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LLh\MLhLXItAx AAx AAx AA$x A$x M"H1DžHL;HLHLASH LhҵAYLhAZ-"fo H fDo fDo0fDo@fofH~fofoD)fDoPfoD)fDopfoD)fofoD) fDo`D)@)PD)0)`)p)e)])U)MD)D)D)D)D)D)))))) )0)@H AfI~ԅx AfoHcE8)H@HH))foL)foH4HxH) foL)0fo)@fo)Pfo)`fo)pfo)Mfo )Mfo0)Mfo@)MLMHHLHLXIff.DHH1H@E~ff.H9uE8L(1LL0H] AAHLLLHL@H LPfMcD9U@ ARH8HcHDH)‹ A9M׉xLHhK2HLHcLHH`HHpD:~1HcIIA<u A:A<IcH(AIDLHA9uxLHhMA9LHHH;tB8foL@HH9)foA)fo)fo )fo0)fo@)foP)fo`) fop)0foM)@foM)PfoM)`foM)pmLNSfDo0L fofofDo fDo@D)fDoPfo)PfDo`foD)fofoD)fofopD) D)0)@)`)p)])U)M)ED)D)D)D)D))LHDžH))))) )0)@HLxLQLxLALLQHLpvHLQL\bHLvQLGLLxLTQHLxLLLx!QLxZfDo0H fofofDo fDo@D)fDoPfo)PfDo`foD)fofoD)fofopD) D)0)@)`)p)])U)M)ED)D)D)D)D))HHDž))))) )0)@`LOHLxOLxgLLxOLx4LLxOLxLpO.QH=HHPTLM8QHHHDžAHDžHDžHDžQH=[HLSLM]oQHHLHDžAHDžHDžE1E1HDžff.A$x A$Htx MtAx A!MtAx A"Hv-H=_<E11mfEuHA8 HHtH;tHA8HHtH;TtHA8,HHtH;)tHA8?Ht!H;tfH~ǸG8HLfo@fo@ fo@0fo@@fo@Pfo@`fo @pfo0fo@foPfo`fopHEdH+%(HHe[A\A]A^A_]L牵xL5LxLH߉xLLxLLωKLKPHXHH)KfoE1E1AHDžHDžHDž8HH)KfoHH)JfoH H)Jfo[P)gJfo7LCLcA=wAA$=wA$x H LE1MAE1LHDžHDžHDžH HfAHHAH;A M0@HDžAHDž1LHMfHHIH;ILxLLx ILhLxvHIfDo0L fofofDo fDo@D)fDoPfo)PfDo`foD)fofoD)fofopD) D)0)@)`)p)])U)M)ED)PD)`D)pD)D))LHDžx)))))))AQH1L0LD0H8LIՅ~?L @ff.ff.9L:L9uLE8LxA1HDžAHXff.LLE HpMKc4"HhIHxHc8LHFE~HPL`L@HIIcIILHAt5A9tuAE9EOD90HXK'ED HcHHDhH8LL)AMcL B49bLxL8L 5Ic:IM)HIƃHcILA419uLǃyLHcLxHHL׃DLxL6L)LL] IHDLLpLE0 fDHcL)IIc|HIALD)HA2LHHHc2HIE 2E HLxHMLVBad call flags for CyFunctionkeywords must be strings__pyx_capi__cannot import name %Sbuffer dtypean integer is requiredbuiltinscython_runtime__builtins__scipy.sparse.csgraph._flowdoes not matchnumpyflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncscipy._cyutilitymemoryview_allocate_bufferarray_cwrappermemoryview_cwrappermemview_sliceslice_memviewslicepybuffer_indexint (__Pyx_memviewslice *)transpose_memslicememoryview_fromsliceget_slice_from_memviewslice_copymemoryview_copymemoryview_copy_from_sliceget_best_orderslice_get_sizefill_contig_strides_arraycopy_data_to_temp_err_extents_err_dimint (PyObject *, PyObject *)_errint (void)_err_no_memorymemoryview_copy_contentsbroadcast_leadingrefcount_copyingrefcount_objects_in_slice_slice_assign_scalarnumpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointernumpy.import_array__module__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutine__dictoffset____vectorcalloffset____weaklistoffset____reduce__const ITYPE_t%.200s() takes %.8s %zd positional argument%.1s (%zd given)scipy/sparse/csgraph/_flow.pyxscipy.sparse.csgraph._flow.MaximumFlowResult.__init__scipy.sparse.csgraph._flow.MaximumFlowResult.__repr__scipy.sparse.csgraph._flow._add_reverse_edgesAcquisition count is %d (line %d)scipy.sparse.csgraph._flow._make_edge_pointersscipy.sparse.csgraph._flow._make_tailsstrings are too large to concattoo many values to unpack (expected %zd)local variable '%s' referenced before assignment'%.200s' object is unsliceablescipy.sparse.csgraph._flow.maximum_flow__name__ must be set to a string object__qualname__ must be set to a string objectfunction's dictionary may not be deletedsetting function's dictionary to a non-dict__annotations__ must be set to a dict objectShared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingUnexpected format string character: '%c'Buffer dtype mismatch, expected %s%s%s but got %sBuffer dtype mismatch, expected '%s' but got %s in '%s.%s'Expected a dimension of size %zu, got %zuExpected %d dimensions, got %dPython 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 ')'Interpreter change detected - this module can only be loaded into one interpreter per process.__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function calls__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function calls%s() got multiple values for keyword argument '%U'%.200s() keywords must be stringsinvalid vtable found for imported type%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject__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.