ó ÚÆ÷Xc@s\ddlZd„Zdefd„ƒYZd„Zd„Zd„Zd„Zd „Z dS( iÿÿÿÿNcCsætj|ƒ}t|jƒdkr=td|jfƒ‚n|jd|jdkri|j}t}nt}t|ƒ}d|jkrdnt }x|dk r´||ƒ}q™W|rÊ|j j}n |j }tj |dkƒS(s’ Solve the linear sum assignment problem. The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a "worker") and vertex j of the second set (a "job"). The goal is to find a complete assignment of workers to jobs of minimal cost. Formally, let X be a boolean matrix where :math:`X[i,j] = 1` iff row i is assigned to column j. Then the optimal assignment has cost .. math:: \min \sum_i \sum_j C_{i,j} X_{i,j} s.t. each row is assignment to at most one column, and each column to at most one row. This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa. The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm. Parameters ---------- cost_matrix : array The cost matrix of the bipartite graph. Returns ------- row_ind, col_ind : array An array of row indices and one of corresponding column indices giving the optimal assignment. The cost of the assignment can be computed as ``cost_matrix[row_ind, col_ind].sum()``. The row indices will be sorted; in the case of a square cost matrix they will be equal to ``numpy.arange(cost_matrix.shape[0])``. Notes ----- .. versionadded:: 0.17.0 Examples -------- >>> cost = np.array([[4, 1, 3], [2, 0, 5], [3, 2, 2]]) >>> from scipy.optimize import linear_sum_assignment >>> row_ind, col_ind = linear_sum_assignment(cost) >>> col_ind array([1, 0, 2]) >>> cost[row_ind, col_ind].sum() 5 References ---------- 1. http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html 2. Harold W. Kuhn. The Hungarian Method for the assignment problem. *Naval Research Logistics Quarterly*, 2:83-97, 1955. 3. Harold W. Kuhn. Variants of the Hungarian method for assignment problems. *Naval Research Logistics Quarterly*, 3: 253-258, 1956. 4. Munkres, J. Algorithms for the Assignment and Transportation Problems. *J. SIAM*, 5(1):32-38, March, 1957. 5. https://en.wikipedia.org/wiki/Hungarian_algorithm is-expected a matrix (2-d array), got a %r arrayiiN( tnptasarraytlentshapet ValueErrortTtTruetFalset_HungarytNonet_step1tmarkedtwhere(t cost_matrixt transposedtstatetstepR ((s8/tmp/pip-build-X4mzal/scipy/scipy/optimize/_hungarian.pytlinear_sum_assignment s E    RcBs eZdZd„Zd„ZRS(sšState of the Hungarian algorithm. Parameters ---------- cost_matrix : 2D matrix The cost matrix. Must have shape[1] >= shape[0]. cCs§|jƒ|_|jj\}}tj|dtƒ|_tj|dtƒ|_d|_d|_ tj ||dfdt ƒ|_ tj ||fdt ƒ|_ dS(Ntdtypeii(tcopytCRRtonestboolt row_uncoveredt col_uncoveredtZ0_rtZ0_ctzerostinttpathR (tselfR tntm((s8/tmp/pip-build-X4mzal/scipy/scipy/optimize/_hungarian.pyt__init__ws  "cCst|j(t|j(dS(sClear all covered matrix cellsN(RRR(R((s8/tmp/pip-build-X4mzal/scipy/scipy/optimize/_hungarian.pyt _clear_covers‚s (t__name__t __module__t__doc__R!R"(((s8/tmp/pip-build-X4mzal/scipy/scipy/optimize/_hungarian.pyRns cCs¸|j|jjddƒdd…tjf8_xvttj|jdkƒŒD]V\}}|j|rP|j|rPd|j||f s  a  & 2