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_* |_ 15;_ {{_at leastat mostmaximum_bipartite_matching%.200s() takes %.8s %zd positional argument%.1s (%zd given)_matching.pyxscipy.sparse.csgraph._matching.maximum_bipartite_matching__init__exactlystringsourceView.MemoryView.Enum.__init__min_weight_full_bipartite_matchingscipy.sparse.csgraph._matching.min_weight_full_bipartite_matchingView.MemoryView.memoryview.setitem_indexedPyObject_GetBuffer: view==NULL argument is obsolete while calling a Python objectNULL result without error in PyObject_CallView.MemoryView.array.__getbuffer__View.MemoryView.array.__reduce_cython__View.MemoryView.array.__setstate_cython__View.MemoryView.memoryview.__getbuffer__View.MemoryView.memoryview.__reduce_cython__View.MemoryView.memoryview.__setstate_cython__View.MemoryView._memoryviewslice.__reduce_cython__View.MemoryView._memoryviewslice.__setstate_cython__View.MemoryView.array.get_memviewView.MemoryView.array.__getitem__name '%U' is not definedView.MemoryView.Enum.__reduce_cython__View.MemoryView.memoryview.__str__View.MemoryView.memoryview.__repr__View.MemoryView.memoryview.get_item_pointerinteger division or modulo by zerovalue too large to perform divisionView.MemoryView.pybuffer_indexView.MemoryView.memoryview.is_sliceView.MemoryView.memoryview.convert_item_to_objectView.MemoryView._memoryviewslice.convert_item_to_objectbytesExpected %.16s, got %.200s'NoneType' is not iterableView.MemoryView.memoryview.assign_item_from_objectView.MemoryView._memoryviewslice.assign_item_from_objectView.MemoryView._err_dimView.MemoryView._err'NoneType' object is not subscriptablehasattr(): attribute name must be stringView.MemoryView.__pyx_unpickle_Enum__set_statetupleView.MemoryView.Enum.__setstate_cython__intan integer is requiredcannot import name %SView.MemoryView.__pyx_unpickle_EnumView.MemoryView.array.__getattr__View.MemoryView.memoryview_cwrapperView.MemoryView._unellipsifyCannot transpose memoryview with indirect dimensionsView.MemoryView.transpose_memsliceAcquisition count is %d (line %d)View.MemoryView.memoryview_fromsliceView.MemoryView.memview_sliceMissing type objectCannot convert %.200s to %.200sIndex out of bounds (axis %d)All dimensions preceding dimension %d must be indexed and not slicedView.MemoryView.slice_memviewsliceStep may not be zero (axis %d)memviewsliceobjlocal variable '%s' referenced before assignmenttoo many values to unpack (expected %zd)need more than %zd value%.1s to unpack'NoneType' object is not iterableView.MemoryView.memoryview.__getitem__View.MemoryView.get_slice_from_memviewView.MemoryView.memoryview.is_c_contigView.MemoryView.memoryview.is_f_contigView.MemoryView.memoryview_copy_from_slicecView.MemoryView.memoryview.copyView.MemoryView.memoryview.copy_fortranView.MemoryView.assert_direct_dimensionsView.MemoryView.memoryview.setitem_slice_assign_scalarView.MemoryView._err_extentsDimension %d is not directView.MemoryView.copy_data_to_tempView.MemoryView.memoryview_copy_contentsvalue too large to convert to intView.MemoryView.memoryview.setitem_slice_assignment__init__.pxdtype.pxdparameters.pxi'bool''char''signed char''unsigned char''short''unsigned short''int''unsigned int''long''unsigned long''long long''unsigned long long''complex float''float''complex double''double''complex long double''long double'a structPython objecta pointera stringendunparseable format string'Buffer dtype mismatch, expected %s%s%s but got %sBuffer dtype mismatch, expected '%s' but got %s in '%s.