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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)C function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)%.200s does not export expected C function %.200sInterpreter change detected - this module can only be loaded into one interpreter per process.Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingvalue too large to convert to intunbound method %.200S() needs an argumentbase class '%.200s' is not a heap typeextension type '%.200s' has no __dict__ slot, but base type '%.200s' has: either add 'cdef dict __dict__' to the extension type or add '__slots__ = [...]' to the base type%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjecttoo many values to unpack (expected %zd)Acquisition count is %d (line %d)Unexpected format string character: '%c'%.200s() keywords must be strings%s() got multiple values for keyword argument '%U'invalid vtable found for imported typejoin() result is too long for a Python string while calling a Python objectNULL result without error in PyObject_Callcan't convert negative value to size_t__annotations__ must be set to a dict object__name__ must be set to a string object__qualname__ must be set to a string object__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 callsfunction's dictionary may not be deletedsetting function's dictionary to a non-dictinstance exception may not have a separate valueraise: exception class must be a subclass of BaseExceptioncalling %R should have returned an instance of BaseException, not %RBuffer 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 ')'Buffer has wrong number of dimensions (expected %d, got %d)Item size of buffer (%zd byte%s) does not match size of '%s' (%zd byte%s)Cannot copy memoryview slice with indirect dimensions (axis %d)Out of bounds on buffer access (axis %d)scipy.sparse.csgraph._shortest_path._johnson_undirectedscipy.sparse.csgraph._shortest_path._johnson_directedscipy.sparse.csgraph._shortest_path._johnson_add_weightsItem size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer not compatible with direct access in dimension %d.Buffer is not indirectly accessible in dimension %d.memviewslice is already initialized!cannot fit '%.200s' into an index-sized integerscipy/sparse/csgraph/_shortest_path.pyxscipy.sparse.csgraph._shortest_path.__defaults__scipy.sparse.csgraph._shortest_path._YenCandidatePaths.min_distancescipy.sparse.csgraph._shortest_path._YenCandidatePaths.max_distance%s() got an unexpected keyword argument '%U'%.200s() takes %.8s %zd positional argument%.1s (%zd given)scipy.sparse.csgraph._shortest_path._YenCandidatePaths.__setstate_cython___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../../tmp/build-env-99e64tom/lib/python3.12/site-packages/numpy/__init__.cython-30.pxdscipy.sparse.csgraph._shortest_path.NegativeCycleError.__init__scipy.sparse.csgraph._shortest_path._YenCandidatePaths.__reduce_cython__scipy.sparse.csgraph._shortest_path._YenCandidatePaths.pop_path_to_memory_view'%.200s' object is not subscriptableCannot convert %.200s to %.200sscipy.sparse.csgraph._shortest_path._floyd_warshallscipy.sparse.csgraph._shortest_path.floyd_warshallscipy.sparse.csgraph._shortest_path._YenCandidatePaths.__cinit__Argument '%.200s' has incorrect type (expected %.200s, got %.200s)need more than %zd value%.1s to unpackscipy.sparse.csgraph._shortest_path._bellman_ford_directedscipy.sparse.csgraph._shortest_path._bellman_ford_undirectedscipy.sparse.csgraph._shortest_path.bellman_fordscipy.sparse.csgraph._shortest_path.shortest_path../scipy/sparse/csgraph/parameters.pxiModule '_shortest_path' has already been imported. Re-initialisation is not supported.scipy.sparse.csgraph._shortest_pathcompile time Python version %d.%d of module '%.100s' %s runtime version %d.%dmultiple bases have vtable conflict: '%.200s' and '%.200s'Unable to initialize pickling for %.200sint (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 *)PyObject *(__Pyx_TypeInfo const *)__mro_entries__ must return a tuplemetaclass conflict: the metaclass of a derived class must be a (non-strict) subclass of the metaclasses of all its basesinit scipy.sparse.csgraph._shortest_pathscipy.sparse.csgraph._shortest_path._YenCandidatePaths.insert_pathscipy.sparse.csgraph._shortest_path._dijkstracarray.to_py.__Pyx_carray_to_py_Py_ssize_tscipy.sparse.csgraph._shortest_path._dijkstra_multi_separatescipy.sparse.csgraph._shortest_path.johnsonscipy.sparse.csgraph._shortest_path._yenlocal variable '%s' referenced before assignment'%.200s' object is unsliceablescipy.sparse.csgraph._shortest_path.yenscipy.sparse.csgraph._shortest_path.dijkstra_cython_3_1_6.cython_function_or_method_cython_3_1_6._common_types_metatypescipy.sparse.csgraph._shortest_path.