# -*- coding: utf-8 -*-
"""
LP solvers for optimal transport using cvxopt
"""

# Author: Remi Flamary <remi.flamary@unice.fr>
#
# License: MIT License

import numpy as np
import scipy as sp
import scipy.sparse as sps

try:
    import cvxopt
    from cvxopt import solvers, matrix, sparse, spmatrix
except ImportError:
    cvxopt=False

def scipy_sparse_to_spmatrix(A):
    """Efficient conversion from scipy sparse matrix to cvxopt sparse matrix"""
    coo = A.tocoo()
    SP = spmatrix(coo.data.tolist(), coo.row.tolist(), coo.col.tolist(), size=A.shape)
    return SP

def barycenter(A, M, weights=None, verbose=False, log=False,solver='interior-point'):
    """Compute the entropic regularized wasserstein barycenter of distributions A

     The function solves the following optimization problem [16]:

    .. math::
       \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{1}(\mathbf{a},\mathbf{a}_i)

    where :

    - :math:`W_1(\cdot,\cdot)` is the Wasserstein distance (see ot.emd.sinkhorn)
    - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}`

    The linear program is solved using the default cvxopt solver if installed. 
    If cvxopt is not installed it uses the lp solver from scipy.optimize.

    Parameters
    ----------
    A : np.ndarray (d,n)
        n training distributions of size d
    M : np.ndarray (d,d)
        loss matrix   for OT
    reg : float
        Regularization term >0
    weights : np.ndarray (n,)
        Weights of each histogram i_i on the simplex          
    verbose : bool, optional
        Print information along iterations
    log : bool, optional
        record log if True
    solver : string, optional
        the solver used, default 'interior-point' use the lp solver from 
        scipy.optimize. None, or 'glpk' or 'mosek' use the solver from cvxopt.

    Returns
    -------
    a : (d,) ndarray
        Wasserstein barycenter
    log : dict
        log dictionary return only if log==True in parameters


    References
    ----------

    .. [16] Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.



    """

    if weights is None:
        weights = np.ones(A.shape[1]) / A.shape[1]
    else:
        assert(len(weights) == A.shape[1])
        
    n_distributions=A.shape[1]
    n=A.shape[0]
        
    n2=n*n
    c=np.zeros((0))
    b_eq1=np.zeros((0))
    for i in range(n_distributions):
        c=np.concatenate((c,M.ravel()*weights[i]))
        b_eq1=np.concatenate((b_eq1,A[:,i]))
    c=np.concatenate((c,np.zeros(n)))    
    
    lst_idiag1=[sps.kron(sps.eye(n),np.ones((1,n))) for i in range(n_distributions)]
    #  row constraints
    A_eq1=sps.hstack((sps.block_diag(lst_idiag1),sps.coo_matrix((n_distributions*n,n))))
    
    # columns constraints
    lst_idiag2=[]
    lst_eye=[]
    for i in range(n_distributions):
        if i==0:
            lst_idiag2.append(sps.kron(np.ones((1,n)),sps.eye(n)))
            lst_eye.append(-sps.eye(n))
        else:
            lst_idiag2.append(sps.kron(np.ones((1,n)),sps.eye(n-1,n)))
            lst_eye.append(-sps.eye(n-1,n))
        
    A_eq2=sps.hstack((sps.block_diag(lst_idiag2),sps.vstack(lst_eye)))
    b_eq2=np.zeros((A_eq2.shape[0]))
    
    # full problem
    A_eq=sps.vstack((A_eq1,A_eq2))
    b_eq=np.concatenate((b_eq1,b_eq2))
    
    if not cvxopt or solver in ['interior-point']: # cvxopt not installed or simplex/interior point
    
        if solver is None:
            solver='interior-point'
            
        options={'sparse':True,'disp': verbose}
        sol=sp.optimize.linprog(c,A_eq=A_eq,b_eq=b_eq,method=solver,options=options)
        x=sol.x
        b=x[-n:]
        
    else:
    
        h=np.zeros((n_distributions*n2+n))
        G=-sps.eye(n_distributions*n2+n)
        
        sol=solvers.lp(matrix(c),scipy_sparse_to_spmatrix(G),matrix(h),A=scipy_sparse_to_spmatrix(A_eq),b=matrix(b_eq),solver=solver)
        
        x=np.array(sol['x'])
        b=x[-n:].ravel()
    
    if log:
        return b, sol
    else:
        return b