%s'Interpreter change detected - this module can only be loaded into one interpreter per process.name__loader__loader__file__origin__package__parent__path__submodule_search_locationsmemviewslice is already initialized!Expected a dimension of size %zu, got %zuExpected %d dimensions, got %dUnexpected format string character: '%c'Python does not define a standard format string size for long double ('g')..Buffer dtype mismatch; next field is at offset %zd but %zd expectedBig-endian buffer not supported on little-endian compilerBuffer acquisition: Expected '{' after 'T'Cannot handle repeated arrays in format stringDoes not understand character buffer dtype format string ('%c')Expected a dimension of size %zu, got %dExpected a comma in format string, got '%c'Expected %d dimension(s), got %dUnexpected end of format string, expected ')'Unable to initialize pickling for %s'%.200s' object is not subscriptablecannot fit '%.200s' into an index-sized integerscipy/sparse/csgraph/_matching.cpython-312-aarch64-linux-gnu.so.p/_matching.c%s (%s:%d)__cinit__View.MemoryView.array.__cinit__shapeArgument '%.200s' has incorrect type (expected %.200s, got %.200s)formatArgument '%.200s' must not be Noneobject of type 'NoneType' has no len()expected bytes, NoneType foundView.MemoryView.array.__setitem__Subscript deletion not supported by %.200sView.MemoryView.array.memview.__get__View.MemoryView.memoryview.__setitem__View.MemoryView.memoryview_copyView.MemoryView.memoryview.T.__get__View.MemoryView.memoryview.shape.__get__View.MemoryView.memoryview.strides.__get__View.MemoryView.memoryview.suboffsets.__get__View.MemoryView.memoryview.ndim.__get__View.MemoryView.memoryview.itemsize.__get__View.MemoryView.memoryview.nbytes.__get__View.MemoryView.memoryview.size.__get__Module '_matching' has already been imported. Re-initialisation is not supported.%d.%dscipy.sparse.csgraph._matchingcompiletime version %s of module '%.100s' does not match runtime version %sbuiltinscython_runtime__builtins__numpydtypeflatiterbroadcastndarraygenericnumbersignedintegerunsignedintegerinexactfloatingcomplexfloatingflexiblecharacterufuncnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is not PyCapsule object_ARRAY_API is NULL pointermodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%x . Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem .FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimenumpy.import_arraygetbuffer(obj, view, flags)init scipy.sparse.csgraph._matchingCannot copy memoryview slice with indirect dimensions (axis %d)View.MemoryView.array_cwrapper__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)View.MemoryView.memoryview.__cinit__value too large to convert to npy_int32Buffer has wrong number of dimensions (expected %d, got %d)buffer dtypeItem size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer exposes suboffsets but no stridesC-contiguous buffer is not indirect in dimension %dBuffer and memoryview are not contiguous in the same dimension.Buffer is not indirectly contiguous in dimension %d.Buffer is not indirectly accessible in dimension %d.Buffer not compatible with direct access in dimension %d.base__reduce_cython____setstate_cython__Tstridessuboffsetsndimitemsizenbytesis_c_contigis_f_contigcopycopy_fortranmemview__getattr__ITYPE_tDTYPE_tscipy.sparse.csgraph._matching._memoryviewsliceInternal class for passing memoryview slices to Pythonscipy.sparse.csgraph._matching.memoryviewscipy.sparse.csgraph._matching.Enumscipy.sparse.csgraph._matching.arraycalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseException%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectco_argcountco_posonlyargcountco_kwonlyargcountco_nlocalsco_stacksizeco_flagsco_codeco_constsco_namesco_varnamesco_freevarsco_cellvarsco_linetablereplacescipy.sparse.csgraph._matching._hopcroft_karpscipy.sparse.csgraph._matching._lapjvsp_single_lscipy.sparse.csgraph._matching._lapjvsp%s() got multiple values for keyword argument '%U'%.200s() keywords must be strings%s() got an unexpected keyword argument '%U'BTYPE_tLXHD@<840,(T  P<99666?