Regina Calculation Engine
Public Member Functions | Static Public Member Functions | Protected Member Functions | Protected Attributes | List of all members
regina::TautEnumeration< LPConstraint, BanConstraint, IntType > Class Template Reference

The main entry point for the tree traversal algorithm to enumerate all taut angle structures in a 3-manifold triangulation. More...

#include <enumerate/treetraversal.h>

Inheritance diagram for regina::TautEnumeration< LPConstraint, BanConstraint, IntType >:
regina::TreeTraversal< LPConstraintNone, BanNone, Integer > regina::BanNone regina::BanConstraintBase

Public Member Functions

 TautEnumeration (const Triangulation< 3 > *tri)
 Creates a new object for running the tree traversal algorithm. More...
 
unsigned long nSolns () const
 Returns the total number of taut angle structures found thus far in the tree traversal search. More...
 
void run (bool(*useSoln)(const TautEnumeration &, void *), void *arg=0)
 Runs the complete tree traversal algorithm to enumerate all taut angle structures. More...
 
bool next (ProgressTracker *tracker=0)
 An incremental step in the enumeration algorithm that runs forward until it finds the next solution. More...
 
bool 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 nVisited () const
 Returns the total number of nodes in the search tree that we have visited thus far in the tree traversal. More...
 
void dumpTypes (std::ostream &out) const
 Writes the current type vector to the given output stream. More...
 
NormalSurfacebuildSurface () const
 Reconstructs the full normal surface that is represented by the type vector at the current stage of the search. More...
 
AngleStructurebuildStructure () const
 Reconstructs the full taut angle structure that is represented by the type vector at the current stage of the search. More...
 
bool verify (const NormalSurface *s, const MatrixInt *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 verify (const AngleStructure *s, const MatrixInt *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...
 

Static Public Member Functions

static bool writeTypes (const TautEnumeration &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 writeStructure (const TautEnumeration &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...
 
static bool supported (NormalCoords coords)
 Indicates whether the given coordinate system is supported by this tree traversal infrastructure. More...
 

Protected Member Functions

void setNext (int nextType)
 Rearranges the search tree so that nextType becomes the next type that we process. More...
 
int nextUnmarkedTriangleType (int startFrom)
 Returns the next unmarked triangle type from a given starting point. More...
 
int 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 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...
 
void init (const int *)
 
template<class LPConstraint , typename IntType >
void enforceBans (LPData< LPConstraint, IntType > &lp) const
 Enforces all bans described by this class in the given tableaux. More...
 

Protected Attributes

const LPInitialTableaux< LPConstraintNoneorigTableaux_
 The original starting tableaux that holds the adjusted matrix of matching equations, before the tree traversal algorithm begins. More...
 
const NormalCoords coords_
 The coordinate system in which we are enumerating or searching for normal surfaces, almost normal surfaces, or taut angle structures. More...
 
const int nTets_
 The number of tetrahedra in the underlying triangulation. More...
 
const int nTypes_
 The total length of a type vector. More...
 
const int nTableaux_
 The maximum number of tableaux that we need to keep in memory at any given time during the backtracking search. More...
 
char * type_
 The current working type vector. More...
 
int * 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 level_
 The current level in the search tree. More...
 
int octLevel_
 The level at which we are enforcing an octagon type (with a strictly positive number of octagons). More...
 
LPData< LPConstraintNone, Integer > * lp_
 Stores tableaux for linear programming at various nodes in the search tree. More...
 
LPData< LPConstraintNone, 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< LPConstraintNone, 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 nVisited_
 Counts the total number of nodes in the search tree that we have visited thus far. More...
 
LPData< LPConstraintNone, IntegertmpLP_ [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...
 
const Triangulation< 3 > * tri_
 The triangulation with which we are working. More...
 
bool * banned_
 Indicates which columns of a tableaux correspond to banned coordinates (e.g., banned normal disc types). More...
 
bool * marked_
 Indicates which columns of a tableaux correspond to marked coordinates (e.g., marked normal disc types). More...
 

Detailed Description

template<class LPConstraint = LPConstraintNone, typename BanConstraint = BanNone, typename IntType = Integer>
class regina::TautEnumeration< LPConstraint, BanConstraint, IntType >

The main entry point for the tree traversal algorithm to enumerate all taut angle structures in a 3-manifold triangulation.

