This routine reserves space for precisely rows * cols elements. In other words, you may later reinitialise the matrix to become smaller if you like, but you cannot reinitialise the matrix to become larger.
rows  the number of rows in the new matrix. This must be strictly positive. 
cols  the number of columns in the new matrix. This must be strictly positive. 

inline 
Computes the inner product of (i) the given row of the given matrix with (ii) the given column of this matrix.
This routine is optimised to use the sparse representation of columns in this matrix.
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. 
thisCol  the column of this matrix to use in the inner product. 

inline 
A variant of multColByRow() that takes into account any adjustments to the tableaux 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 where we are using additional constraints through the template parameter LPConstraint, these extra constraints 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 given row of the given matrix with (ii) the given column of this matrix. We assume that the given column of this matrix describes one of the two quadrilateral coordinates in some tetrahedron that together form an octagon type, and (via the helper routine LPConstraint::Coefficients::innerProductOct) we implicitly adjust the coefficients of our extra constraints accordingly.
This routine is optimised to use the sparse representation of columns in this matrix.
This routine is not used with angle structure coordinates.
m  the matrix whose row we will use in the adjusted inner product. 
mRow  the row of the matrix m to use in the adjusted inner product. 
thisCol  the column of this matrix to use in the adjusted inner product. 

inline 
Negates all elements in the given row of this matrix.
row  the row whose elements should be negated. This must be between 0 and rows()1 inclusive. 
bool regina::NTreeEnumeration< LPConstraint, BanConstraint, Integer >::next  (  NProgressTracker *  tracker = 0  ) 
An incremental step in the tree traversal algorithm that runs forward until it finds the next solution.
Specifically: this continues the tree traversal from the current point until either it finds the next vertex normal or almost normal surface (in which case it returns true
), or until the tree traversal is completely finished with no more solutions to be found (in which case it returns false
).
If you simply wish to find and process all vertex surfaces, you may wish to consider the allinone routine run() instead. By using next() to step through one solution at a time however, you obtain more finegrained control: for instance, you can "pause" and restart the search, or have tighter control over multithreading.
If next() does return true
because it found a solution, you can extract details of the solution directly from this tree enumeration object: for instance, you can dump the type vector using dumpTypes(), or you can reconstruct the full normal or almost normal surface using buildSurface() and perform some other operations upon it. If you do call buildSurface(), remember to delete the normal surface once you are finished with it.
An optional progress tracker may be passed. If so, this routine will update the percentage progress and poll for cancellation requests. It will be assumed that an appropriate stage has already been declared via NProgressTracker::newStage() before this routine is called, and that NProgressTracker::setFinished() will be called after this routine returns (and presumably not until the entire search tree is exhausted). The percentage progress will be given in the context of a complete enumeration of the entire search tree (i.e., it will typically start at a percentage greater than 0, and end at a percentage less than 100).
true
(indicating that it has not yet finished the search).tracker  a progress tracker through which progress will be reported, or 0 if no progress reporting is required. 
true
if we found another vertex surface, or false
if the search has now finished and no more vertex surfaces were found. bool regina::NTautEnumeration< LPConstraint, BanConstraint, Integer >::next  (  NProgressTracker *  tracker = 0  ) 
An incremental step in the enumeration algorithm that runs forward until it finds the next solution.
Specifically: this continues the enumeration from the current point until either it finds the next taut angle structure (in which case it returns true
), or until the enumeration algorithm is completely finished with no more solutions to be found (in which case it returns false
).
If you simply wish to find and process all taut angle structures, you may wish to consider the allinone routine run() instead. By using next() to step through one solution at a time however, you obtain more finegrained control: for instance, you can "pause" and restart the search, or have tighter control over multithreading.
If next() does return true
because it found a solution, you can extract details of the solution directly from this enumeration object: for instance, you can dump the type vector using dumpTypes(), or you can reconstruct the full taut angle structure using buildStructure() and perform some other operations upon it. If you do call buildStructure(), remember to delete the angle structure once you are finished with it.
An optional progress tracker may be passed. If so, this routine will update the percentage progress and poll for cancellation requests. It will be assumed that an appropriate stage has already been declared via NProgressTracker::newStage() before this routine is called, and that NProgressTracker::setFinished() will be called after this routine returns (and presumably not until the entire search tree is exhausted). The percentage progress will be given in the context of a complete enumeration of the entire search tree (i.e., it will typically start at a percentage greater than 0, and end at a percentage less than 100).
true
(indicating that it has not yet finished the search).tracker  a progress tracker through which progress will be reported, or 0 if no progress reporting is required. 
true
if we found another vertex surface, or false
if the search has now finished and no more taut angle strutures were found.