`E<996LLLLLLL9LLLLLLLLLLLLLLLLLLLLL 9LLLLLLzerosyxwarningswarnupdateunpackunable to allocate shape and strides.unable to allocate array data.uint8tocsr__test__sumstructstringsourcestopstepstartsizeshape__setstate_cython____setstate__scipy.sparse.csgraph._matchingscipy.sparserow__reduce_ex____reduce_cython____reduce__range__pyx_vtable____pyx_unpickle_Enum__pyx_type__pyx_state__pyx_result__pyx_getbuffer__pyx_checksum__pyx_PickleErrorpickleperm_typepackobjnumpy.core.umath failed to importnumpy.core.multiarray failed to importnumpynpno full matching existsno default __reduce__ due to non-trivial __cinit____new__ndim__name__namemodemin_weight_full_bipartite_matching (line 283)min_weight_full_bipartite_matching min_weight_full_bipartite_matching(biadjacency_matrix, maximize=False) Returns the minimum weight full matching of a bipartite graph. .. versionadded:: 1.6.0 Parameters ---------- biadjacency_matrix : sparse matrix Biadjacency matrix of the bipartite graph: A sparse matrix in CSR, CSC, or COO format whose rows represent one partition of the graph and whose columns represent the other partition. An edge between two vertices is indicated by the corresponding entry in the matrix, and the weight of the edge is given by the value of that entry. This should not be confused with the full adjacency matrix of the graph, as we only need the submatrix defining the bipartite structure. maximize : bool (default: False) Calculates a maximum weight matching if true. Returns ------- row_ind, col_ind : array An array of row indices and one of corresponding column indices giving the optimal matching. The total weight of the matching can be computed as ``graph[row_ind, col_ind].sum()``. The row indices will be sorted; in the case of a square matrix they will be equal to ``numpy.arange(graph.shape[0])``. Notes ----- Let :math:`G = ((U, V), E)` be a weighted bipartite graph with non-zero weights :math:`w : E \to \mathbb{R} \setminus \{0\}`. This function then produces a matching :math:`M \subseteq E` with cardinality .. math:: \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert), which minimizes the sum of the weights of the edges included in the matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such matching exists. When :math:`\lvert U \rvert = \lvert V \rvert`, this is commonly referred to as a perfect matching; here, since we allow :math:`\lvert U \rvert` and :math:`\lvert V \rvert` to differ, we follow Karp [1]_ and refer to the matching as *full*. This function implements the LAPJVsp algorithm [2]_, short for "Linear assignment problem, Jonker--Volgenant, sparse". The problem it solves is equivalent to the rectangular linear assignment problem. [3]_ As such, this function can be used to solve the same problems as :func:`scipy.optimize.linear_sum_assignment`. That function may perform better when the input is dense, or for certain particular types of inputs, such as those for which the :math:`(i, j)`'th entry is the distance between two points in Euclidean space. If no full matching exists, this function raises a ``ValueError``. For determining the size of the largest matching in the graph, see :func:`maximum_bipartite_matching`. We require that weights are non-zero only to avoid issues with the handling of explicit zeros when converting between different sparse representations. Zero weights can be handled by adding a constant to all weights, so that the resulting matrix contains no zeros. References ---------- .. [1] Richard Manning Karp: An algorithm to Solve the m x n Assignment Problem in Expected Time O(mn log n). Networks, 10(2):143-152, 1980. .. [2] Roy Jonker and Anton Volgenant: A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems. Computing 38:325-340, 1987. .. [3] https://en.wikipedia.org/wiki/Assignment_problem Examples -------- >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.csgraph import min_weight_full_bipartite_matching Let us first consider an example in which all weights are equal: >>> biadjacency_matrix = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]]) Here, all we get is a perfect matching of the graph: >>> print(min_weight_full_bipartite_matching(biadjacency_matrix)[1]) [2 0 1] That is, the first, second, and third rows are matched with the third, first, and