__int__ returned non-int (type %.200s)unbound method %.200S() needs an argument%.200s does not export expected C function %.200sC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s) while calling a Python objectNULL result without error in PyObject_Callcalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseException%s() got an unexpected keyword argument '%U'need more than %zd value%.1s to unpackcannot fit '%.200s' into an index-sized integer'%.200s' object is not subscriptableBuffer has wrong number of dimensions (expected %d, got %d)Item size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer is not indirectly contiguous in dimension %d.Buffer and memoryview are not contiguous in the same dimension.C-contiguous buffer is not contiguous in dimension %dC-contiguous buffer is not indirect in dimension %dBuffer exposes suboffsets but no stridesBuffer not compatible with direct access in dimension %d.Buffer is not indirectly accessible in dimension %d.memviewslice is already initialized!value too large to convert to npy_int32scipy.sparse.csgraph._flow._edmonds_karpscipy.sparse.csgraph._flow._dinicvalue too large to convert to int../../../scipy/sparse/csgraph/parameters.pxiModule '_flow' has already been imported. Re-initialisation is not supported.compile time Python version %d.%d of module '%.100s' %s runtime version %d.%dint (struct __pyx_array_obj *)struct __pyx_array_obj *(PyObject *, Py_ssize_t, char *, char const *, char *)PyObject *(PyObject *, int, int, __Pyx_TypeInfo const *)struct __pyx_memoryview_obj *(struct __pyx_memoryview_obj *, PyObject *)int (__Pyx_memviewslice *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int, int)char *(Py_buffer *, char *, Py_ssize_t, Py_ssize_t)PyObject *(__Pyx_memviewslice, int, PyObject *(*)(char *), int (*)(char *, PyObject *), int)__Pyx_memviewslice *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)void (struct __pyx_memoryview_obj *, __Pyx_memviewslice *)PyObject *(struct __pyx_memoryview_obj *)PyObject *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)char (__Pyx_memviewslice *, int)Py_ssize_t (__Pyx_memviewslice *, int)Py_ssize_t (Py_ssize_t *, Py_ssize_t *, Py_ssize_t, int, char)void *(__Pyx_memviewslice *, __Pyx_memviewslice *, char, int)int (int, Py_ssize_t, Py_ssize_t)int (PyObject *, PyObject *, int)int (__Pyx_memviewslice, __Pyx_memviewslice, int, int, int)void (__Pyx_memviewslice *, int, int)void (__Pyx_memviewslice *, int, int, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, int)void (__Pyx_memviewslice *, int, size_t, void *, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, size_t, void *)_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 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 runtime../../../../../../usr/lib/python3/dist-packages/numpy/__init__.cython-30.pxdinit scipy.sparse.csgraph._flow_cython_3_1_6._common_types_metatype_cython_3_1_6.cython_function_or_method?@&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&`&/'`&`&&`&`&`&`&`&'&`&`&&`&`&&&&`&`&&`&`&`&`&`&`&`&`&`&`&`&`&`&&''&`&&' '&`&`&&`&`&`&p&x&`&p&0000000001100000000000000000000000000001 1110000000000 322 322222$44224224442222222222222222 3 3424.4$44224222 342 3 411 41111104`5114114441111111111111111 4 4`41M4:404`5114111 441 47999999999K599K5999999999999999999K579999996899999999999999999 89E577zerostypetranspose__test__target_formattailssum__spec__source value ({}) must be between source_flowsource and sink vertices must differsourcesorted_indicessink value ({}) must be between sinkshape__set_name__selfscipy/sparse/csgraph/_flow.pyxscipy.sparse.csgraph._flowscipy.sparse._sputilsscipy.sparsesafely_cast_index_arraysrev_edge_ptrres_ptrres_indptr_viewres_indptrres_indices_viewres_indicesres_data_viewres_data__repr__range__qualname____pyx_vtable__pydata_sparse_cls__prepare__popnumpy._core.umath