__pyx_defaultsscipy.sparse.csgraph._shortest_path._YenCandidatePathstakes no arguments%.200s() %s (%zd given)takes exactly one argumentBad call flags for CyFunctiontakes no keyword arguments%.200s() %s_cython_3_1_6Unknown exceptioncannot import name %S__pyx_capi____loader__loader__file__origin__package__parent__path__submodule_search_locationsneeds an argumentan integer is requiredkeywords must be strings__pyx_fatalerrorendunparsable format string'complex double''signed char''unsigned char''short''unsigned short''int''unsigned int''long''unsigned long''long long''unsigned long long''double''complex long double''bool''char''complex float''float'a structPython objecta pointera string'long double'buffer dtype__setstate_cython__exactlynumpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointernumpy.import_arrayat mostat least__init__name '%U' is not defined__reduce_cython__Missing type objectfloyd_warshall__cinit__Kbellman_fordbuiltinscython_runtime__builtins__does not match__debug__numpyflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncscipy._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_scalarformat_from_typeinfo__orig_bases__(n)fortranjohnsonjohnson_dist_arrayyendijkstra__reduce____module____dictoffset____vectorcalloffset____weaklistoffset__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_shortest_pathconst doubleconst intITYPE_tDTYPE_tc͐9^͑44T44DT44   h t      4 N   Zfr~4N- VV- VVVVV- - VV- VVN N - VVVVVVVVVVVVVVVV 7 V7 7 VV VVV V        $  7    H@@H@@@@@ @@ @@ @@@@@@@@@@@@@@@@HH @0 @@ @@@H @H    6  a  q`N<*"БǑ, ` W M z r]D'*[#2###!*A-0-- -,,111101ucݵ˵ڻԻ00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899 yen(csgraph, source, sink, K, *, directed=True, return_predecessors=False, unweighted=False) Yen's K-Shortest Paths algorithm on a directed or undirected graph. .. versionadded:: 1.14.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. source : int The index of the starting node for the paths. sink : int The index of the ending node for the paths. K : int The number of shortest paths to find. directed : bool, optional If ``True`` (default), then find the shortest path on a directed graph: only move from point ``i`` to point ``j`` along paths ``csgraph[i, j]``. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along ``csgraph[i, j]`` or ``csgraph[j, i]``. return_predecessors : bool, optional If ``True``, return the size ``(M, N)`` predecessor matrix. Default: ``False``. unweighted : bool, optional If ``True``, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. Default: ``False``. Returns ------- dist_array : ndarray Array of size ``M`` of shortest distances between the source and sink nodes. ``dist_array[i]`` gives the i-th shortest distance from the source to the sink along the graph. ``M`` is the number of shortest paths found, which is less than or equal to `K`. predecessors : ndarray Returned only if ``return_predecessors == True``. The M x N matrix of predecessors, which can be used to reconstruct the shortest paths. ``M`` is the number of shortest paths found, which is less than or equal to `K`. Row ``i`` of the predecessor matrix contains information on the ``i``-th shortest path from the source to the sink: each entry ``predecessors[i, j]`` gives the index of the previous node in the path from the source to node ``j``. If the path does not pass via node ``j``, then ``predecessors[i, j] = -9999``. Raises ------ NegativeCycleError: If there are negative cycles in the graph Notes ----- Yen's algorithm is a graph search algorithm that finds single-source `K`-shortest loopless paths for a graph with nonnegative edge cost. The algorithm was published by Jin Y. Yen in 1971 and employs any shortest path algorithm to find the best path, then proceeds to find ``K - 1`` deviations of the best path. The algorithm is based on Dijsktra's algorithm for finding each shortest path. In case there are negative edges in the graph, Johnson's algorithm is applied. If multiple valid solutions are possible, output may vary with SciPy and Python version. References ---------- .. [1] https://en.wikipedia.org/wiki/Yen%27s_algorithm .. [2] https://www.ams.org/journals/qam/1970-27-04/S0033-569X-1970-0253822-7/ Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import yen >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_array, predecessors = yen(csgraph=graph, source=0, sink=3, K=2, ... directed=False, return_predecessors=True) >>> dist_array array([2., 5.]) >>> predecessors array([[-9999, 0, -9999, 1], [-9999, -9999, 0, 2]], dtype=int32) johnson(csgraph, directed=True, indices=None, return_predecessors=False, unweighted=False) Compute the shortest path lengths using Johnson's algorithm. Johnson's algorithm combines the Bellman-Ford algorithm and Dijkstra's algorithm to quickly find shortest paths in a way that is robust to the presence of negative cycles. If a negative cycle is detected, an error is raised. For graphs without negative edge weights, dijkstra may be faster. .