second column respectively. Note that in this example, the 0 in the input matrix does *not* correspond to an edge with weight 0, but rather a pair of vertices not paired by an edge. Note also that in this case, the output matches the result of applying :func:`maximum_bipartite_matching`: >>> from scipy.sparse.csgraph import maximum_bipartite_matching >>> biadjacency = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]]) >>> print(maximum_bipartite_matching(biadjacency, perm_type='column')) [2 0 1] When multiple edges are available, the ones with lowest weights are preferred: >>> biadjacency = csr_matrix([[3, 3, 6], [4, 3, 5], [10, 1, 8]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(col_ind) [0 2 1] The total weight in this case is :math:`3 + 5 + 1 = 9`: >>> print(biadjacency[row_ind, col_ind].sum()) 9 When the matrix is not square, i.e. when the two partitions have different cardinalities, the matching is as large as the smaller of the two partitions: >>> biadjacency = csr_matrix([[0, 1, 1], [0, 2, 3]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [0 1] [2 1] >>> biadjacency = csr_matrix([[0, 1], [3, 1], [1, 4]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [0 2] [1 0] When one or both of the partitions are empty, the matching is empty as well: >>> biadjacency = csr_matrix((2, 0)) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [] [] In general, we will always reach the same sum of weights as if we had used :func:`scipy.optimize.linear_sum_assignment` but note that for that one, missing edges are represented by a matrix entry of ``float('inf')``. Let us generate a random sparse matrix with integer entries between 1 and 10: >>> import numpy as np >>> from scipy.sparse import random >>> from scipy.optimize import linear_sum_assignment >>> sparse = random(10, 10, random_state=42, density=.5, format='coo') * 10 >>> sparse.data = np.ceil(sparse.data) >>> dense = sparse.toarray() >>> dense = np.full(sparse.shape, np.inf) >>> dense[sparse.row, sparse.col] = sparse.data >>> sparse = sparse.tocsr() >>> row_ind, col_ind = linear_sum_assignment(dense) >>> print(dense[row_ind, col_ind].sum()) 28.0 >>> row_ind, col_ind = min_weight_full_bipartite_matching(sparse) >>> print(sparse[row_ind, col_ind].sum()) 28.0 minmemviewmaximum_bipartite_matching (line 16) maximum_bipartite_matching(graph, perm_type='row') Returns a matching of a bipartite graph whose cardinality is as least that of any given matching of the graph. Parameters ---------- graph : sparse matrix Input sparse in CSR format whose rows represent one partition of the graph and whose columns represent the other partition. An edge between two vertices is indicated by the corresponding entry in the matrix existing in its sparse representation. perm_type : str, {'row', 'column'} Which partition to return the matching in terms of: If ``'row'``, the function produces an array whose length is the number of columns in the input, and whose :math:`j`'th element is the row matched to the :math:`j`'th column. Conversely, if ``perm_type`` is ``'column'``, this returns the columns matched to each row. Returns ------- perm : ndarray A matching of the vertices in one of the two partitions. Unmatched vertices are represented by a ``-1`` in the result. Notes ----- This function implements the Hopcroft--Karp algorithm [1]_. Its time complexity is :math:`O(\lvert E \rvert \sqrt{\lvert V \rvert})`, and its space complexity is linear in the number of rows. In practice, this asymmetry between rows and columns means that it can be more efficient to transpose the input if it contains more columns than rows. By Konig's theorem, the cardinality of the matching is also the number of vertices appearing in a minimum vertex cover of the graph. Note that if the sparse representation contains explicit zeros, these are still counted as edges. The implementation was changed in SciPy 1.4.0 to allow matching of general bipartite graphs, where previous versions would assume that a perfect matching existed. As such, code written against 1.4.0 will not necessarily work on older versions. References ---------- .. