/* Copyright (c) 1997-2021
   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.
--------------------------------------------------------------------------------
*/

#include "polymake/client.h"
#include "polymake/Matrix.h"
#include "polymake/PowerSet.h"
#include "polymake/Array.h"
#include "polymake/Integer.h"
#include "polymake/linalg.h"
#include "polymake/matroid/deletion_contraction.h"

namespace polymake { namespace matroid {

namespace {

Int get_index(Int element, const Array<Int>& index)
{
   auto where = std::lower_bound(index.begin(), index.end(), element);
   return where != index.end() && *where==element ? where - index.begin() : -1;
}

// Produced a potential matrix for binary representation.
// Also returns the basis for which the unit matrix was chosen.
std::pair<Matrix<Int>, Set<Int>>
produce_binary_matrix(const Int n_elements, const Int rank, const Array<Set<Int>>& bases)
{
   const Set<Int> basis = bases.front();
   const Array<Int> basis_index(basis);

   Matrix<Int> bpoints(n_elements, rank);

   // Write the unit matrix for the first basis
   Int i = 0;
   for (auto b = entire(basis); !b.at_end(); ++b, ++i) {
      bpoints(*b, i)=1;
   }

   // Insert 1's according to basis exchange axiom
   for (auto it = entire(bases); !it.at_end(); ++it) {
      const Int intersection_size = (basis * (*it)).size();
      if (intersection_size == rank-1) {
         Int temp = (*it-basis).front();
         Int basis_element_index = get_index((basis-*it).front(), basis_index);
         bpoints(temp, basis_element_index)=1;
      }
   }

   return std::make_pair(bpoints, basis);
}

// Computes the bases of a subset of points mod p
// Minor is assumed to have full (column) rank.
Set<Set<Int>> bases_for_finite_field(const Matrix<Int>& points, const Set<Int>& minor, Int p)
{
   Set<Set<Int>> result;
   for (auto i = entire(all_subsets_of_k(minor, points.cols())); !i.at_end(); ++i) {
      if (det(points.minor(*i, All)) % p != 0)
         result.push_back(*i);  // all_subsets_of_k come in lexicographically ascending order
   }
   return result;
}


// Take a ternary column (i.e. containing 0,1,-1) and changes it to the "next"
// vector in the sense that the support remains unchanged and the 
// supporting +-1-part is treated as encoding a binary number, which is incremented by 1.
// However, the first nonzero entry has to remain 1 always and if the row cannot be increased anymore,
// the row is resetted to contain only 1's in its support and returns false, otherwise true.
bool increase_ternary_row(Matrix<Int>& points, Int row)
{
   const Set<Int> supp = support(points.row(row));
   if (supp.size() <= 1) return false;
   Int fr = supp.front();

   for (auto r_it = entire<reversed>(supp); *r_it != fr; ++r_it) {
      points(row, *r_it) *= -1;
      if (points(row, *r_it) < 0) return true;
   }

   points.row(row).slice(supp).fill(1);
   // If we arrive here, we can't actually increase the vector 
   return false;         
}

} //end namespace

void binary_representation(BigObject matroid)
{
   const Array<Set<Int>> bases = matroid.give("BASES");
   const Int r = matroid.give("RANK");
   const Int n = matroid.give("N_ELEMENTS");

   if (r == 0) {
      matroid.take("BINARY") << 1;
      matroid.take("BINARY_VECTORS") << Matrix<Int>(n, 1);
      return;
   }

   const Matrix<Int> bpoints = produce_binary_matrix(n, r, bases).first;

   const Set<Set<Int>> bpoint_basis_set = bases_for_finite_field(bpoints, sequence(0, bpoints.rows()), 2);
   if (bpoint_basis_set == Set<Set<Int>>(bases)) {
      matroid.take("BINARY_VECTORS") << bpoints;
      matroid.take("BINARY") << true;
   } else {
      matroid.take("BINARY") << false;
   }
}

void ternary_representation(BigObject matroid)
{
   const Array<Set<Int>> bases = matroid.give("BASES");
   const Int r = matroid.give("RANK");
   const Int n = matroid.give("N_ELEMENTS");

   if (r == 0) {
      matroid.take("TERNARY") << true;
      matroid.take("TERNARY_VECTORS") << Matrix<Int>(n, 1);
      return;
   }

   // First we compute the support of the matrix representation.
   std::pair<Matrix<Int>, Set<Int>> tpoint_prototype = produce_binary_matrix(n, r, bases);
   Matrix<Int> tpoints = tpoint_prototype.first;
   const Set<Int>& unit_basis = tpoint_prototype.second;
   const Array<Int> remaining_elements(sequence(0, n) - unit_basis);

   Map<Int, Int> label_identity;
   for (auto li = entire(sequence(0, n)); !li.at_end(); ++li) {
      label_identity[*li] = *li;
   }

   // Compute bases of restrictions to unit_basis + remaining_elements[0],...,unit_basis + (all remaining_elements)
   Map<Int, Set<Set<Int>>> restriction_bases;
   Map<Int, Set<Int>> restricted_ground_sets;
   restriction_bases[ remaining_elements.size()-1] = Set<Set<Int>>{bases};
   restricted_ground_sets[ remaining_elements.size()-1] = sequence(0,n);
   Set<Int> deletion_set;
   for (Int re_index = remaining_elements.size()-1; re_index >= 1; --re_index) {
      restriction_bases[ re_index-1 ] = minor_bases(Deletion(), restriction_bases[re_index], 
                                                    scalar2set(remaining_elements[re_index]), 
                                                    label_identity);
      restricted_ground_sets[ re_index-1 ] = restricted_ground_sets[ re_index ] 
                                             - remaining_elements[re_index];
   }

   // Index of the "remaining_elements" element currently being added
   Int backtrack_index = 0;
   bool need_to_increase = false;

   // Backtracking algorithm for iterating all potential presentations.
   while (backtrack_index >= 0) {
      if (need_to_increase) {
         bool have_increased = false;
         // The first vector after the unit matrix is never changed.
         if (backtrack_index > 0) 
            have_increased = increase_ternary_row( tpoints, remaining_elements[ backtrack_index ]);

         if (!have_increased) {
            --backtrack_index;
            continue;
         } else {
            need_to_increase = false;
         }
      }

      if (bases_for_finite_field(tpoints,restricted_ground_sets[backtrack_index], 3)
          == restriction_bases[ backtrack_index ]) {
         ++backtrack_index; 
         if (backtrack_index == remaining_elements.size()) {
            // This is a presentation!
            matroid.take("TERNARY") << true;
            matroid.take("TERNARY_VECTORS") << tpoints;
            return;
         }
      } else {
         need_to_increase = true;
      }
   } // END while

   // If we arrive here, it's not ternary
   matroid.take("TERNARY") << false;
}

Function4perl(&binary_representation, "binary_representation(Matroid)");
Function4perl(&ternary_representation, "ternary_representation(Matroid)");

} }

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