For the analogous algorithm to enumerate vertex normal or almost normal surfaces, see the class TreeEnumeration instead.

This class follows a similar structure to the enumeration of vertex normal surfaces, as described in "A tree traversal algorithm for decision problems in knot theory and 3-manifold topology", Burton and Ozlen, Algorithmica 65:4 (2013), pp. 772-801.

To enumerate all taut angle structures on a given 3-manifold triangulation, simply construct a TautEnumeration object and call run().

Alternatively, you can have more fine-grained control over the search. Instead of calling run(), you can construct a TautEnumeration object and repeatedly call next() to step through each taut angle structure one at a time. This allows you to pause and resume the search as you please.

By using appropriate template parameters LPConstraint and/or BanConstraint, it is possible to impose additional linear constraints on the angle structure solution space, and/or explicitly force particular angles to be zero. See the LPConstraintBase and BanConstraintBase class notes for details.

The template argument IntType indicates the integer type that will be used throughout the underlying linear programming machinery. Unless you have a good reason to do otherwise, you should use the arbitrary-precision Integer class (in which integers can grow arbitrarily large, and overflow can never occur).

Precondition
The parameters LPConstraint and BanConstraint must be subclasses of LPConstraintSubspace and BanConstraintBase respectively. Note in particular that the base class LPConstraintBase is not enough here. See the LPConstraintBase, LPConstraintSubspace and BanConstraintBase class notes for further details.
The default constructor for the template class IntType must intialise each new integer to zero. The classes Integer and NativeInteger, for instance, have this property.
Headers
Parts of this template class are implemented in a separate header (treetraversal-impl.h), which is not included automatically by this file. Most end users should not need this extra header, since Regina's calculation engine already includes explicit instantiations for common combinations of template arguments.
Warning
The API for this class has not yet been finalised. This means that the class interface may change in new versions of Regina, without maintaining backward compatibility. If you use this class directly in your own code, please watch the detailed changelogs upon new releases to see if you need to make changes to your code.
Python
Not present.

Member Function Documentation

◆ buildStructure()

AngleStructure* regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::buildStructure ( ) const
inherited

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).

Precondition
This tree traversal is at a point in the search where it has found a feasible solution that represents a taut angle structure. This condition is always true after TautEnumeration::next() returns true, or any time that TautEnumeration::run() calls its callback function.
We are working with angle structure coordinates; that is, the coordinate system passed to the TreeTraversal constructor was NS_ANGLE.
Returns
the taut angle structure that has been found at the current stage of the search.

◆ buildSurface()

NormalSurface* regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::buildSurface ( ) const
inherited

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 TreeSingleSoln::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.

Precondition
This tree traversal is at a point in the search where it has found a feasible solution that represents a normal surface (though this need not be a vertex surface). This condition is always true after TreeEnumeration::next() or TreeSingleSoln::find() returns true, or any time that TreeEnumeration::run() calls its callback function.
We are working with normal or almost normal surfaces. That is, the coordinate system passed to the TreeTraversal constructor was not NS_ANGLE.
Returns
a normal surface that has been found at the current stage of the search.

◆ constraintsBroken()

bool regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::constraintsBroken
inlineinherited

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 TreeTraversal 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).

Returns
true if the constraints were not added successfully, or false if the constraints were added successfully.

◆ dumpTypes()

void regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::dumpTypes ( std::ostream &  out) const
inlineinherited

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.

Parameters
outthe output stream to which to write.

◆ feasibleBranches()

int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::feasibleBranches ( int  quadType)
protectedinherited

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 TreeEnumeration, not merged together as in TreeSingleSoln.

Precondition
The given quadrilateral or angle type has not yet been processed in the search tree (i.e., it has not had an explicit value selected).
When using angle structure coordinates, the final scaling coordinate has already been enforced as positive. (This is because, for angle structures, this routine does nothing to eliminate the zero solution.)
Parameters
quadTypethe quadrilateral or angle type to examine.
Returns
the number of type values 0, 1, 2 or 3 that yield a feasible system; this will be between 0 and 4 inclusive for quadrilateral types, or between 0 and 3 inclusive for angle types.

◆ nextUnmarkedTriangleType()

int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nextUnmarkedTriangleType ( int  startFrom)
inlineprotectedinherited

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.