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for Maximum Matchings in Bipartite Graphs" In: SIAM Journal of Computing 2.4 (1973), pp. 225--231. :doi:`10.1137/0202019` Examples -------- >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.csgraph import maximum_bipartite_matching As a simple example, consider a bipartite graph in which the partitions contain 2 and 3 elements respectively. Suppose that one partition contains vertices labelled 0 and 1, and that the other partition contains vertices labelled A, B, and C. Suppose that there are edges connecting 0 and C, 1 and A, and 1 and B. This graph would then be represented by the following sparse matrix: >>> graph = csr_matrix([[0, 0, 1], [1, 1, 0]]) Here, the 1s could be anything, as long as they end up being stored as elements in the sparse matrix. We can now calculate maximum matchings as follows: >>> print(maximum_bipartite_matching(graph, perm_type='column')) [2 0] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [ 1 -1 0] The first output tells us that 1 and 2 are matched with C and A respectively, and the second output tells us that A, B, and C are matched with 1, nothing, and 0 respectively. Note that explicit zeros are still converted to edges. This means that a different way to represent the above graph is by using the CSR structure directly as follows: >>> data = [0, 0, 0] >>> indices = [2, 0, 1] >>> indptr = [0, 1, 3] >>> graph = csr_matrix((data, indices, indptr)) >>> print(maximum_bipartite_matching(graph, perm_type='column')) [2 0] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [ 1 -1 0] When one or both of the partitions are empty, the matching is empty as well: >>> graph = csr_matrix((2, 0)) >>> print(maximum_bipartite_matching(graph, perm_type='column')) [-1 -1] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [] When the input matrix is square, and the graph is known to admit a perfect matching, i.e. a matching with the property that every vertex in the graph belongs to some edge in the matching, then one can view the output as the permutation of rows (or columns) turning the input matrix into one with the property that all diagonal elements are non-empty: >>> a = [[0, 1, 2, 0], [1, 0, 0, 1], [2, 0, 0, 3], [0, 1, 3, 0]] >>> graph = csr_matrix(a) >>> perm = maximum_bipartite_matching(graph, perm_type='row') >>> print(graph[perm].toarray()) [[1 0 0 1] [0 1 2 0] [0 1 3 0] [2 0 0 3]] maximum_bipartite_matchingmaximizemaxmatching__main__jitemsize <= 0 for cython.arrayitemsizeissubdtypeissparseisposinfint32indptrindices__import__iinfoidigraph must be sparsegraph must be in CSC, CSR, or COO format.graphgot differing extents in dimension %d (got %d and %d)got __getstate__fortranformatfloat64flagsexplicit zero weights are removed before matchingexpected a matrix containing numerical entries, errorenumerateencodeemptyeliminate_zerosdtype_is_objectdtypedouble__dict__datacsrcsccoocolumncline_in_traceback__class__c/build/scipy-CRQwve/scipy-1.11.4/scipy/sparse/csgraph/_matching.pyxbool_biadjacency_matrix_tbiadjacency_matrixbasebastypeasarrayargsortarangeallocate_bufferalla_View.MemoryViewValueErrorUnable to convert item to objectTypeErrorTPickleErrorOut of bounds on buffer access (axis %d)OMemoryErrorInvalid shape in axis %d: %d.Invalid mode, expected 'c' or 'fortran', got %sIndirect dimensions not supportedIndexErrorIncompatible checksums (0x%x vs (0xb068931, 0x82a3537, 0x6ae9995) = (name))ImportErrorITYPEEmpty shape tuple for cython.arrayEllipsisDTYPECannot index with type '%s'Cannot create writable memory view from read-only memoryviewCannot assign to read-only memoryviewCan only create a buffer that is contiguous in memory.Buffer view does not expose stridesBTYPEASCII@  __pyx_fatalerror;4mmoH pxpp|shs&*+/$T $HTtĘԜ\$pĤ4(DXTdt\ T0 Hx $ Dx  $L T  0 D Dtxd$ldpHt$HD( !"H#lD$h%%&,&P'd,\9tTTHUUPHVtXY4Z D[D[h\]4]T] ]T^xD^_d` $aHdcefpHgl\rDstXDtt8DuudwxTty{ |d  4 !