failed to importnumpy._core.multiarray failed to importnumpynpnnz__name__nmsgmove_atmove_a__module__{} method is not supported yet.method__metaclass__maximum_flow (line 35) maximum_flow(csgraph, source, sink) Maximize the flow between two vertices in a graph. .. versionadded:: 1.4.0 Parameters ---------- csgraph : csr_array The square matrix representing a directed graph whose (i, j)'th entry is an integer representing the capacity of the edge between vertices i and j. source : int The source vertex from which the flow flows. sink : int The sink vertex to which the flow flows. method: {'edmonds_karp', 'dinic'}, optional The method/algorithm to be used for computing the maximum flow. Following methods are supported, * 'edmonds_karp': Edmonds Karp algorithm in [1]_. * 'dinic': Dinic's algorithm in [4]_. Default is 'dinic'. .. versionadded:: 1.8.0 Returns ------- res : MaximumFlowResult A maximum flow represented by a ``MaximumFlowResult`` which includes the value of the flow in ``flow_value``, and the flow graph in ``flow``. Raises ------ TypeError: if the input graph is not in CSR format. ValueError: if the capacity values are not integers, or the source or sink are out of bounds. Notes ----- This solves the maximum flow problem on a given directed weighted graph: A flow associates to every edge a value, also called a flow, less than the capacity of the edge, so that for every vertex (apart from the source and the sink vertices), the total incoming flow is equal to the total outgoing flow. The value of a flow is the sum of the flow of all edges leaving the source vertex, and the maximum flow problem consists of finding a flow whose value is maximal. By the max-flow min-cut theorem, the maximal value of the flow is also the total weight of the edges in a minimum cut. To solve the problem, we provide Edmonds--Karp [1]_ and Dinic's algorithm [4]_. The implementation of both algorithms strive to exploit sparsity. The time complexity of the former :math:`O(|V|\,|E|^2)` and its space complexity is :math:`O(|E|)`. The latter achieves its performance by building level graphs and finding blocking flows in them. Its time complexity is :math:`O(|V|^2\,|E|)` and its space complexity is :math:`O(|E|)`. The maximum flow problem is usually defined with real valued capacities, but we require that all capacities are integral to ensure convergence. When dealing with rational capacities, or capacities belonging to :math:`x\mathbb{Q}` for some fixed :math:`x \in \mathbb{R}`, it is possible to reduce the problem to the integral case by scaling all capacities accordingly. Solving a maximum-flow problem can be used for example for graph cuts optimization in computer vision [3]_. References ---------- .. [1] Edmonds, J. and Karp, R. M. Theoretical improvements in algorithmic efficiency for network flow problems. 1972. Journal of the ACM. 19 (2): pp. 248-264 .. [2] Cormen, T. H. and Leiserson, C. E. and Rivest, R. L. and Stein C. Introduction to Algorithms. Second Edition. 2001. MIT Press. .. [3] https://en.wikipedia.org/wiki/Graph_cuts_in_computer_vision .. [4] Dinic, Efim A. Algorithm for solution of a problem of maximum flow in networks with power estimation. In Soviet Math. Doklady, vol. 11, pp. 1277-1280. 