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/* Copyright (c) 1997-2024
Ewgenij Gawrilow, Michael Joswig, and the polymake team
Technische Universität Berlin, Germany
https://polymake.org
This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 2, or (at your option) any
later version: http://www.gnu.org/licenses/gpl.txt.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
--------------------------------------------------------------------------------
*/
#pragma once
#include "polymake/Integer.h"
#include "polymake/Vector.h"
#include "polymake/Map.h"
#include "polymake/PowerSet.h"
#include "polymake/permutations.h"
namespace pm {
namespace {
template <typename T, typename Iterator>
void make_index(Iterator it, Map<T, Int>& index_of)
{
Int index = 0;
while (!it.at_end()) {
index_of[*it] = index++;
++it;
}
}
Vector<Int> squeeze(const Vector<Int>& v, const Set<Int>& s)
{
Map<Int, Int> index_of;
make_index(entire(s), index_of);
Vector<Int> w(v.size());
auto wit = entire(w);
for (auto vit = entire(v); !vit.at_end(); ++vit)
*wit++ = index_of[*vit];
return w;
}
} // end anonymous namespace
template <typename T>
class Plucker { // a class to hold and process Plücker coordinates of a subspace ("flat")
public:
typedef Map<Set<Int>, T> CooType;
protected:
Int d_; // the dimension of the ambient space
Int k_; // the dimension of the flat
CooType coos_; // the binom{d_,k_}-tuple of Plücker coordinates of the flat
public:
template <typename>
friend struct spec_object_traits;
Plucker()
: d_(0)
, k_(0)
, coos_(CooType()) {}
template <typename E>
explicit Plucker(const Vector<E>& v)
: d_(v.size())
, k_(1)
, coos_(CooType()) {
auto vit = entire(v);
for (Int i = 0; i < d_; ++i, ++vit)
coos_[scalar2set(i)] = *vit;
}
template <typename E>
Plucker(Int d, Int k, const Vector<E>& v)
: d_(d)
, k_(k)
, coos_(CooType()) {
if (v.size() != Integer::binom(d, k))
throw std::runtime_error("The number of coordinates is not the expected one, binom(d,k)");
auto vit = entire(v);
for (auto fit = entire(all_subsets_of_k(sequence(0,d_), k_)); !fit.at_end(); ++fit, ++vit)
coos_[*fit] = *vit;
}
Plucker(Int d, Int k)
: d_(d)
, k_(k)
, coos_(CooType()) {}
Int d() const { return d_; }
Int k() const { return k_; }
const T& operator[] (const Set<Int>& s) const { return coos_[s]; }
Vector<T> coordinates() const
{
Vector<T> v(static_cast<Int>(Integer::binom(d_,k_)));
auto vit = entire(v);
for (auto cit = entire(coos_); !cit.at_end(); ++cit, ++vit)
*vit = cit->second;
return v;
}
Vector<T> point() const
{
if (k_!=1) {
throw std::runtime_error("The dimension of the flat " + std::to_string(k_) + " > 1, it can't be converted to a point");
}
return coordinates();
}
template <typename Permutation>
Plucker<T> permuted(const Permutation& perm) const
{
if(perm.size()!=d_)
throw std::runtime_error("The size of the permutation is not the expected one.");
Plucker<T> plucker(d_,k_);
for (auto cit = entire(coos_); !cit.at_end(); ++cit)
plucker.coos_[pm::permuted(cit->first,perm)] = cit->second;
return plucker;
}
friend Plucker join(const Plucker& p1, const Plucker& p2)
{
if (p1.d() != p2.d())
throw std::runtime_error("Ambient dimensions of p1 and p2 are not equal");
const Int d = p1.d(), k = p1.k() + p2.k();
if (k > d)
throw std::runtime_error("The sum of the dimensions of the flats " + std::to_string(k) +
" is greater than the dimension of the ambient space " + std::to_string(d) +
", they can't be joined");
// We iterate over all pairs of sets (A,B), A in binom{[d],k1}, B in binom{[d],k2}
// such that A and B are disjoint.