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] indices : array_like or int, optional if specified, only compute the paths from the points at the given indices. return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray, shape (n_indices, n_nodes,) Returned only if return_predecessors == True. If `indices` is None then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph Notes ----- This routine is specially designed for graphs with negative edge weights. If all edge weights are positive, then Dijkstra's algorithm is a better choice. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import johnson >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = johnson(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) bellman_ford(csgraph, directed=True, indices=None, return_predecessors=False, unweighted=False) Compute the shortest path lengths using the Bellman-Ford algorithm. The Bellman-Ford algorithm can robustly deal with graphs with negative weights. If a negative cycle is detected, an error is raised. For graphs without negative edge weights, Dijkstra's algorithm may be faster. .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] indices : array_like or int, optional if specified, only compute the paths from the points at the given indices. return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray, shape (n_indices, n_nodes,) Returned only if ``return_predecessors=True``. If `indices` is None then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph Notes ----- This routine is specially designed for graphs with negative edge weights. If all edge weights are positive, then Dijkstra's algorithm is a better choice. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import bellman_ford >>> graph = [ ... [0, 1 ,2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = bellman_ford(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) dijkstra(csgraph, directed=True, indices=None, return_predecessors=False, unweighted=False, limit=np.inf, min_only=False) Dijkstra algorithm using priority queue .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of non-negative distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j] and from point j to i along paths csgraph[j, i]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j or j to i along either csgraph[i, j] or csgraph[j, i]. .. warning:: Refer the notes below while using with ``directed=False``. indices : array_like or int, optional if specified, only compute the paths from the points at the given indices. return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. limit : float, optional The maximum distance to calculate, must be >= 0. Using a smaller limit will decrease computation time by aborting calculations between pairs that are separated by a distance > limit. For such pairs, the distance will be equal to np.inf (i.e., not connected). .. versionadded:: 0.14.0 min_only : bool, optional If False (default), for every node in the graph, find the shortest path from every node in indices. If True, for every node in the graph, find the shortest path from any of the nodes in indices (which can be substantially faster). .. versionadded:: 1.3.0 Returns ------- dist_matrix : ndarray, shape ([n_indices, ]n_nodes,) The matrix of distances between graph nodes. If min_only=False, dist_matrix has shape (n_indices, n_nodes) and dist_matrix[i, j] gives the shortest distance from point i to point j along the graph. If min_only=True, dist_matrix has shape (n_nodes,) and contains for a given node the shortest path to that node from any of the nodes in indices. predecessors : ndarray, shape ([n_indices, ]n_nodes,) If ``min_only=False``, this has shape ``(n_indices, n_nodes)``, otherwise it has shape ``(n_nodes,)``. If `indices` is None and ``min_only=False`` then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. Returned only if return_predecessors == True. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 sources : ndarray, shape (n_nodes,) Returned only if min_only=True and return_predecessors=True. Contains the index of the source which had the shortest path to each target. If no path exists within the limit, this will contain -9999. The value at the indices passed will be equal to that index (i.e. the fastest way to reach node i, is to start on node i). Notes ----- As currently implemented, Dijkstra's algorithm does not work for graphs with direction-dependent distances when directed == False. i.e., if csgraph[i,j] and csgraph[j,i] are not equal and both are nonzero, setting directed=False will not yield the correct result. Also, this routine does not work for graphs with negative distances. Negative distances can lead to infinite cycles that must be handled by specialized algorithms such as Bellman-Ford's algorithm or Johnson's algorithm. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import dijkstra >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [0, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 3) 3 >>> dist_matrix, predecessors = dijkstra(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) floyd_warshall(csgraph, directed=True, return_predecessors=False, unweighted=False, overwrite=False) Compute the shortest path lengths using the Floyd-Warshall algorithm .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. overwrite : bool, optional If True, overwrite csgraph with the result. This applies only if csgraph is a dense, c-ordered array with dtype=float64. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray Returned only if return_predecessors == True. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph Notes ----- If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import floyd_warshall >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = floyd_warshall(csgraph=graph, directed=False, return_predecessors=True) >>> dist_matrix array([[0., 1., 2., 2.], [1., 0., 3., 1.], [2., 3., 0., 3.], [2., 1., 3., 0.]]) >>> predecessors array([[-9999, 0, 0, 1], [ 1, -9999, 0, 1], [ 2, 0, -9999, 2], [ 1, 3, 3, -9999]], dtype=int32) shortest_path(csgraph, method='auto', directed=True, return_predecessors=False, unweighted=False, overwrite=False, indices=None) Perform a shortest-path graph search on a positive directed or undirected graph. .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. method : string ['auto'|'FW'|'D'], optional Algorithm to use for shortest paths. Options are: 'auto' -- (default) select the best among 'FW', 'D', 'BF', or 'J' based on the input data. 'FW' -- Floyd-Warshall algorithm. Computational cost is approximately ``O[N^3]``. The input csgraph will be converted to a dense representation. 'D' -- Dijkstra's algorithm with priority queue. Computational cost is approximately ``O[I * (E + N) * log(N)]``, where ``E`` is the number of edges in the graph, and ``I = len(indices)`` if ``indices`` is passed. Otherwise, ``I = N``. The input csgraph will be converted to a csr representation. 'BF' -- Bellman-Ford algorithm. This algorithm can be used when weights are negative. If a negative cycle is encountered, an error will be raised. Computational cost is approximately ``O[N(N^2 k)]``, where ``k`` is the average number of connected edges per node. The input csgraph will be converted to a csr representation. 'J' -- Johnson's algorithm. Like the Bellman-Ford algorithm, Johnson's algorithm is designed for use when the weights are negative. It combines the Bellman-Ford algorithm with Dijkstra's algorithm for faster computation. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. overwrite : bool, optional If True, overwrite csgraph with the result. This applies only if method == 'FW' and csgraph is a dense, c-ordered array with dtype=float64. indices : array_like or int, optional If specified, only compute the paths from the points at the given indices. Incompatible with method == 'FW'. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray, shape (n_indices, n_nodes,) Returned only if return_predecessors == True. If `indices` is None then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph See Also -------- :ref:`word-ladders-example` : An illustratation of the ``shortest_path`` API with a meaninful example. It also reconstructs the shortest path by using predecessors matrix returned by this function. Notes ----- As currently implemented, Dijkstra's algorithm and Johnson's algorithm do not work for graphs with direction-dependent distances when directed == False. i.e., if csgraph[i,j] and csgraph[j,i] are non-equal edges, method='D' may yield an incorrect result. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import shortest_path >>> graph = [ ... [0, 0, 7, 0], ... [0, 0, 8, 5], ... [7, 8, 0, 0], ... [0, 5, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 2) 7 (1, 2) 8 (1, 3) 5 (2, 0) 7 (2, 1) 8 (3, 1) 5 >>> sources = [0, 2] >>> dist_matrix, predecessors = shortest_path(csgraph=graph, directed=False, indices=sources, return_predecessors=True) >>> dist_matrix array([[ 0., 15., 7., 20.], [ 7., 8., 0., 13.]]) >>> predecessors array([[-9999, 2, 0, 1], [ 2, 2, -9999, 1]], dtype=int32) Reconstructing shortest paths from sources to all the nodes of the graph. >>> shortest_paths = {} >>> for idx in range(len(sources)): ... for node in range(4): ... curr_node = node # start from the destination node ... path = [] ... while curr_node != -9999: # no previous node available, exit the loop ... path = [curr_node] + path # prefix the previous node obtained from the last iteration ... curr_node = int(predecessors[idx][curr_node]) # set current node to previous node ... shortest_paths[(sources[idx], node)] = path ... Computing the length of the shortest path from node 0 to node 3 of the graph. It can be observed that computed length and the ``dist_matrix`` value are exactly same. >>> shortest_paths[(0, 3)] [0, 2, 1, 3] >>> path03 = shortest_paths[(0, 3)] >>> sum([graph[path03[0], path03[1]], graph[path03[1], path03[2]], graph[path03[2], path03[3]]]) np.int64(20) >>> dist_matrix[0][3] np.float64(20.0) Another example of computing shortest path length from node 2 to node 3. Here, ``dist_matrix[1][3]`` is used to get the length of the path returned by ``shortest_path``. This is because node 2 is the second source, so the lengths of the path from it to other nodes in the graph will be at index 1 in ``dist_matrix``. >>> shortest_paths[(2, 3)] [2, 1, 3] >>> path23 = shortest_paths[(2, 3)] >>> sum([graph[path23[0], path23[1]], graph[path23[1], path23[2]]]) np.int64(13) >>> dist_matrix[1][3] np.float64(13.0) scipy/sparse/csgraph/_shortest_path.pyxscipy.sparse.csgraph._validationscipy.sparse.csgraph._shortest_pathnumpy._core.umath failed to importnumpy._core.multiarray failed to importno default __reduce__ due to non-trivial __cinit__Note that Cython is deliberately stricter than PEP-484 and rejects subclasses of builtin types. If you need to pass subclasses then set the 'annotation_typing' directive to False.Not enough rows in sources matrix. Got Not enough rows in distances matrix. Got Invalid predecessors array shape Graph has negative weights: dijkstra will give inaccurate results if the graph contains negative cycles. Consider johnson or bellman_ford.Cannot specify indices with method == 'FW'. yen(csgraph, source, sink, K, *, directed=True, return_predecessors=False, unweighted=False) Yen's K-Shortest Paths algorithm on a directed or undirected graph. .. versionadded:: 1.14.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. source : int The index of the starting node for the paths. sink : int The index of the ending node for the paths. K : int The number of shortest paths to find. directed : bool, optional If ``True`` (default), then find the shortest path on a directed graph: only move from point ``i`` to point ``j`` along paths ``csgraph[i, j]``. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along ``csgraph[i, j]`` or ``csgraph[j, i]``. return_predecessors : bool, optional If ``True``, return the size ``(M, N)`` predecessor matrix. Default: ``False``. unweighted : bool, optional If ``True``, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. Default: ``False``. Returns ------- dist_array : ndarray Array of size ``M`` of shortest distances between the source and sink nodes. ``dist_array[i]`` gives the i-th shortest distance from the source to the sink along the graph. ``M`` is the number of shortest paths found, which is less than or equal to `K`. predecessors : ndarray Returned only if ``return_predecessors == True``. The M x N matrix of predecessors, which can be used to reconstruct the shortest paths. ``M`` is the number of shortest paths found, which is less than or equal to `K`. Row ``i`` of the predecessor matrix contains information on the ``i``-th shortest path from the source to the sink: each entry ``predecessors[i, j]`` gives the index of the previous node in the path from the source to node ``j``. If the path does not pass via node ``j``, then ``predecessors[i, j] = -9999``. Raises ------ NegativeCycleError: If there are negative cycles in the graph Notes ----- Yen's algorithm is a graph search algorithm that finds single-source `K`-shortest loopless