ć!$ #Dl##`$$t$$4p%%%&&88'l''Կ8($\(D(()L)|)))**(H+h+T+D4<.6.hP.(/t//zRx \0,x< @HA BNdБ |ȑ A-ACA-(đ<A-AF A-B AA- ؑA-Aa A-A (dA-Ae A-D AA-TDA-A`CBBo A-A M A-A MOIAOM A BB<tA-APCBG\ A-A sLABB@p$A-APCBBo A-A M A-A P,A-AGA-4XA-A B\ A-A L A-A pTA-A@CDBn BAAA-A dBBAA-A@-[ BAA [,, C-B0BBa A-A , C-B0BBa A-A T(pA-A0CCpBAA-A0-w CA F BA S, C-B0BBa A-A , C-B0BBa A-A ,Ġ C-B0BBa A-A , C-B0BBa A-A p@A-A0BEk CA-A Q B MKA-A0-CCA-A0-B@ A-A DY A-A E A-A S A-A A-AP CFBe DAA-A JDAA-AP -LA-AP -s BA DA-A0Ew A-A L A-A M A-A `ȫTA-A@CFmNT A-A \ A-A MBI A-A GEdJD4 A-Ap B CCF c A-A P|A-ACBDt A-A MVDIHP A h A kAĺpA-AC CD I A-A KFojr A F A GGHA D CLRPD E T D LB A C,tlA-AG A-A J A-B t0A-A` B CEBEO A-A CTWk A C A V A K D D C e B ,A-AO A-C Q A-B 8LA-AP BCBf A-A TTA-A@BC[ A-A H A-A Bo A c C e C LA-AP BEV`EDA-AP -EBIA-AP -BK A CIBNBBACD A Z BB ` A E BA 0 HA-A CV A-A L A-A P A-A` B BBCGOE A-A QA-$ A-A`CBGIdBTcA-A`-b A ~ A C A u B Z A z B } B W B qAi A H A H( <A-AH A-A AA-@ A-A DZ A-A E A-A T A-A 4 A-ABA-lX XA-A@BBFw AA-A Q B MLA-A@-F AA-A CBD D A-ABG G A-A @ (A-A@BEPg AA-A ADA-T p4A-AB EBBB AADA-A -ZF L CABA-A s BAA tA-AYA-L \$A-A` BG D; 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An edge between two vertices is indicated by the corresponding entry in the matrix, and the weight of the edge is given by the value of that entry. This should not be confused with the full adjacency matrix of the graph, as we only need the submatrix defining the bipartite structure. maximize : bool (default: False) Calculates a maximum weight matching if true. Returns ------- row_ind, col_ind : array An array of row indices and one of corresponding column indices giving the optimal matching. The total weight of the matching can be computed as ``graph[row_ind, col_ind].sum()``. The row indices will be sorted; in the case of a square matrix they will be equal to ``numpy.arange(graph.shape[0])``. Notes ----- Let :math:`G = ((U, V), E)` be a weighted bipartite graph with non-zero weights :math:`w : E \to \mathbb{R} \setminus \{0\}`. This function then produces a matching :math:`M \subseteq E` with cardinality .. math:: \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert), which minimizes the sum of the weights of the edges included in the matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such matching exists. When :math:`\lvert U \rvert = \lvert V \rvert`, this is commonly referred to as a perfect matching; here, since we allow :math:`\lvert U \rvert` and :math:`\lvert V \rvert` to differ, we follow Karp [1]_ and refer to the matching as *full*. This function implements the LAPJVsp algorithm [2]_, short for "Linear assignment problem, Jonker--Volgenant, sparse". The problem it solves is equivalent to the rectangular linear assignment problem. [3]_ As such, this function can be used to solve the same problems as :func:`scipy.optimize.linear_sum_assignment`. That function may perform better when the input is dense, or for certain particular types of inputs, such as those for which the :math:`(i, j)`'th entry is the distance between two points in Euclidean space. If no full matching exists, this function raises a ``ValueError``. For determining the size of the largest matching in the graph, see :func:`maximum_bipartite_matching`. We require that weights are non-zero only to avoid issues with the handling of explicit zeros when converting between different sparse representations. Zero weights can be handled by adding a constant to all weights, so that the resulting matrix contains no zeros. References ---------- .. [1] Richard Manning Karp: An algorithm to Solve the m x n Assignment Problem in Expected Time O(mn log n). Networks, 10(2):143-152, 1980. .. [2] Roy Jonker and Anton Volgenant: A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems. Computing 38:325-340, 