Plucker result(d,k);
for (auto Ait = entire(all_subsets_of_k(sequence(0,p1.d()), p1.k())); !Ait.at_end(); ++Ait) {
Set<Int> rest(sequence(0, d) - *Ait);
for (auto Bit = entire(all_subsets_of_k(rest, p2.k())); !Bit.at_end(); ++Bit) {
Set<Int> U(*Ait); U += *Bit;
const Vector<Int> perm = Vector<Int>(p1.k(), entire(*Ait)) | Vector<Int>(p2.k(), entire(*Bit));
result.coos_[U] += permutation_sign(squeeze(perm, U)) * p1[*Ait] * p2[*Bit];
}
}
return result;
}
friend Plucker meet(const Plucker& p1, const Plucker& p2)
{
if (p1.d() != p2.d())
throw std::runtime_error("Ambient dimensions of p1 and p2 are not equal");
const Int d = p1.d(), k = p1.k() + p2.k() - d;
if (k < 0) {
throw std::runtime_error("The sum of the dimensions of the flats " + std::to_string(p1.k() + p2.k()) +
" is less than the dimension of the ambient space " + std::to_string(d) +
", they can't be intersected");
}
// We iterate over all pairs of sets (A,B), A in binom{[d],k1}, B in binom{[d],k2},
// such that |A cap B| = k = k1+k2-d.
// For this, we decompose A into A = A1 cup S with S = A cap B, so that B = S cup B1
// for a (d-k1)-set B1.
Plucker result(d, k);
for (auto Ait = entire(all_subsets_of_k(sequence(0, p1.d()), p1.k())); !Ait.at_end(); ++Ait) {
const Set<Int> A{*Ait};
const Set<Int> rest = sequence(0, d) - A;
for (auto Sit = entire(all_subsets_of_k(A, k)); !Sit.at_end(); ++Sit) {
const Set<Int> S{*Sit};
for (auto B1it = entire(all_subsets_of_k(rest, d-p1.k())); !B1it.at_end(); ++B1it) {
const Set<Int> B1{*B1it};
const Set<Int> B = S+B1;
const Set<Int> A1 = A-S;
const Vector<Int> perm = Vector<Int>(A1.size(), entire(A1)) | Vector<Int>(B1.size(), entire(B1));
result.coos_[*Sit] += permutation_sign(squeeze(perm, A1+B1)) * p1[A] * p2[B];
}
}
}
return result;
}
friend Plucker operator+ (const Plucker& p1, const Plucker& p2) { return join(p1,p2); }
friend Plucker operator* (const Plucker& p1, const Plucker& p2) { return meet(p1,p2); }
/** This function takes a 2-flat F and a vector v that is supposed to be contained in it,
* and gives back a vector that spans the orthogonal complement of v in F.
*/
Vector<T> project_out(const Vector<T>& v) const
{
if (k_ != 2)
throw std::runtime_error("Only projecting from planes is implemented");
SparseMatrix<T> M(static_cast<Int>(Integer::binom(d_, 2)+1), d_);
Int row_ct = 0;
for (auto fit = entire(all_subsets_of_k(sequence(0, d_), k_)); !fit.at_end(); ++fit, ++row_ct) {
M(row_ct, fit->front()) = -v[fit->back()];
M(row_ct, fit->back()) = v[fit->front()];
}
M.row(row_ct) = v;
const Vector<T> vs = coordinates() | 1;
return lin_solve(M, vs).dehomogenize();
}
SparseVector<T> project_out(const Plucker& p) const
{
return project_out(p.point());
}
template <typename Output> friend
Output& operator<< (GenericOutput<Output>& outs, const Plucker& e)
{
return outs.top() << "(" << e.d() << " " << e.k() << " [" << e.coordinates() << "])";
}
};
} // end namespace pm
namespace polymake {
using pm::Plucker;
}
/*
namespace std {
template <typename T>
struct numeric_limits<pm::Plucker<T> > {
// static const bool is_integer = false;
// static const bool is_signed = false;
};
}
*/
// Local Variables:
// mode:C++
// c-basic-offset:3
// indent-tabs-mode:nil
// End:
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