paths for a graph with nonnegative edge cost. The algorithm was published by Jin Y. Yen in 1971 and employs any shortest path algorithm to find the best path, then proceeds to find ``K - 1`` deviations of the best path. The algorithm is based on Dijsktra's algorithm for finding each shortest path. In case there are negative edges in the graph, Johnson's algorithm is applied. If multiple valid solutions are possible, output may vary with SciPy and Python version. References ---------- .. [1] https://en.wikipedia.org/wiki/Yen%27s_algorithm .. [2] https://www.ams.org/journals/qam/1970-27-04/S0033-569X-1970-0253822-7/ Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import yen >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_array, predecessors = yen(csgraph=graph, source=0, sink=3, K=2, ... directed=False, return_predecessors=True) >>> dist_array array([2., 5.]) >>> predecessors array([[-9999, 0, -9999, 1], [-9999, -9999, 0, 2]], dtype=int32) shortest_path(csgraph, method='auto', directed=True, return_predecessors=False, unweighted=False, overwrite=False, indices=None) Perform a shortest-path graph search on a positive directed or undirected graph. .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. method : string ['auto'|'FW'|'D'], optional Algorithm to use for shortest paths. Options are: 'auto' -- (default) select the best among 'FW', 'D', 'BF', or 'J' based on the input data. 'FW' -- Floyd-Warshall algorithm. Computational cost is approximately ``O[N^3]``. The input csgraph will be converted to a dense representation. 'D' -- Dijkstra's algorithm with priority queue. Computational cost is approximately ``O[I * (E + N) * log(N)]``, where ``E`` is the number of edges in the graph, and ``I = len(indices)`` if ``indices`` is passed. Otherwise, ``I = N``. The input csgraph will be converted to a csr representation. 'BF' -- Bellman-Ford algorithm. This algorithm can be used when weights are negative. If a negative cycle is encountered, an error will be raised. Computational cost is approximately ``O[N(N^2 k)]``, where ``k`` is the average number of connected edges per node. The input csgraph will be converted to a csr representation. 'J' -- Johnson's algorithm. Like the Bellman-Ford algorithm, Johnson's algorithm is designed for use when the weights are negative. It combines the Bellman-Ford algorithm with Dijkstra's algorithm for faster computation. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. overwrite : bool, optional If True, overwrite csgraph with the result. This applies only if method == 'FW' and csgraph is a dense, c-ordered array with dtype=float64. indices : array_like or int, optional If specified, only compute the paths from the points at the given indices. Incompatible with method == 'FW'. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray, shape (n_indices, n_nodes,) Returned only if return_predecessors == True. If `indices` is None then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph See Also -------- :ref:`word-ladders-example` : An illustratation of the ``shortest_path`` API with a meaninful example. It also reconstructs the shortest path by using predecessors matrix returned by this function. Notes ----- As currently implemented, Dijkstra's algorithm and Johnson's algorithm do not work for graphs with direction-dependent distances when directed == False. i.e., if csgraph[i,j] and csgraph[j,i] are non-equal edges, method='D' may yield an incorrect result. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import shortest_path >>> graph = [ ... [0, 0, 7, 0], ... [0, 0, 8, 5], ... [7, 8, 0, 0], ... [0, 5, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 2) 7 (1, 2) 8 (1, 3) 5 (2, 0) 7 (2, 1) 8 (3, 1) 5 >>> sources = [0, 2] >>> dist_matrix, predecessors = shortest_path(csgraph=graph, directed=False, indices=sources, return_predecessors=True) >>> dist_matrix array([[ 0., 15., 7., 20.], [ 7., 8., 0., 13.]]) >>> predecessors array([[-9999, 2, 0, 1], [ 2, 2, -9999, 1]], dtype=int32) Reconstructing shortest paths from sources to all the nodes of the graph. >>> shortest_paths = {} >>> for idx in range(len(sources)): ... for node in range(4): ... curr_node = node # start from the destination node ... path = [] ... while curr_node != -9999: # no previous node available, exit the loop ... path = [curr_node] + path # prefix the previous node obtained from the last iteration ... curr_node = int(predecessors[idx][curr_node]) # set current node to previous node ... shortest_paths[(sources[idx], node)] = path ... Computing the length of the shortest path