inlineprotected 
Returns the next unmarked triangle type from a given starting point.
Specifically, this routine returns the first unmarked triangle type whose type number is greater than or equal to startFrom. For more information on marking, see the BanConstraintBase class notes.
This routine simply searches through types by increasing index into the type vector; in particular, it does not make any use of the reordering defined by the typeOrder_ array.
startFrom  the index into the type vector of the triangle type from which we begin searching. 

inline 
Returns the total number of vertex normal or almost normal surfaces found thus far in the tree traversal search.
If you called run(), then this will simply be the total number of vertex surfaces. If you are calling next() one surface at time, this will be the partial count of how many vertex surfaces have been found until now.

inline 
Returns the total number of taut angle structures found thus far in the tree traversal search.
If you called run(), then this will simply be the total number of taut angle structures. If you are calling next() one surface at time, this will be the partial count of how many taut angle structures have been found until now.

inline 
Creates a new object for running the tree traversal algorithm.
This prepares the algorithm; in order to run the algorithm and enumerate taut angle structures, you can either:
tri  the triangulation in which we wish to enumerate taut angle structures. 

inline 
Creates a new object for running the tree traversal algorithm.
This prepares the algorithm; in order to run the algorithm and enumerate vertex surfaces, you can either:
tri  the triangulation in which we wish to enumerate vertex surfaces. 
coords  the coordinate system in which wish to enumerate vertex surfaces. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, or NS_AN_STANDARD. 

inline 
Creates a new object for running the tree traversal / branching algorithm to locate a nontrivial surface that satisfies the chosen constraints.
This constructor prepares the algorithm; in order to run the algorithm you should call find(), which returns true
or false
according to whether or not such a surface was found.
tri  the triangulation in which we wish to search for a nontrivial surface. 
coords  the normal or almost normal coordinate system in which to work. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, or NS_AN_STANDARD. 

protected 
Initialises a new base object for running the tree traversal algorithm.
This routine may only be called by subclass constructors; for more information on how to create and run a tree traversal, see subclasses such as NTreeEnumeration, NTautEnumeration or NTreeSingleSoln instead.
tri  the triangulation in which we wish to search for normal surfaces or taut angle structures. 
coords  the coordinate system in which wish to search for normal surfaces or taut angle structures. This must be one of NS_QUAD, NS_STANDARD, NS_AN_QUAD_OCT, NS_AN_STANDARD, or NS_ANGLE. 
branchesPerQuad  the maximum number of branches we spawn in the search tree for each quadrilateral or angle type (e.g., 4 for a vanilla normal surface tree traversal algorithm, or 3 for enumerating taut angle structures). 
branchesPerTri  the maximum number of branches we spawn in the search tree for each triangle type (e.g., 2 for a vanilla normal surface tree traversal algorithm). If the underlying coordinate system does not support triangles then this argument will be ignored. 
enumeration  true if we should optimise the tableaux for a full enumeration of vertex surfaces or taut angle structures, or false if we should optimise the tableaux for an existence test (such as searching for a nontrivial normal disc or sphere). 

inline 
Initialises an empty trie.

