/*
	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
	of the License, or (at your option) any later version.

	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.

	You should have received a copy of the GNU General Public License
	along with this program; if not, write to the Free Software
	Foundation, Inc., 51 Franklin Street, Fifth Floor,
	Boston, MA  02110-1301, USA.

	---
	Copyright (C) 2011 - 2015, Simon Hampe <simon.hampe@googlemail.com>

	Contains functions dealing with finding the diagonal as a piecwise divisor
	in a simplicial fan.
	*/

#include "polymake/client.h"
#include "polymake/Matrix.h"
#include "polymake/Rational.h"
#include "polymake/Vector.h"
#include "polymake/IncidenceMatrix.h"
#include "polymake/Array.h"
#include "polymake/PowerSet.h"
#include "polymake/tropical/divisor.h"
#include "polymake/tropical/skeleton.h"
#include "polymake/tropical/misc_tools.h"
#include "polymake/tropical/thomog.h"

namespace polymake { namespace tropical {



	///////////////////////////////////////////////////////////////////////////////////////

	//Documentation see perl wrapper
	template <typename Addition>
		perl::Object simplicial_with_diagonal(perl::Object fan) {

			//Extract values
			Matrix<Rational> rays = fan.give("VERTICES");
			IncidenceMatrix<> cones = fan.give("MAXIMAL_POLYTOPES");
			Vector<Integer> weights = fan.give("WEIGHTS");

			//Dehomogenize rays and remove leading coordinate
			Set<int> nonfar = far_and_nonfar_vertices(rays).second;
			rays = rays.minor(~nonfar,~scalar2set(0));
			rays = tdehomog(rays,0,false);
			cones = cones.minor(All,~nonfar);

			//Compute new rays (first the first component, then the second, then the diagonal rays)
			Matrix<Rational> p_rays(3*rays.rows(), 2* rays.cols());
			p_rays.minor(sequence(0,rays.rows()), sequence(0, rays.cols())) = 
				p_rays.minor(sequence(rays.rows(), rays.rows()), sequence(rays.cols(), rays.cols())) = 
				p_rays.minor(sequence(2*rays.rows(), rays.rows()), sequence(0,rays.cols())) =
				p_rays.minor(sequence(2*rays.rows(), rays.rows()), sequence(rays.cols(), rays.cols())) = rays;

			//Compute translation maps for ray indices
			Map<int,int> second_comp;
			Map<int,int> diagon_comp;
			for(int r = 0; r < rays.rows(); r++) {
				second_comp[r] = r+rays.rows();
				diagon_comp[r] = r+ 2* rays.rows();
			}

			Vector<Set<int> > p_cones;
			Vector<Integer> p_weights;
			//Go through pairs of cones
			for(int c1 = 0; c1 < cones.rows(); c1++) {
				Set<int> cone1 = cones.row(c1);
				for(int c2 = 0; c2 < cones.rows(); c2++) {
					Set<int> cone2 = cones.row(c2);
					Set<int> inter = cone1 * cone2;
					//If the cones share no ray, their product is not subdivided
					if (inter.size() == 0) {
                                           Set<int> cone2_translate{ second_comp.map(cone2) };
                                           p_cones |= (cone1 + cone2_translate);
                                           p_weights |= weights[c1] * weights[c2];
					}
					else {
						//Otherwise we get a subdivision cone for each subset of the intersection
                                                Array<Set<int>> subdivisions{ pm::all_subsets(inter) };
						for(int sd = 0; sd < subdivisions.size(); sd++) {
							Set<int> complement = inter - subdivisions[sd];
							//We replace the elements from the subset of the intersection in the FIRST cone
							//by the corresponding diagonal rays and replace the elements of the rest of the
							//intersection in the SECOND cone by the corresponding diagonal rays
							Set<int> first_cone_remaining = cone1 - subdivisions[sd];
							Set<int> second_cone_remaining{ second_comp.map(cone2 - complement) };
							Set<int> first_cone_diagonal{ diagon_comp.map(subdivisions[sd]) };
							Set<int> second_cone_diagonal{ diagon_comp.map(complement) };