from node 0 to node 3 of the graph. It can be observed that computed length and the ``dist_matrix`` value are exactly same. >>> shortest_paths[(0, 3)] [0, 2, 1, 3] >>> path03 = shortest_paths[(0, 3)] >>> sum([graph[path03[0], path03[1]], graph[path03[1], path03[2]], graph[path03[2], path03[3]]]) np.int64(20) >>> dist_matrix[0][3] np.float64(20.0) Another example of computing shortest path length from node 2 to node 3. Here, ``dist_matrix[1][3]`` is used to get the length of the path returned by ``shortest_path``. This is because node 2 is the second source, so the lengths of the path from it to other nodes in the graph will be at index 1 in ``dist_matrix``. >>> shortest_paths[(2, 3)] [2, 1, 3] >>> path23 = shortest_paths[(2, 3)] >>> sum([graph[path23[0], path23[1]], graph[path23[1], path23[2]]]) np.int64(13) >>> dist_matrix[1][3] np.float64(13.0) oQ  %Rq tnAYj}ArQgV2QQa vQaxs!"G1CvQ 1 A 7'A"F!9F%vWE 1 A 7'A"Kq!q& 2T("Cs"DA *AQ!wbjqbQc6q1KqbRs!:T6q1BgQc+[q 1 !64vQ e1A BfBd& aq !63azV1 e1A BfBc*Da 1 !6#V1 BfASa !63azV1 BfBc*Daq2U!7%q7!6ayt1qRvQcq"F!3fA 1 Qa Q**;1##7q $A Q**;1##7q wban$>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import johnson >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = johnson(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) floyd_warshall(csgraph, directed=True, return_predecessors=False, unweighted=False, overwrite=False) Compute the shortest path lengths using the Floyd-Warshall algorithm .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. overwrite : bool, optional If True, overwrite csgraph with the result. This applies only if csgraph is a dense, c-ordered array with dtype=float64. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray Returned only if return_predecessors == True. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph Notes ----- If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import floyd_warshall >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = floyd_warshall(csgraph=graph, directed=False, return_predecessors=True) >>> dist_matrix array([[0., 1., 2., 2.], [1., 0., 3., 1.], [2., 3., 0., 3.], [2., 1., 3., 0.]]) >>> predecessors array([[-9999, 0, 0, 1], [ 1, -9999, 0, 1], [ 2, 0, -9999, 2], [ 1, 3, 3, -9999]], dtype=int32) bellman_ford(csgraph, directed=True, indices=None, return_predecessors=False, unweighted=False) Compute the shortest path lengths using the Bellman-Ford algorithm. The Bellman-Ford algorithm can robustly deal with graphs with negative weights. If a negative cycle is detected, an error is raised. For graphs without negative edge weights, Dijkstra's algorithm may be faster. .. versionadded:: 0.11.0 Parameters ---------- csgraph : array_like, or sparse array or matrix, 2 dimensions The N x N array of distances representing the input graph. directed : bool, optional If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i] indices : array_like or int, optional if specified, only compute the paths from the points at the given indices. return_predecessors : bool, optional If True, return the size (N, N) predecessor matrix. unweighted : bool, optional If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized. Returns ------- dist_matrix : ndarray The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph. predecessors : ndarray, shape (n_indices, n_nodes,) Returned only if ``return_predecessors=True``. If `indices` is None then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 Raises ------ NegativeCycleError: if there are negative cycles in the graph Notes ----- This routine is specially designed for graphs with negative edge weights. If all edge weights are positive, then Dijkstra's algorithm is a better choice. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import bellman_ford >>> graph = [ ... [0, 1 ,2, 0], ... [0, 0, 0, 1], ... [2, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 0) 2 (2, 3) 3 >>> dist_matrix, predecessors = bellman_ford(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) _YenCandidatePaths.__setstate_cython___YenCandidatePaths.__reduce_cython__ Routines for performing shortest-path graph searches The main interface is in the function :func:`shortest_path`. This calls cython routines that compute the shortest path using the Floyd-Warshall algorithm, Dijkstra's algorithm with priority queue, the Bellman-Ford algorithm, or Johnson's Algorithm. Yen's k-Shortest Path Algorithm is available for finding the k-shortest paths between two nodes in a graph. Not enough rows in predecessors matrix. Got Negative cycle detected on node %i1Z,AYjBfBa!9Ja"$a#4q wc G6!0HAQ 1  whd'qvRq s. a G;a G1Biq E&1 a 87%s#Rr2Rq rQfBa Qwc 87! *AQ~Qiq2!)