inline 
Returns the total number of nodes in the search tree that we have visited thus far in the tree traversal.
This figure might grow much faster than the number of solutions, since it also counts traversals through "dead ends" in the search tree.
This counts all nodes that we visit, including those that fail any or all of the domination, feasibility and zero tests. The precise way that this number is calculated is subject to change in future versions of Regina.
If you called an "all at once" routine such as NTreeEnumeration::run() or NTreeSingleSoln::find(), then this will be the total number of nodes that were visited in the entire tree traversal. If you are calling an "incremental" routine such as NTreeEnumeration::next() (i.e., you are generating one solution at time), then this will be the partial count of how many nodes have been visited so far.

protected 
Gives a rough estimate as to what percentage of the way the current type vector is through a full enumeration of the search tree.
This is useful for progress tracking.
This routine only attemps to determine the percentage within a reasonable range of error (at the time of writing, 0.01%). This allows it to be more efficient (in particular, by only examining the branches closest to the root of the search tree).

inline 
Adds the given entry in the given row to this column.
row  the row containing the given value. 
val  the value at this location in the matrix. 

inline 
Returns the rank of this matrix.
Note that, if we are imposing extra constraints through the template parameter LPConstraint, then there will be extra variables to enforce these, and so the rank will be larger than the rank of the original matching equation matrix.

inline 
Reserves enough space to store the elements of a maxRows by maxCols matrix.
This is just an upper bound: your matrix may end up using fewer elements than this, but it cannot use more.
This matrix will still not be initialised until you call either initClone() or initIdentity(). See the class notes for details.
maxRows  an upper bound on the number of rows that you will need for this matrix. This must be strictly positive. 
maxCols  an upper bound on the number of columns that you will need for this matrix. This must be strictly positive. 

inline 
Reserves enough memory for this tableaux to work with.
You must call this routine before doing anything else with this tableaux.
The data in this tableaux will not be initialised, and the contents and behaviour of this tableaux will remain undefined until you call one of the initialisation routines initStart() or initClone().
origTableaux  the original starting tableaux that holds the adjusted matrix of matching equations, before the tree traversal algorithm began. 

inline 
Returns the number of rows in this matrix.
This relates to the currently assigned matrix size, not the total amount of memory that was originally reserved.

inline 
Runs the complete tree traversal algorithm to enumerate vertex normal or almost normal surfaces.
For each vertex surface that is found, this routine will call the function useSoln. It will pass two arguments to this function: (i) this tree enumeration object, and (ii) an arbitrary piece of data that you can supply via the argument arg.
You can extract details of the solution directly from the tree enumeration object: for instance, you can dump the type vector using dumpTypes(), or you can reconstruct the full normal or almost normal surface using buildSurface() and perform some other operations upon it. If you do call buildSurface(), remember to delete the normal surface once you are finished with it.
The tree traversal will block until your callback function useSoln returns. If the callback function returns true
, then run() will continue the tree traversal. If it returns false
, then run() will abort the search and return immediately.
The usual way of using this routine is to construct a NTreeEnumeration object and then immediately call run(). However, if you prefer, you may call run() after one or more calls to next(). In this case, run() will continue the search from the current point and run it to its completion. In other words, run() will locate and call useSoln for all vertex surfaces that had not yet been found, but it will not call useSoln on those surfaces that had previously been found during earlier calls to next().
true
(indicating that it has not yet finished the search).useSoln  a callback function that will be called each time we locate a vertex surface, as described above. 
arg  the second argument to pass to the callback function; this may be any type of data that you like. 