							Set<int> final_cone = first_cone_remaining + second_cone_remaining + 
								first_cone_diagonal + second_cone_diagonal;

							p_cones |= final_cone;
							p_weights |= weights[c1] * weights[c2];

						}//END iterate subdivisions
					}//END compute subdivision cones

				}//END iterate second cone
			}//END iterate first cone

			//Re-add a vertex at the origin 
			p_rays = zero_vector<Rational>() | p_rays;
			p_rays = p_rays / unit_vector<Rational>(p_rays.cols(),0);
			for(int c = 0; c < p_cones.dim(); c++) 
				p_cones[c] += (p_rays.rows()-1);

			//Return result
			perl::Object result(perl::ObjectType::construct<Addition>("Cycle"));
			result.take("VERTICES") << thomog(p_rays);
			result.take("MAXIMAL_POLYTOPES") << p_cones;
			result.take("WEIGHTS") << p_weights;
			return result;
		}//END function simplicial_with_diagonal

	///////////////////////////////////////////////////////////////////////////////////////

	//Documentation see perl wrapper
	template <typename Addition>
		Matrix<Rational> simplicial_piecewise_system(perl::Object fan) {

			//Extract values
			int fan_dim = fan.give("PROJECTIVE_DIM");

			//Compute diagonal subdivision and skeleton
			perl::Object diag_fan = simplicial_with_diagonal<Addition>(fan);
			perl::Object skeleton = skeleton_complex<Addition>(diag_fan,fan_dim,true);

			//Extract values
			Matrix<Rational> old_rays = diag_fan.give("SEPARATED_VERTICES");
			old_rays = tdehomog(old_rays);
			Set<int> nonfar = far_and_nonfar_vertices(old_rays).second;
			IncidenceMatrix<> skeleton_cones = skeleton.give("SEPARATED_MAXIMAL_POLYTOPES");

			//Prepare solution matrix
			Matrix<Rational> result(skeleton_cones.rows(),0);

			//Iterate through all cones of the skeleton
			for(int tc = 0; tc < skeleton_cones.rows(); tc++) {
				//Create the function(s) Psi_tau
				Vector<int> tc_rays(skeleton_cones.row(tc) - nonfar);
				Matrix<Rational> psi_tau(0,old_rays.rows());
				for(int r = 0; r < tc_rays.dim(); r++) {
					psi_tau /= (unit_vector<Rational>(old_rays.rows(), tc_rays[r]));
				}

				//Compute the divisor
				perl::Object divisor = divisorByValueMatrix<Addition>(diag_fan, psi_tau);

				//Extract cones, rays and weights
				Matrix<Rational> div_rays = divisor.give("VERTICES");
				div_rays = tdehomog(div_rays);
				IncidenceMatrix<> div_cones = divisor.give("MAXIMAL_POLYTOPES");
				Vector<Integer> div_weights = divisor.give("WEIGHTS");

				//Associate to each ray its original index 
				Map<int,int> div_ray_to_old;
				for(int dr = 0; dr < div_rays.rows(); dr++) {
					for(int oray = 0; oray < old_rays.rows(); oray++) {
						if(old_rays.row(oray) == div_rays.row(dr)) {
							div_ray_to_old[dr] = oray; break;
						}
					}
				}//END translate ray indices

				//Now go through all d-dimensional cones in the divisor and insert their weight at the appropriate point
				Vector<Integer> tau_column(skeleton_cones.rows());
				for (int rho = 0; rho < div_cones.rows(); rho++) {
					//Map rho rays to old rays
                                        Set<int> rho_old{ div_ray_to_old.map(div_cones.row(rho)) };
					//Find the original cone equal to that
					for (int oc = 0; oc < skeleton_cones.rows(); oc++) {
						if ((skeleton_cones.row(oc) * rho_old).size() == fan_dim+1) {
							tau_column[oc] = div_weights[rho]; break;
						}
					}
				}//END iterate divisor cones
				result |= tau_column;

			}//END iterate skeleton cones

			return result;

		}//END simplicial_piecewise_kernel

	///////////////////////////////////////////////////////////////////////////////////////