1987. .. [3] https://en.wikipedia.org/wiki/Assignment_problem Examples -------- >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.csgraph import min_weight_full_bipartite_matching Let us first consider an example in which all weights are equal: >>> biadjacency_matrix = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]]) Here, all we get is a perfect matching of the graph: >>> print(min_weight_full_bipartite_matching(biadjacency_matrix)[1]) [2 0 1] That is, the first, second, and third rows are matched with the third, first, and second column respectively. Note that in this example, the 0 in the input matrix does *not* correspond to an edge with weight 0, but rather a pair of vertices not paired by an edge. Note also that in this case, the output matches the result of applying :func:`maximum_bipartite_matching`: >>> from scipy.sparse.csgraph import maximum_bipartite_matching >>> biadjacency = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]]) >>> print(maximum_bipartite_matching(biadjacency, perm_type='column')) [2 0 1] When multiple edges are available, the ones with lowest weights are preferred: >>> biadjacency = csr_matrix([[3, 3, 6], [4, 3, 5], [10, 1, 8]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(col_ind) [0 2 1] The total weight in this case is :math:`3 + 5 + 1 = 9`: >>> print(biadjacency[row_ind, col_ind].sum()) 9 When the matrix is not square, i.e. when the two partitions have different cardinalities, the matching is as large as the smaller of the two partitions: >>> biadjacency = csr_matrix([[0, 1, 1], [0, 2, 3]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [0 1] [2 1] >>> biadjacency = csr_matrix([[0, 1], [3, 1], [1, 4]]) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [0 2] [1 0] When one or both of the partitions are empty, the matching is empty as well: >>> biadjacency = csr_matrix((2, 0)) >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) >>> print(row_ind, col_ind) [] [] In general, we will always reach the same sum of weights as if we had used :func:`scipy.optimize.linear_sum_assignment` but note that for that one, missing edges are represented by a matrix entry of ``float('inf')``. Let us generate a random sparse matrix with integer entries between 1 and 10: >>> import numpy as np >>> from scipy.sparse import random >>> from scipy.optimize import linear_sum_assignment >>> sparse = random(10, 10, random_state=42, density=.5, format='coo') * 10 >>> sparse.data = np.ceil(sparse.data) >>> dense = sparse.toarray() >>> dense = np.full(sparse.shape, np.inf) >>> dense[sparse.row, sparse.col] = sparse.data >>> sparse = sparse.tocsr() >>> row_ind, col_ind = linear_sum_assignment(dense) >>> print(dense[row_ind, col_ind].sum()) 28.0 >>> row_ind, col_ind = min_weight_full_bipartite_matching(sparse) >>> print(sparse[row_ind, col_ind].sum()) 28.0 maximum_bipartite_matching(graph, perm_type='row') Returns a matching of a bipartite graph whose cardinality is as least that of any given matching of the graph. Parameters ---------- graph : sparse matrix Input sparse in CSR format whose rows represent one partition of the graph and whose columns represent the other partition. An edge between two vertices is indicated by the corresponding entry in the matrix existing in its sparse representation. perm_type : str, {'row', 'column'} Which partition to return the matching in terms of: If ``'row'``, the function produces an array whose length is the number of columns in the input, and whose :math:`j`'th element is the row matched to the :math:`j`'th column. Conversely, if ``perm_type`` is ``'column'``, this returns the columns matched to each row. Returns ------- perm : ndarray A matching of the vertices in one of the two partitions. Unmatched vertices are represented by a ``-1`` in the result. Notes ----- This function implements the Hopcroft--Karp algorithm [1]_. Its time complexity is :math:`O(\lvert E \rvert \sqrt{\lvert V \rvert})`, and its space