{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "No Docstrings" } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/sparse/csgraph/__init__.py", "item_line": 0, "item_type": "module", "aliases": [ "scipy.sparse.csgraph" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "see_also": [], "signature": null, "references": null, "qa": "scipy.sparse.csgraph", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Fast graph algorithms based on sparse matrix representations." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Compressed sparse graph routines (" }, { "__type": "InlineRole", "__tag": 4003, "value": "scipy.sparse.csgraph", "domain": null, "role": "mod", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ")" } ], "level": 0, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": ".. autosummary:: \n :toctree:generated/\n connected_components -- determine connected components of a graph\n laplacian -- compute the laplacian of a graph\n shortest_path -- compute the shortest path between points on a positive graph\n dijkstra -- use Dijkstra's algorithm for shortest path\n floyd_warshall -- use the Floyd-Warshall algorithm for shortest path\n bellman_ford -- use the Bellman-Ford algorithm for shortest path\n johnson -- use Johnson's algorithm for shortest path\n yen -- use Yen's algorithm for K-shortest paths between to nodes.\n breadth_first_order -- compute a breadth-first order of nodes\n depth_first_order -- compute a depth-first order of nodes\n breadth_first_tree -- construct the breadth-first tree from a given node\n depth_first_tree -- construct a depth-first tree from a given node\n minimum_spanning_tree -- construct the minimum spanning tree of a graph\n reverse_cuthill_mckee -- compute permutation for reverse Cuthill-McKee ordering\n maximum_flow -- solve the maximum flow problem for a graph\n maximum_bipartite_matching -- compute a maximum matching of a bipartite graph\n min_weight_full_bipartite_matching - compute a minimum weight full matching of a bipartite graph\n structural_rank -- compute the structural rank of a graph\n NegativeCycleError", "execution_status": null }, { "__type": "Code", "__tag": 4050, "value": ".. autosummary:: \n :toctree:generated/\n construct_dist_matrix\n csgraph_from_dense\n csgraph_from_masked\n csgraph_masked_from_dense\n csgraph_to_dense\n csgraph_to_masked\n reconstruct_path", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Contents" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This module uses graphs which are stored in a matrix format. A graph with N nodes can be represented by an (N x N) adjacency matrix G. If there is a connection from node i to node j, then G[i, j] = w, where w is the weight of the connection. For nodes i and j which are not connected, the value depends on the representation:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for dense array representations, non-edges are represented by G[i, j] = 0, infinity, or NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for dense masked representations (of type np.ma.MaskedArray), non-edges are represented by masked values. This can be useful when graphs with zero-weight edges are desired." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for sparse array representations, non-edges are represented by non-entries in the matrix. This sort of sparse representation also allows for edges with zero weights." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "As a concrete example, imagine that you would like to represent the following undirected graph " } ] }, { "__type": "Code", "__tag": 4050, "value": " G\n\n (0)\n / \\\n 1 2\n / \\\n(2) (1)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This graph has three nodes, where node 0 and 1 are connected by an edge of weight 2, and nodes 0 and 2 are connected by an edge of weight 1. We can construct the dense, masked, and sparse representations as follows, keeping in mind that an undirected graph is represented by a symmetric matrix " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> import numpy as np\n>>> G_dense = np.array([[0, 2, 1],\n... [2, 0, 0],\n... [1, 0, 0]])\n>>> G_masked = np.ma.masked_values(G_dense, 0)\n>>> from scipy.sparse import csr_array\n>>> G_sparse = csr_array(G_dense)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This becomes more difficult when zero edges are significant. For example, consider the situation when we slightly modify the above graph " } ] }, { "__type": "Code", "__tag": 4050, "value": " G2\n\n (0)\n / \\\n 0 2\n / \\\n(2) (1)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is identical to the previous graph, except nodes 0 and 2 are connected by an edge of zero weight. In this case, the dense representation above leads to ambiguities: how can non-edges be represented if zero is a meaningful value? In this case, either a masked or sparse representation must be used to eliminate the ambiguity " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> import numpy as np\n>>> G2_data = np.array([[np.inf, 2, 0 ],\n... [2, np.inf, np.inf],\n... [0, np.inf, np.inf]])\n>>> G2_masked = np.ma.masked_invalid(G2_data)\n>>> from scipy.sparse.csgraph import csgraph_from_dense\n>>> # G2_sparse = csr_array(G2_data) would give the wrong result\n>>> G2_sparse = csgraph_from_dense(G2_data, null_value=np.inf)\n>>> G2_sparse.data\narray([ 2., 0., 2., 0.])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Here we have used a utility routine from the csgraph submodule in order to convert the dense representation to a sparse representation which can be understood by the algorithms in submodule. By viewing the data array, we can see that the zero values are explicitly encoded in the graph." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Graph Representations" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Matrices may represent either directed or undirected graphs. This is specified throughout the csgraph module by a boolean keyword. Graphs are assumed to be directed by default. In a directed graph, traversal from node i to node j can be accomplished over the edge G[i, j], but not the edge G[j, i]. Consider the following dense graph " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> import numpy as np\n>>> G_dense = np.array([[0, 1, 0],\n... [2, 0, 3],\n... [0, 4, 0]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "When " }, { "__type": "InlineCode", "__tag": 4051, "value": "directed=True" }, { "__type": "Text", "__tag": 4046, "value": " we get the graph " } ] }, { "__type": "Code", "__tag": 4050, "value": " ---1--> ---3-->\n(0) (1) (2)\n <--2--- <--4---", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In a non-directed graph, traversal from node i to node j can be accomplished over either G[i, j] or G[j, i]. If both edges are not null, and the two have unequal weights, then the smaller of the two is used." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "So for the same graph, when " }, { "__type": "InlineCode", "__tag": 4051, "value": "directed=False" }, { "__type": "Text", "__tag": 4046, "value": " we get the graph " } ] }, { "__type": "Code", "__tag": 4050, "value": "(0)--1--(1)--3--(2)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that a symmetric matrix will represent an undirected graph, regardless of whether the 'directed' keyword is set to True or False. In this case, using " }, { "__type": "InlineCode", "__tag": 4051, "value": "directed=True" }, { "__type": "Text", "__tag": 4046, "value": " generally leads to more efficient computation." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The routines in this module accept as input either scipy.sparse representations (csr, csc, or lil format), masked representations, or dense representations with non-edges indicated by zeros, infinities, and NaN entries." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Directed vs. undirected" } ], "level": 2, "target": null } ], "local_refs": [] }