Regina Calculation Engine
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Stores an intermediate tableaux for the dual simplex method, and contains all of the core machinery for using the dual simplex method. More...
#include <enumerate/ntreelp.h>
Public Member Functions | |
LPData () | |
Constructs a new tableaux. More... | |
~LPData () | |
Destroys this tableaux. More... | |
void | reserve (const LPInitialTableaux< LPConstraint > *origTableaux) |
Reserves enough memory for this tableaux to work with. More... | |
void | initStart () |
Initialises this tableaux by beginning at the original starting tableaux and working our way to any feasible basis. More... | |
void | initClone (const LPData &parent) |
Initialises this tableaux to be a clone of the given tableaux. More... | |
unsigned | columns () const |
Returns the number of columns in this tableaux. More... | |
unsigned | coordinateColumns () const |
Returns the number of columns in this tableaux that correspond to normal coordinates or angle structure coordinates. More... | |
bool | isFeasible () const |
Returns whether or not this system is feasible. More... | |
bool | isActive (unsigned pos) const |
Determines whether the given variable is currently active. More... | |
int | sign (unsigned pos) const |
Returns the sign of the given variable under the current basis. More... | |
void | constrainZero (unsigned pos) |
Constrains this system further by setting the given variable to zero and deactivating it. More... | |
void | constrainPositive (unsigned pos) |
Constrains this system further by constraining the given variable to be strictly positive. More... | |
void | 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 | dump (std::ostream &out) const |
Writes details of this tableaux to the given output stream. More... | |
void | 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... | |
Stores an intermediate tableaux for the dual simplex method, and contains all of the core machinery for using the dual simplex method.
This class forms part of the tree traversal algorithms for enumerating and locating 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, and "A fast branching algorithm for unknot recognition with experimental polynomial-time behaviour", Burton and Ozlen, arXiv:1211.1079. It is also used for locating a single strict angle structure, and for enumerating all taut angle structures.
This class is designed to represent a state partway through the tree traversal algorithm, where the tableaux has been altered to constrain some variables:
We do not store the full tableaux (which is dense and slow to work with). Instead we store the matrix of row operations that were applied to the original starting tableaux (in the notation of Burton and Ozlen, we store the matrix M_beta^{-1}, where M is the original matrix stored in the class LPInitialTableaux, and beta is the current basis).
If the system is infeasible (because the constraints on variables as described above are too severe), then the contents of the internal data members are undefined (other than the data member feasible_, which is guaranteed to be false
). This is because the code is optimised to abort any operation as soon as infeasibility is detected, which may leave the data members in a broken state. If you are not sure, you should always call isFeasible() before performing any other query or operation on this tableaux.
This class is designed to be used in a backtracking search, which means the API is cumbersome but we can quickly rewrite and copy data. The rules are as follows:
Like LPInitialTableaux, this class can enforce additional linear constraints (such as positive Euler characteristic) through the template parameter LPConstraint. If there are no such constraints, simply use the template parameter LPConstraintNone.
In the context of normal surfaces (not angle structures): Although the underlying coordinate system is based on quadrilaterals and (optionally) triangles, this class has elementary support for octagons also, as seen in almost normal surface theory. For the purposes of this class, an octagon is represented as a pair of quadrilaterals of different types in the same tetrahedron: these meet the boundary of the tetrahedron in the same arcs as a single octagon, and therefore interact with the matching equations in the same way.
To declare that you will be using octagons in some tetrahedron, you must call constrainOct(quad1, quad2), where quad1 and quad2 are the two corresponding quadrilateral columns. This will have the following effects, all of which may alter the tableaux:
This class has been optimised to ensure that you only have one octagon type declared at any given time (which is consistent with the constraints of almost normal surface theory).
All tableaux elements are of the integer class Integer, which is supplied as a template argument. This same integer class will be used as a template argument for LPConstraint.