inline 
Runs the complete tree traversal algorithm to enumerate all taut angle structures.
For each taut angle structure that is found, this routine will call the function useSoln. It will pass two arguments to this function: (i) this enumeration object, and (ii) an arbitrary piece of data that you can supply via the argument arg.
You can extract details of the solution directly from the enumeration object: for instance, you can dump the type vector using dumpTypes(), or you can reconstruct the full taut angle structure using buildStructure() and perform some other operations upon it. If you do call buildStructure(), remember to delete the angle structure once you are finished with it.
The enumeration will block until your callback function useSoln returns. If the callback function returns true
, then run() will continue the enumeration. If it returns false
, then run() will abort the search and return immediately.
The usual way of using this routine is to construct an NTautEnumeration object and then immediately call run(). However, if you prefer, you may call run() after one or more calls to next(). In this case, run() will continue the search from the current point and run it to its completion. In other words, run() will locate and call useSoln for all taut angle structures that had not yet been found, but it will not call useSoln on those solutions that had previously been found during earlier calls to next().
true
(indicating that it has not yet finished the search).useSoln  a callback function that will be called each time we locate a taut angle structure, as described above. 
arg  the second argument to pass to the callback function; this may be any type of data that you like. 

protected 
Rearranges the search tree so that nextType becomes the next type that we process.
Specifically, this routine will set typeOrder_[level_ + 1] to nextType_, and will move other elements of typeOrder_ back by one position to make space as required.
nextType  the next type to process. 

inline 
Returns the sign of the given variable under the current basis.
This does not attempt to "undo" any changes of variable caused by prior calls to constrainPositive() or constrainOct(); it simply tests the sign of the variable in the given column of the tableaux in its current form.
Specifically: if the given variable is inactive or nonbasic, this routine returns zero. If the given variable is in the basis, this routine returns the sign of the corresponding integer on the righthand side of the tableaux.
pos  the index of the variable to query. This must be between 0 and origTableaux_>columns()1 inclusive. 

inlinestatic 
Indicates whether the given coordinate system is supported by this tree traversal infrastructure.
Currently this is true only for NS_STANDARD and NS_QUAD (for normal surfaces), NS_AN_STANDARD and NS_AN_QUAD_OCT (for almost normal surfaces), and NS_ANGLE (for taut angle structures). Any additional restrictions imposed by LPConstraint and BanConstraint will also be taken into account.
coords  the coordinate system being queried. 
true
if and only if this coordinate system is supported.

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.

inline 
Swaps the two given rows of this matrix.
The two arguments r1 and r2 may be equal (in which case the matrix will be left unchanged).

inline 
Returns the underlying 3manifold triangulation from which the matching equations were derived.

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>getEulerChar() 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. bool regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::verify  (  const NNormalSurface *  s, 
const NMatrixInt *  matchingEqns = 0 

)  const 
Ensures that the given normal or almost normal surface satisfies the matching equations, as well as any additional constraints from the template parameter LPConstraint.
This routine is for use only with normal (or almost normal) surfaces, not angle structures.
This routine is provided for diagnostic, debugging and verification purposes.
Instead of using the initial tableaux to verify the matching equations, this routine goes back to the original matching equations matrix as constructed by regina::makeMatchingEquations(). This ensures that the test is independent of any potential problems with the tableaux. You are not required to pass your own matching equations (if you don't, they will be temporarily reconstructed for you); however, you may pass your own if you wish to use a nonstandard matching equation matrix, and/or reuse the same matrix to avoid the overhead of reconstructing it every time this routine is called.
s  the normal surface to verify. 
matchingEqns  the matching equations to check against the given surface; this may be 0, in which case the matching equations will be temporarily reconstructed for you using regina::makeMatchingEquations(). 
true
if the given surface passes all of the tests described above, or false
if it fails one or more tests (indicating a problem or error). bool regina::NTreeTraversal< LPConstraint, BanConstraint, Integer >::verify  (  const NAngleStructure *  s, 
const NMatrixInt *  angleEqns = 0 