1970. Examples -------- Perhaps the simplest flow problem is that of a graph of only two vertices with an edge from source (0) to sink (1):: (0) --5--> (1) Here, the maximum flow is simply the capacity of the edge: >>> import numpy as np >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import maximum_flow >>> graph = csr_array([[0, 5], [0, 0]]) >>> maximum_flow(graph, 0, 1).flow_value 5 >>> maximum_flow(graph, 0, 1, method='edmonds_karp').flow_value 5 If, on the other hand, there is a bottleneck between source and sink, that can reduce the maximum flow:: (0) --5--> (1) --3--> (2) >>> graph = csr_array([[0, 5, 0], [0, 0, 3], [0, 0, 0]]) >>> maximum_flow(graph, 0, 2).flow_value 3 A less trivial example is given in [2]_, Chapter 26.1: >>> graph = csr_array([[0, 16, 13, 0, 0, 0], ... [0, 0, 10, 12, 0, 0], ... [0, 4, 0, 0, 14, 0], ... [0, 0, 9, 0, 0, 20], ... [0, 0, 0, 7, 0, 4], ... [0, 0, 0, 0, 0, 0]]) >>> maximum_flow(graph, 0, 5).flow_value 23 It is possible to reduce the problem of finding a maximum matching in a bipartite graph to a maximum flow problem: Let :math:`G = ((U, V), E)` be a bipartite graph. Then, add to the graph a source vertex with edges to every vertex in :math:`U` and a sink vertex with edges from every vertex in :math:`V`. Finally, give every edge in the resulting graph a capacity of 1. Then, a maximum flow in the new graph gives a maximum matching in the original graph consisting of the edges in :math:`E` whose flow is positive. Assume that the edges are represented by a :math:`\lvert U \rvert \times \lvert V \rvert` matrix in CSR format whose :math:`(i, j)`'th entry is 1 if there is an edge from :math:`i \in U` to :math:`j \in V` and 0 otherwise; that is, the input is of the form required by :func:`maximum_bipartite_matching`. Then the CSR representation of the graph constructed above contains this matrix as a block. Here's an example: >>> graph = csr_array([[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]]) >>> print(graph.toarray()) [[0 1 0 1] [1 0 1 0] [0 1 1 0]] >>> i, j = graph.shape >>> n = graph.nnz >>> indptr = np.concatenate([[0], ... graph.indptr + i, ... np.arange(n + i + 1, n + i + j + 1), ... [n + i + j]]) >>> indices = np.concatenate([np.arange(1, i + 1), ... graph.indices + i + 1, ... np.repeat(i + j + 1, j)]) >>> data = np.ones(n + i + j, dtype=int) >>> >>> graph_flow = csr_array((data, indices, indptr)) >>> print(graph_flow.toarray()) [[0 1 1 1 0 0 0 0 0] [0 0 0 0 0 1 0 1 0] [0 0 0 0 1 0 1 0 0] [0 0 0 0 0 1 1 0 0] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0]] At this point, we can find the maximum flow between the added sink and the added source and the desired matching can be obtained by restricting the flow function to the block corresponding to the original graph: >>> result = maximum_flow(graph_flow, 0, i+j+1, method='dinic') >>> matching = result.flow[1:i+1, i+1:i+j+1] >>> print(matching.toarray()) [[0 1 0 0] [1 0 0 0] [0 0 1 0]] This tells us that the first, second, and third vertex in :math:`U` are matched with the second, first, and third vertex in :math:`V` respectively. While this solves the maximum bipartite matching problem in general, note that algorithms specialized to that problem, such as :func:`maximum_bipartite_matching`, will generally perform better. This approach can also be used to solve various common generalizations of the maximum bipartite matching problem. If, for instance, some vertices can be matched with more than one other vertex, this may be handled by modifying the capacities of the new graph appropriately. maximum_flowmax_make_tails_make_edge_pointers__main__mjissparseis_pydata_spmatrixis_pydata_sparse_is_coroutineint32_initializing__init__indptrindicesiinfoidx_dtypeihas_sorted_indicesgraph must be specified as a square matrix.graph must be in CSR formatgraph capacities must be integers__func__from_scipy_sparseformatflow_valueflow_matrixflow_arrayflowfloat64emptyedmonds_karpdtype__doc__dinicdatacsr_matrixcsr_arraycsrcsgraph_origcsgraphconvert_pydata_sparse_to_scipycline_in_traceback__class_getitem____class__b_databat_ptrat_indptr_viewat_indices_viewat_endatasyncio.coroutinesastypeasarrayarange_add_reverse_edgesa_ptra_indptr_viewa_indices_viewa_enda_data_viewa?ValueErrorTypeErrorRepresents the result of a maximum flow calculation. Attributes ---------- flow_value : int The value of the maximum flow. flow : csr_array The maximum flow. MaximumFlowResult with value of %dMaximumFlowResult.