( 1xq ,A#= 0. Using a smaller limit will decrease computation time by aborting calculations between pairs that are separated by a distance > limit. For such pairs, the distance will be equal to np.inf (i.e., not connected). .. versionadded:: 0.14.0 min_only : bool, optional If False (default), for every node in the graph, find the shortest path from every node in indices. If True, for every node in the graph, find the shortest path from any of the nodes in indices (which can be substantially faster). .. versionadded:: 1.3.0 Returns ------- dist_matrix : ndarray, shape ([n_indices, ]n_nodes,) The matrix of distances between graph nodes. If min_only=False, dist_matrix has shape (n_indices, n_nodes) and dist_matrix[i, j] gives the shortest distance from point i to point j along the graph. If min_only=True, dist_matrix has shape (n_nodes,) and contains for a given node the shortest path to that node from any of the nodes in indices. predecessors : ndarray, shape ([n_indices, ]n_nodes,) If ``min_only=False``, this has shape ``(n_indices, n_nodes)``, otherwise it has shape ``(n_nodes,)``. If `indices` is None and ``min_only=False`` then ``n_indices = n_nodes`` and the shape of the matrix becomes ``(n_nodes, n_nodes)``. Returned only if return_predecessors == True. The matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999 sources : ndarray, shape (n_nodes,) Returned only if min_only=True and return_predecessors=True. Contains the index of the source which had the shortest path to each target. If no path exists within the limit, this will contain -9999. The value at the indices passed will be equal to that index (i.e. the fastest way to reach node i, is to start on node i). Notes ----- As currently implemented, Dijkstra's algorithm does not work for graphs with direction-dependent distances when directed == False. i.e., if csgraph[i,j] and csgraph[j,i] are not equal and both are nonzero, setting directed=False will not yield the correct result. Also, this routine does not work for graphs with negative distances. Negative distances can lead to infinite cycles that must be handled by specialized algorithms such as Bellman-Ford's algorithm or Johnson's algorithm. If multiple valid solutions are possible, output may vary with SciPy and Python version. Examples -------- >>> from scipy.sparse import csr_array >>> from scipy.sparse.csgraph import dijkstra >>> graph = [ ... [0, 1, 2, 0], ... [0, 0, 0, 1], ... [0, 0, 0, 3], ... [0, 0, 0, 0] ... ] >>> graph = csr_array(graph) >>> print(graph) Coords Values (0, 1) 1 (0, 2) 2 (1, 3) 1 (2, 3) 3 >>> dist_matrix, predecessors = dijkstra(csgraph=graph, directed=False, indices=0, return_predecessors=True) >>> dist_matrix array([0., 1., 2., 2.]) >>> predecessors array([-9999, 0, 0, 1], dtype=int32) convert_pydata_sparse_to_scipy TnAYj}Aaq$A5JfA5#T6q2U!7%q7%uA 2T)2Q #1 !6#V1 q'q '(/y)*G1*1! t3a()NbPQqAZwjqq1BfASa7"F! 1  q$AXWHA%-Ya '7!8:XQAq(r1C1Ac+Rq%Qq ",HBa""$4HAQ :R'xrAr.*A!,A!/t1t81A1L5q1ARvQ.aqRvQk&,G6RvRs$fA1AqrQk#Rq !#2V1Ky2Rqq}Aq_A Bqxq 1-QnAYj*!vQaxs!"G1CvQwgS"F!9F%vQwgS"Kq!q& 2T("Cs"DA *AQ"F"Cq $fAuARqq'#QkAqRvRs!:Tq%qRvRs!:Tq6#V1we56ayt1 qq -|1!*MQt3a !Fb= A"F"Cq $fAqRvQcq"F!3fA  m1 1 --B"A9Bj Q&gRvQAXWHA%Ya  ]! G8:XQ -Q a1:Taqq 81A"(!1{(!1Invalid sources array shape Negative cycle in nodes %sNegativeCycleError.__init__safely_cast_index_arraysindices out of range 0...Nfloyd_warshall (line 290)unrecognized method '%s'shortest_path (line 47)scipy.sparse._sputilsbellman_ford (line 870)No edge between nodes has_negative_weightsreturn_predecessorsdummy_source_matrixpredecessor_matrixjohnson_dist_arraydummy_double_arraycline_in_tracebackasyncio.coroutinesNegativeCycleErrorjohnson (line 1111)dijkstra (line 483)_YenCandidatePaths__setstate_cython__num_paths_foundlimit must be >= 0dummy_int_arrayNo paths to popvalidate_graphfloyd_warshallcopy_if_sparseAssertionErroryen (line 1393)source_matrixshortest_path__reduce_cython__isMaskedArraycopy_if_dense__class_getitem__scipy.sparsereturn_shape_is_coroutine_initializingdense_outputcsrT_indicesbellman_fordRuntimeError__mro_entries__dist_matrixcsr_indicescsrT_indptrImportErrorunweighted__pyx_vtable__dist_arraycsr_outputcsr_indptrcompressedatleast_1dValueError__reduce_ex____pyx_stateoverwrite__metaclass__isenabledis_sparsecsr_arraycsrT_dataaccept_fvTypeErrorwarnings__setstate____set_name____qualname__min_onlyissparseisfinite__getstate__, expected directeddijkstradiagonalcsr_datacsgraphTadd_note. 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