)  const 
Ensures that the given angle structure satisfies the angle equations, as well as any additional constraints from the template parameter LPConstraint.
This routine is for use only with angle structures, not normal (or almost normal) surfaces.
This routine is provided for diagnostic, debugging and verification purposes.
Instead of using the initial tableaux to verify the angle equations, this routine goes back to the original angle equations matrix as constructed by NAngleStructureVector::makeAngleEquations(). This ensures that the test is independent of any potential problems with the tableaux. You are not required to pass your own angle equations (if you don't, they will be temporarily reconstructed for you); however, you may pass your own if you wish to use a nonstandard angle equation matrix, and/or reuse the same matrix to avoid the overhead of reconstructing it every time this routine is called.
s  the angle structure to verify. 
angleEqns  the angle equations to check against the given angle structure; this may be 0, in which case the angle equations will be temporarily reconstructed for you using NAngleStructureVector::makeMatchingEquations(). 
true
if the given angle structure passes all of the tests described above, or false
if it fails one or more tests (indicating a problem or error).

static 
A callback function that writes to standard output the full angle structure coordinates of the taut angle structure at the current point in the given tree traversal search.
You can use this as the callback function useSoln that is passed to run().
The angle structure coordinates will be written on a single line, with spaces and punctuation separating them, a prefix indicating which solution we are up to, and a final newline appended. The final scaling coordinate (used to projectivise the angle structure polytope) will also be written. This output format is subject to change in future versions of Regina.
The second (void*) argument is ignored. It is only present for compatibility with run().
true
, or any time that run() calls its callback function.tree  the tree traversal object from which we are extracting the current taut angle structure. 
true
(which indicates to run() that we should continue the tree traversal).

inlinestatic 
A callback function that writes to standard output the full trianglequadrilateral coordinates of the vertex normal or almost normal surface at the current point in the given tree traversal search.
You can use this as the callback function useSoln that is passed to run().
The normal surface coordinates will be written on a single line, with spaces and punctuation separating them, a prefix indicating which solution we are up to, and a final newline appended. This output format is subject to change in future versions of Regina.
The second (void*) argument is ignored. It is only present for compatibility with run().
true
, or any time that run() calls its callback function.tree  the tree traversal object from which we are extracting the current vertex normal or almost normal surface. 
true
(which indicates to run() that we should continue the tree traversal).

inlinestatic 
A callback function that writes to standard output the type vector at the current point in the given tree traversal search.
You can use this as the callback function useSoln that is passed to run().
The type vector will be written on a single line, with no spaces between types, with a prefix indicating which solution we are up to, and with a final newline appended. This output format is subject to change in future versions of Regina.
The second (void*) argument is ignored. It is only present for compatibility with run().
true
, or any time that run() calls its callback function.tree  the tree traversal object from which we are extracting the current type vector. 
true
(which indicates to run() that we should continue the tree traversal).

inlinestatic 
A callback function that writes to standard output the type vector at the current point in the given tree traversal search.
You can use this as the callback function useSoln that is passed to run().
The type vector will be written on a single line, with no spaces between types, with a prefix indicating which solution we are up to, and with a final newline appended. This output format is subject to change in future versions of Regina.
The second (void*) argument is ignored. It is only present for compatibility with run().
true
, or any time that run() calls its callback function.tree  the tree traversal object from which we are extracting the current type vector. 
true
(which indicates to run() that we should continue the tree traversal).

inlineprotected 
Destroys this object and all associated data.

inline 
Destroys this tableaux.
This is safe even if reserve() was never called.

inline 
Destroys this matrix.

inline 
Destroys this matrix and all of the data it contains.
You can safely destroy a matrix that is uninitialised or only partially initialised (i.e., space has been reserved but the matrix size is not set).

protected 
Destroys this object.

inline 
Destroys this trie.

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 coordinate system in which we are enumerating or searching for normal surfaces, almost normal surfaces, or taut angle structures.
This must be one of NS_QUAD or NS_STANDARD if we are only supporting normal surfaces, one of NS_AN_QUAD_OCT or NS_AN_STANDARD if we are allowing octagons in almost normal surfaces, or NS_ANGLE if we are searching for taut angle structures.