	//Documentation see perl wrapper
	template <typename Addition>
		Matrix<Rational> simplicial_diagonal_system(perl::Object fan) {
			//Extract values
			int fan_dim = fan.give("PROJECTIVE_DIM");
			int fan_ambient_dim = fan.give("PROJECTIVE_AMBIENT_DIM");
			Matrix<Rational> fan_rays = fan.give("VERTICES");

			//Compute diagonal subdivision and skeleton
			perl::Object diag_fan = simplicial_with_diagonal<Addition>(fan);
			perl::Object skeleton = skeleton_complex<Addition>(diag_fan,fan_dim,true);

			Matrix<Rational> nspace = simplicial_piecewise_system<Addition>(fan);

			//Identify diagonal cones
			Matrix<Rational> sk_rays = skeleton.give("SEPARATED_VERTICES");
				sk_rays = tdehomog(sk_rays);
			IncidenceMatrix<> sk_cones = skeleton.give("SEPARATED_MAXIMAL_POLYTOPES");
			Set<int> diag_rays; 
			for (int r = 0; r < sk_rays.rows(); r++) {
				if (sk_rays.row(r).slice(1,fan_ambient_dim) ==
						sk_rays.row(r).slice(fan_ambient_dim+1, fan_ambient_dim) && sk_rays(r,0) == 0) 
					diag_rays += r;
				if (is_zero(sk_rays.row(r).slice(1)))
					diag_rays += r; // Add the origin as well.
			}
			Set<int> diag_cones;
			for(int c = 0; c < sk_cones.rows(); c++) {
				if((diag_rays * sk_cones.row(c)).size() == sk_cones.row(c).size()) {
					diag_cones += c;
				}
			}

			Vector<Rational> v(nspace.rows());
			v.slice(diag_cones) = ones_vector<Rational>(diag_cones.size());

			nspace |= v;
			return nspace;
		}

	// ------------------------- PERL WRAPPERS ---------------------------------------------------

	UserFunctionTemplate4perl("# @category Inverse problems"
			"# This function takes a simplicial fan F (without "
			"# lineality space) and computes the coarsest subdivision of F x F containing all "
			"# diagonal rays (r,r)"
			"# @param Cycle<Addition> F A simplicial fan without lineality space."
			"# @return Cycle<Addition> The product complex FxF subdivided such that it contains "
			"# all diagonal rays",
			"simplicial_with_diagonal<Addition>(Cycle<Addition>)");

	UserFunctionTemplate4perl("# @category Inverse problems"
			"# This function takes a d-dimensional simplicial fan F and computes the linear system "
			"# defined in the following way: For each d-dimensional cone t in the diagonal subdivision of FxF, let psi_t be the "
			"# piecewise polynomial defined by subsequently applying the rational functions that "
			"# are 1 one exactly one ray of t and 0 elsewhere. Now for which coefficients a_t"
			"# is sum_t a_t psi_t * (FxF) = 0?"
			"# @param Cycle<Addition> F  A simplicial fan without lineality space"
			"# @return Matrix<Rational> The above mentioned linear system. The rows "
			"# are equations, the columns correspond to d-dimensional cones of FxF in the order given "
			"# by skeleton_complex(simplicial_with_diagonal(F), d, 1)",
			"simplicial_piecewise_system<Addition>(Cycle<Addition>)");

	UserFunctionTemplate4perl("# @category Inverse problems"
			"# This function computes the inhomogeneous version of simplicial_piecewise_system"
			"# in the sense that it computes the result of the above mentioned function (i.e. "
			"# which coefficients for the piecewise polynomials yield the zero divisor)"
			"# and adds another column at the end where only the entries corresponding to the "
			"# diagonal cones are 1, the rest is zero. This can be seen as asking for a "
			"# solution to the system that cuts out the diagonal (all solutions whose last entry is 1)"
			"# @param Cycle<Addition> fan. A simplicial fan without lineality space."
			"# @return Matrix<Rational>",
			"simplicial_diagonal_system<Addition>(Cycle<Addition>)");

}}