Precondition
We are working in standard normal or almost normal coordinates. That is, the coordinate system passed to the TreeTraversal constructor was one of NS_STANDARD or NS_AN_STANDARD.
The argument startFrom is at least nTets_ (i.e., it is at least as large as the index of the first triangle type).
Parameters
startFromthe index into the type vector of the triangle type from which we begin searching.
Returns
the index into the type vector of the next unmarked triangle type from startFrom onwards, or -1 if there are no more remaining.

◆ nVisited()

unsigned long regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nVisited
inlineinherited

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 TreeEnumeration::run() or TreeSingleSoln::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 TreeEnumeration::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.

Returns
the number of nodes visited so far.

◆ percent()

double regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::percent ( ) const
protectedinherited

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).

Returns
the percentage, as a number between 0 and 100 inclusive.

◆ setNext()

void regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::setNext ( int  nextType)
protectedinherited

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.

Precondition
nextType is in the range 0,...,nTypes-1 inclusive.
nextType is still waiting to be processed; that is, nextType does not appear in the list typeOrder_[0,...,level_].
Parameters
nextTypethe next type to process.

◆ supported()

bool regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::supported ( NormalCoords  coords)
inlinestaticinherited

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.

Parameters
coordsthe coordinate system being queried.
Returns
true if and only if this coordinate system is supported.

◆ verify() [1/2]

bool regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::verify ( const AngleStructure s,
const MatrixInt angleEqns = 0 
) const
inherited

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 AngleStructureVector::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 non-standard angle equation matrix, and/or reuse the same matrix to avoid the overhead of reconstructing it every time this routine is called.

Precondition
The coordinate system passed to the TreeTraversal constructor was NS_ANGLE.
If angleEqns is non-null, then the number of columns in angleEqns is equal to the number of coordinates in the underlying angle structure coordinate system.
Parameters
sthe angle structure to verify.
angleEqnsthe 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 AngleStructureVector::makeAngleEquations().
Returns
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).

◆ verify() [2/2]

bool regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::verify ( const NormalSurface s,
const MatrixInt matchingEqns = 0 
) const
inherited

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 non-standard matching equation matrix, and/or reuse the same matrix to avoid the overhead of reconstructing it every time this routine is called.

If you do not pass your own matching equations and Regina is unable to construct them (which can happen when the underlying triangulation is not supported by the normal coordinate system), then this routine will return false.

Precondition
The normal or almost normal surface s uses the same coordinate system as was passed to the TreeTraversal constructor. Moreover, this coordinate system is in fact a normal or almost normal coordinate system (i.e., not NS_ANGLE).
If matchingEqns is non-null, then the number of columns in matchingEqns is equal to the number of coordinates in the underlying normal or almost normal coordinate system.
Parameters
sthe normal surface to verify.
matchingEqnsthe 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().
Returns
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).

Member Data Documentation

◆ coords_

const NormalCoords regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::coords_
protectedinherited

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.

◆ level_

int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::level_
protectedinherited

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.

◆ lp_

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 pre-prepared for future processing. See the documentation within routines such as TreeEnumeration::next() for details of when and how these tableaux are constructed.

◆ lpSlot_

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 TreeEnumeration::next() for details of when and how these tableaux are constructed and later overwritten.

◆ nextSlot_

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.

◆ nTableaux_

const int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nTableaux_
protectedinherited

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

◆ nTets_

const int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nTets_
protectedinherited

The number of tetrahedra in the underlying triangulation.

◆ nTypes_

const int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nTypes_
protectedinherited

The total length of a type vector.

◆ nVisited_

unsigned long regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::nVisited_
protectedinherited

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.

◆ octLevel_

int regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::octLevel_
protectedinherited

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.

◆ origTableaux_

const LPInitialTableaux<LPConstraintNone > regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::origTableaux_
protectedinherited

The original starting tableaux that holds the adjusted matrix of matching equations, before the tree traversal algorithm begins.

◆ tmpLP_

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.

◆ type_

char* regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::type_
protectedinherited

The current working type vector.

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

◆ typeOrder_

int* regina::TreeTraversal< LPConstraintNone , BanNone , Integer >::typeOrder_
protectedinherited

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).


The documentation for this class was generated from the following file:

Copyright © 1999-2021, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@maths.uq.edu.au).