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.

protected 
The current level in the search tree.
As the search runs, this holds the index into typeOrder_ corresponding to the last type that we chose.
int regina::LPConstraintNonSpun::Coefficients::longitude 
The coefficient of the longitude equation for the corresponding column of the matching equation matrix.

protected 
Stores tableaux for linear programming at various nodes in the search tree.
We only store a limited number of tableaux at any given time, and as the search progresses we overwrite old tableaux with new tableaux.
More precisely, we store a linear number of tableaux, essentially corresponding to the current node in the search tree and all of its ancestores, all the way up to the root node. In addition to these tableaux, we also store other immediate children of these ancestores that we have preprepared for future processing. See the documentation within routines such as NTreeEnumeration::next() for details of when and how these tableaux are constructed.

protected 
Recall from above that the array lp_ stores tableaux for the current node in the search tree and all of its ancestors.
This means we have one tableaux for the root node, as well as additional tableaux at each level 0,1,...,level_.
The array lpSlot_ indicates which element of the array lp_ holds each of these tableaux. Specifically: lpSlot_[0] points to the tableaux for the root node, and for each level i in the range 0,...,level_, the corresponding tableaux is *lpSlot_[i+1]. Again, see the documentation within routines such as NTreeEnumeration::next() for details of when and how these tableaux are constructed and later overwritten.

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.
unsigned regina::LPInitialTableaux< LPConstraint >::Col::minus[4] 
The rows containing these 1 entries, in any order.
The same row may appear in this list more than once (indicating a 2, 3 or 4 entry in the matrix).

protected 
Points to the next available tableaux in lp_ that is free to use at each level of the search tree.
Specifically: nextSlot_[0] points to the next free tableaux at the root node, and for each level i in the range 0,...,level_, the corresponding next free tableaux is *nextSlot_[i+1].
The precise layout of the nextSlot_ array depends on the order in which we process quadrilateral, triangle and/or angle types.
unsigned regina::LPInitialTableaux< LPConstraint >::Col::nMinus 
The total number of 1 entries in this column.
unsigned regina::LPInitialTableaux< LPConstraint >::Col::nPlus 
The total number of +1 entries in this column.

protected 
The maximum number of tableaux that we need to keep in memory at any given time during the backtracking search.

protected 
The number of tetrahedra in the underlying triangulation.

protected 
The total length of a type vector.

protected 
Counts the total number of nodes in the search tree that we have visited thus far.
This may grow much faster than the number of solutions, since it also counts traversals through "dead ends" in the search tree.

protected 
The level at which we are enforcing an octagon type (with a strictly positive number of octagons).
If we are working with angle structures or normal surfaces only (and so we do not allow octagons at all), then octLevel_ = nTypes_. If we are allowing almost normal surfaces but we have not yet chosen an octagon type, then octLevel_ = 1.

protected 
The original starting tableaux that holds the adjusted matrix of matching equations, before the tree traversal algorithm begins.
unsigned regina::LPInitialTableaux< LPConstraint >::Col::plus[4] 
The rows containing these +1 entries, in any order.
The same row may appear in this list more than once (indicating a +2, +3 or +4 entry in the matrix).

protected 
Temporary tableaux used by the function feasibleBranches() to determine which quadrilateral types or angle types have good potential for pruning the search tree.
Other routines are welcome to use these temporary tableaux also (as "scratch space"); however, be aware that any call to feasibleBranches() will overwrite them.

protected 
The triangulation with which we are working.

protected 
The current working type vector.
As the search runs, we modify this type vector inplace. Any types beyond the current level in the search tree will always be set to zero.

protected 
A permutation of 0,...,nTypes_1 that indicates in which order we select types: the first type we select (at the root of the tree) is type_[typeOrder_[0]], and the last type we select (at the leaves of the tree) is type_[typeOrder_[nTypes_1]].
This permutation is allowed to change as the algorithm runs (though of course you can only change sections of the permutation that correspond to types not yet selected).