__repr__MaximumFlowResult.__init__MaximumFlowResultImportErrorITYPEDTYPE0 and {}.Create for each edge pointers to its tail.Create for each edge pointers to its reverse.Add reversed edges to all edges in a graph. This adds to a given directed weighted graph all edges in the reverse direction and give them weight 0, unless they already exist. Parameters ---------- a : csr_array The square matrix in CSR format representing a directed graph Returns ------- res : csr_array A new matrix in CSR format in which the missing edges are represented by explicit zeros. maximum_flow(csgraph, source, sink) Maximize the flow between two vertices in a graph. .. versionadded:: 1.4.0 Parameters ---------- csgraph : csr_array The square matrix representing a directed graph whose (i, j)'th entry is an integer representing the capacity of the edge between vertices i and j. source : int The source vertex from which the flow flows. sink : int The sink vertex to which the flow flows. method: {'edmonds_karp', 'dinic'}, optional The method/algorithm to be used for computing the maximum flow. Following methods are supported, * 'edmonds_karp': Edmonds Karp algorithm in [1]_. * 'dinic': Dinic's algorithm in [4]_. Default is 'dinic'. .. versionadded:: 1.8.0 Returns ------- res : MaximumFlowResult A maximum flow represented by a ``MaximumFlowResult`` which includes the value of the flow in ``flow_value``, and the flow graph in ``flow``. Raises ------ TypeError: if the input graph is not in CSR format. ValueError: if the capacity values are not integers, or the source or sink are out of bounds. Notes ----- This solves the maximum flow problem on a given directed weighted graph: A flow associates to every edge a value, also called a flow, less than the capacity of the edge, so that for every vertex (apart from the source and the sink vertices), the total incoming flow is equal to the total outgoing flow. The value of a flow is the sum of the flow of all edges leaving the source vertex, and the maximum flow problem consists of finding a flow whose value is maximal. By the max-flow min-cut theorem, the maximal value of the flow is also the total weight of the edges in a minimum cut. To solve the problem, we provide Edmonds--Karp [1]_ and Dinic's algorithm [4]_. The implementation of both algorithms strive to exploit sparsity. The time complexity of the former :math:`O(|V|\,|E|^2)` and its space complexity is :math:`O(|E|)`. The latter achieves its performance by building level graphs and finding blocking flows in them. Its time complexity is :math:`O(|V|^2\,|E|)` and its space complexity is :math:`O(|E|)`. The maximum flow problem is usually defined with real valued capacities, but we require that all capacities are integral to ensure convergence. When dealing with rational capacities, or capacities belonging to :math:`x\mathbb{Q}` for some fixed :math:`x \in \mathbb{R}`, it is possible to reduce the problem to the integral case by scaling all capacities accordingly. Solving a maximum-flow problem can be used for example for graph cuts optimization in computer vision [3]_. References ---------- .. [1] Edmonds, J. and Karp, R. M. Theoretical improvements in algorithmic efficiency for network flow problems. 1972. Journal of the ACM. 19 (2): pp. 248-264 .. [2] Cormen, T. H. and Leiserson, C. E. and Rivest, R. L. and Stein C. Introduction to Algorithms. Second Edition. 2001. MIT Press. .. [3] https://en.wikipedia.org/wiki/Graph_cuts_in_computer_vision .. [4] Dinic, Efim A. Algorithm for solution of a problem of maximum flow in networks with power estimation. In Soviet Math. Doklady, vol. 11, pp. 1277-1280. 