Regina Calculation Engine
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A class that constraints the tableaux of normal surface matching equations to ensure that Euler characteristic is strictly positive.
More...
#include <enumerate/treeconstraint.h>
Classes | |
struct | Coefficients |
Stores the extra coefficients in the tableaux associated with this constraint class (in this case, one extra integer per column). More... | |
Public Types | |
enum | { nConstraints = 1 } |
enum | { nConstraints } |
Static Public Member Functions | |
static bool | addRows (LPCol< regina::LPConstraintEuler > *col, const int *columnPerm, const Triangulation< 3 > *tri) |
template<typename IntType > | |
static void | constrain (LPData< regina::LPConstraintEuler, IntType > &lp, unsigned numCols) |
static bool | verify (const NormalSurface *s) |
static bool | verify (const AngleStructure *) |
static bool | supported (NormalCoords coords) |
static bool | addRows (LPCol< LPConstraintBase > *col, const int *columnPerm, const Triangulation< 3 > *tri) |
Explicitly constructs equations for the linear function(s) constrained by this class. More... | |
template<typename IntType > | |
static void | constrain (LPData< LPConstraintNone, IntType > &lp, unsigned numCols) |
Explicitly constraints each of these linear functions to an equality or inequality in the underlying tableaux. More... | |
A class that constraints the tableaux of normal surface matching equations to ensure that Euler characteristic is strictly positive.
There are many ways of writing Euler characteritic as a linear function. The function constructed here has integer coefficients, but otherwise has no special properties of note.
This constraint can work with either normal or almost normal coordinates. In the case of almost normal coordinates, the function is modified to measure Euler characteristic minus the number of octagons (a technique of Casson, also employed by Jaco and Rubinstein, that is used to ensure we do not have more than two octagons when searching for a normal or almost normal sphere in the 3-sphere recognition algorithm).
See the LPConstraintBase class notes for details on all member functions and structs.
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inherited |
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staticinherited |
Explicitly constructs equations for the linear function(s) constrained by this class.
Specifically, this routine takes an array of Coefficients objects (one for each column of the initial tableaux) and fills in the necessary coefficient data.
The precise form of the linear function(s) will typically depend upon the underlying triangulation. For this reason, the triangulation is explicitly passed, along with the permutation that indicates which columns of the initial tableaux correspond to which normal or angle structure coordinates.
More precisely: recall that, for each linear function, the initial tableaux acquires one new variable x_i that evaluates this linear function f(x). This routine must create the corresponding row that sets f(x) - x_i = 0
. Thus it must construct the coefficients of f(x) in the columns corresponding to normal coordinates, and it must also set a coefficient of -1 in the column for the corresponding new variable.
For each subclass S of LPConstraintBase, the array col must be an array of objects of type LPCol<S>. The class LPCol<S> is itself a larger subclass of the Coefficients class. This exact type must be used because the compiler must know how large each column object is in order to correct access each element of the given array.
As described in the LPInitialTableaux class notes, it might not be possible to construct the linear functions (since the triangulation might not satisfy the necessary requirements). In this case, this routine should ensure that the linear functions are in fact the zero functions, and should return false
(but it must still set -1 coefficients for the new variables as described above). Otherwise (if the linear function were successfully constructed) this routine should return true
.
If you are implementing this routine in a subclass that works with angle structure coordinates, remember that your linear constraints must not interact with the scaling coordinate (the final angle structure coordinate that is used to projectivise the angle structure polytope into a polyhedral cone). Your implementation of this routine must ensure that your linear constraints all have coefficient zero in this column.
col | the array of columns as stored in the initial tableaux (i.e., the data member LPInitialTableaux::col_). |
columnPerm | the corresponding permutation of columns that describes how columns of the tableaux correspond to normal or angle structure coordinates in the underlying triangulation (i.e., the data member LPInitialTableaux::columnPerm_). |
tri | the underlying triangulation. |
true
if the linear functions were successfully constructed, or false
if not (in which case they will be replaced with the zero functions instead).
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staticinherited |
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. |