complexity is linear in the number of rows. In practice, this asymmetry between rows and columns means that it can be more efficient to transpose the input if it contains more columns than rows. By Konig's theorem, the cardinality of the matching is also the number of vertices appearing in a minimum vertex cover of the graph. Note that if the sparse representation contains explicit zeros, these are still counted as edges. The implementation was changed in SciPy 1.4.0 to allow matching of general bipartite graphs, where previous versions would assume that a perfect matching existed. As such, code written against 1.4.0 will not necessarily work on older versions. References ---------- .. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for Maximum Matchings in Bipartite Graphs" In: SIAM Journal of Computing 2.4 (1973), pp. 225--231. :doi:`10.1137/0202019` Examples -------- >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.csgraph import maximum_bipartite_matching As a simple example, consider a bipartite graph in which the partitions contain 2 and 3 elements respectively. Suppose that one partition contains vertices labelled 0 and 1, and that the other partition contains vertices labelled A, B, and C. Suppose that there are edges connecting 0 and C, 1 and A, and 1 and B. This graph would then be represented by the following sparse matrix: >>> graph = csr_matrix([[0, 0, 1], [1, 1, 0]]) Here, the 1s could be anything, as long as they end up being stored as elements in the sparse matrix. We can now calculate maximum matchings as follows: >>> print(maximum_bipartite_matching(graph, perm_type='column')) [2 0] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [ 1 -1 0] The first output tells us that 1 and 2 are matched with C and A respectively, and the second output tells us that A, B, and C are matched with 1, nothing, and 0 respectively. Note that explicit zeros are still converted to edges. This means that a different way to represent the above graph is by using the CSR structure directly as follows: >>> data = [0, 0, 0] >>> indices = [2, 0, 1] >>> indptr = [0, 1, 3] >>> graph = csr_matrix((data, indices, indptr)) >>> print(maximum_bipartite_matching(graph, perm_type='column')) [2 0] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [ 1 -1 0] When one or both of the partitions are empty, the matching is empty as well: >>> graph = csr_matrix((2, 0)) >>> print(maximum_bipartite_matching(graph, perm_type='column')) [-1 -1] >>> print(maximum_bipartite_matching(graph, perm_type='row')) [] When the input matrix is square, and the graph is known to admit a perfect matching, i.e. a matching with the property that every vertex in the graph belongs to some edge in the matching, then one can view the output as the permutation of rows (or columns) turning the input matrix into one with the property that all diagonal elements are non-empty: >>> a = [[0, 1, 2, 0], [1, 0, 0, 1], [2, 0, 0, 3], [0, 1, 3, 0]] >>> graph = csr_matrix(a) >>> perm = maximum_bipartite_matching(graph, perm_type='row') >>> print(graph[perm].toarray()) [[1 0 0 1] [0 1 2 0] [0 1 3 0] [2 0 0 3]]  _>ZZZZZpR>h>4TX>==|xTDDHN(=TOHpU<<<DIN;@:`WDI`Y99 ȕ ^^^صRеI`R(@З0RHPT`hxPТt0(``P(H(@@Сp@`Q8bgZ0^(\$\p7\@& \=Y^[ [#^`[ HY0L[ 0\"Z0Zp[ XpXPZ@X)8Y _H[ \!X[ ^\x\_Zp_`_X_P_H\@X8W p\_`\DZXX hZ_x\`\@x_8p_0h_(`_ xX X_P_^H_@_Zx[ X8_10_@2^8^0Z(Z X 0Z (_Y6W[*[pP\h[` _X_H \@_8^0_  ] _ Z [H\_ h\ ^X X\p8^@^%pX^H^\#P^`.ZPWHX8 Y0XY(\3[о]]['[`"^XYPW@ @Y8\  Y`^\ (Yؽ YȽ X(ZP[@Z HZp8ZXWP]@  ^ PZ XZZ [YؼYмYȼ\\!\`]P XH^@X^0 ^((^ [[л&XȻY^] @\8\^`U/usr/lib/debug/.dwz/aarch64-linux-gnu/python3-scipy.debug~\-ׇ!1\I!9f64efc07bdfb458473e0b5e978f3d31711be5.debugp.shstrtab.note.gnu.build-id.gnu.hash.dynsym.dynstr.gnu.version.gnu.version_r.rela.dyn.rela.plt.init.text.fini.rodata.eh_frame_hdr.eh_frame.init_array.fini_array.data.rel.ro.dynamic.got.got.plt.data.bss.gnu_debugaltlink.gnu_debuglink $o$( 0 8oTEo@T((x-^BLL` hYYc Y Y`naa4tz^LLT+(08  SWW WNX4LX