1970. Examples -------- Perhaps the simplest flow problem is that of a graph of only two vertices with an edge from source (0) to sink (1):: (0) --5--> (1) Here, the maximum flow is simply the capacity of the edge: >>> import numpy as np >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import maximum_flow >>> graph = csr_array([[0, 5], [0, 0]]) >>> maximum_flow(graph, 0, 1).flow_value 5 >>> maximum_flow(graph, 0, 1, method='edmonds_karp').flow_value 5 If, on the other hand, there is a bottleneck between source and sink, that can reduce the maximum flow:: (0) --5--> (1) --3--> (2) >>> graph = csr_array([[0, 5, 0], [0, 0, 3], [0, 0, 0]]) >>> maximum_flow(graph, 0, 2).flow_value 3 A less trivial example is given in [2]_, Chapter 26.1: >>> graph = csr_array([[0, 16, 13, 0, 0, 0], ... [0, 0, 10, 12, 0, 0], ... [0, 4, 0, 0, 14, 0], ... [0, 0, 9, 0, 0, 20], ... [0, 0, 0, 7, 0, 4], ... [0, 0, 0, 0, 0, 0]]) >>> maximum_flow(graph, 0, 5).flow_value 23 It is possible to reduce the problem of finding a maximum matching in a bipartite graph to a maximum flow problem: Let :math:`G = ((U, V), E)` be a bipartite graph. Then, add to the graph a source vertex with edges to every vertex in :math:`U` and a sink vertex with edges from every vertex in :math:`V`. Finally, give every edge in the resulting graph a capacity of 1. Then, a maximum flow in the new graph gives a maximum matching in the original graph consisting of the edges in :math:`E` whose flow is positive. Assume that the edges are represented by a :math:`\lvert U \rvert \times \lvert V \rvert` matrix in CSR format whose :math:`(i, j)`'th entry is 1 if there is an edge from :math:`i \in U` to :math:`j \in V` and 0 otherwise; that is, the input is of the form required by :func:`maximum_bipartite_matching`. Then the CSR representation of the graph constructed above contains this matrix as a block. Here's an example: >>> graph = csr_array([[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]]) >>> print(graph.toarray()) [[0 1 0 1] [1 0 1 0] [0 1 1 0]] >>> i, j = graph.shape >>> n = graph.nnz >>> indptr = np.concatenate([[0], ... graph.indptr + i, ... np.arange(n + i + 1, n + i + j + 1), ... [n + i + j]]) >>> indices = np.concatenate([np.arange(1, i + 1), ... graph.indices + i + 1, ... np.repeat(i + j + 1, j)]) >>> data = np.ones(n + i + j, dtype=int) >>> >>> graph_flow = csr_array((data, indices, indptr)) >>> print(graph_flow.toarray()) [[0 1 1 1 0 0 0 0 0] [0 0 0 0 0 1 0 1 0] [0 0 0 0 1 0 1 0 0] [0 0 0 0 0 1 1 0 0] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0]] At this point, we can find the maximum flow between the added sink and the added source and the desired matching can be obtained by restricting the flow function to the block corresponding to the original graph: >>> result = maximum_flow(graph_flow, 0, i+j+1, method='dinic') >>> matching = result.flow[1:i+1, i+1:i+j+1] >>> print(matching.toarray()) [[0 1 0 0] [1 0 0 0] [0 0 1 0]] This tells us that the first, second, and third vertex in :math:`U` are matched with the second, first, and third vertex in :math:`V` respectively. While this solves the maximum bipartite matching problem in general, note that algorithms specialized to that problem, such as :func:`maximum_bipartite_matching`, will generally perform better. This approach can also be used to solve various common generalizations of the maximum bipartite matching problem. If, for instance, some vertices can be matched with more than one other vertex, this may be handled by modifying the capacities of the new graph appropriately. __pyx_fatalerror&BfAQe6$fA$AQU!1 Ea}ATar1 %q 1a ]& RwaqV1Da 81Jaz4vQ !:Q 1AAp S&QfAQ"!1%Qa$AQ  !1Ja&b%Rqrq"AV1$A"F!2Rqa'q6"Bc&aQ1 "Cq x}ATar1 qN!2RqfCvSs! V3aWCqawc gSV3aq>! q!N!1QkAQ q O1A! q QqQ 9Bj ]'Aa+1D1)!qG10.uHAYd'Aiqt:QgV9Aj s!'wcjwfAS7&jwb#WCwfAQj=WAXQ#7!7&BauBb57&j;7!6#7!7&Bat7!' Y.ay 'QxwgQ)2WG9A"'wa  !1&aqwc 1A}AQiwaqg^81 1vQay1G1aj:'!81A)2\*AZvQaq''9! 1N!j*AQgQhawawb A[D ZA4Bd!A N! 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