regina::HilbertCD Class Reference

Implements a modified Contejean-Devie algorithm for enumerating Hilbert bases. More...

#include <enumerate/hilbertcd.h>

Static Public Member Functions
template<class RayClass , typename Action >
static void	enumerate (Action &&action, const MatrixInt &subspace, const ValidityConstraints &constraints)
	Determines the Hilbert basis that generates all integer points in the intersection of the n-dimensional non-negative orthant with some linear subspace. More...

Detailed Description

Implements a modified Contejean-Devie algorithm for enumerating Hilbert bases.

This is based on the stack-based algorithm described in "An efficient incremental algorithm for solving systems of linear Diophantine equations", Inform. and Comput. 113 (1994), 143-172, and has been modified to allow for additional constraints (such as the quadrilateral constraints from normal surface theory).

All routines of interest within this class are static; no object of this class should ever be created.

Warning

For normal surface theory, the Contejean-Devie algorithm is extremely slow, even when modified to incorporate admissibility constraints. Consider using the much faster HilbertPrimal or HilbertDual instead.

Member Function Documentation

enumerate()

template<class RayClass , typename Action >

static void regina::HilbertCD::enumerate ( Action &&  action, const MatrixInt &  subspace, const ValidityConstraints &  constraints  )

static

Determines the Hilbert basis that generates all integer points in the intersection of the n-dimensional non-negative orthant with some linear subspace.

The resulting basis elements will be of the class RayClass, and will be passed into the given action function one at a time.

The non-negative orthant is an n-dimensional cone with its vertex at the origin. The extremal rays of this cone are the n non-negative coordinate axes. This cone also has n facets, where the ith facet is the non-negative orthant of the plane perpendicular to the ith coordinate axis.

This routine takes a linear subspace, defined by the intersection of a set of hyperplanes through the origin (this subspace is described as a matrix, with each row giving the equation for one hyperplane).

The purpose of this routine is to compute the Hilbert basis of the set of all integer points in the intersection of the original cone with this linear subspace. The resulting list of basis vectors will contain no duplicates or redundancies.

The parameter constraints may contain a set of validity constraints, in which case this routine will only return valid basis elements. Each validity constraint is of the form "at most one of these coordinates may be non-zero"; see the ValidityConstraints class for details. These contraints have the important property that, although validity is not preserved under addition, invalidity is.

For each of the resulting basis elements, this routine will call action (which must be a function or some other callable object). This action should return void, and must take exactly one argument, which will be the basis element stored using RayClass. The argument will be passed as an rvalue; a typical action would take it as an rvalue reference (RayClass&&) and move its contents into some other more permanent storage.

Precondition

The template argument RayClass is derived from (or equal to) Vector<T>, where T is one of Regina's arbitrary-precision integer classes (Integer or LargeInteger).

Warning

For normal surface theory, the Contejean-Devie algorithm is extremely slow, even when modified to incorporate admissibility constraints. Consider using the much faster HilbertPrimal or HilbertDual instead.

Python

There are two versions of this function available in Python. The first version is the same as the C++ function; here you must pass action, which may be a pure Python function. The second form does not have an action argument; instead you call enumerate(subspace, constraints), and it returns a Python list containing all Hilbert basis elements. In both versions, the argument RayClass is fixed as VectorInt.

Parameters

action	a function (or other callable object) that will be called for each basis element. This function must take a single argument, which will be passed as an rvalue of type RayClass.
subspace	a matrix defining the linear subspace to intersect with the given cone. Each row of this matrix is the equation for one of the hyperplanes whose intersection forms this linear subspace. The number of columns in this matrix must be the dimension of the overall space in which we are working.
constraints	a set of validity constraints as described above, or ValidityConstraints::none if none should be imposed.

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

• enumerate/hilbertcd.h