UndirectedGraph¶

class menpo.shape.UndirectedGraph(adjacency_matrix, copy=True, skip_checks=False)[source]

Bases: Graph

Class for Undirected Graph definition and manipulation.

Parameters:
• adjacency_matrix ((n_vertices, n_vertices, ) ndarray or csr_matrix) –

The adjacency matrix of the graph. The non-edges must be represented with zeros and the edges can have a weight value.

Note: adjacency_matrix must be symmetric.
• copy (bool, optional) – If False, the adjacency_matrix will not be copied on assignment.
• skip_checks (bool, optional) – If True, no checks will be performed.
Raises:
• ValueError – adjacency_matrix must be either a numpy.ndarray or a scipy.sparse.csr_matrix.
• ValueError – Graph must have at least two vertices.
• ValueError – adjacency_matrix must be square (n_vertices, n_vertices, ), ({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given instead.
• ValueError – The adjacency matrix of an undirected graph must be symmetric.

Examples

The following undirected graph

|---0---|
|       |
|       |
1-------2
|       |
|       |
3-------4
|
|
5


can be defined as

import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 1, 1],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0]])


or

from scipy.sparse import csr_matrix
([1] * 14,
([0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 3, 4, 3, 5],
[1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 4, 3, 5, 3])),
shape=(6, 6))


The adjacency matrix of the following graph with isolated vertices

       0---|
|
|
1       2
|
|
3-------4

5


can be defined as

import numpy as np
adjacency_matrix = np.array([[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])


or

from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 6, ([0, 2, 2, 4, 3, 4],
[2, 0, 4, 2, 4, 3])),
shape=(6, 6))

find_all_paths(start, end, path=[])

Returns a list of lists with all the paths (without cycles) found from start vertex to end vertex.

Parameters: start (int) – The vertex from which the paths start. end (int) – The vertex from which the paths end. path (list, optional) – An existing path to append to. paths (list of list) – The list containing all the paths from start to end.
find_all_shortest_paths(algorithm='auto', unweighted=False)

Returns the distances and predecessors arrays of the graph’s shortest paths.

Parameters:
• algorithm ('str', see below, optional) –

The algorithm to be used. Possible options are:

 ‘dijkstra’ Dijkstra’s algorithm with Fibonacci heaps ‘bellman-ford’ Bellman-Ford algorithm ‘johnson’ Johnson’s algorithm ‘floyd-warshall’ Floyd-Warshall algorithm ‘auto’ Select the best among the above
• unweighted (bool, optional) – If True, then find unweighted distances. That is, rather than finding the path between each vertex such that the sum of weights is minimized, find the path such that the number of edges is minimized.
Returns:

• distances ((n_vertices, n_vertices,) ndarray) – The matrix of distances between all graph vertices. distances[i,j] gives the shortest distance from vertex i to vertex j along the graph.
• predecessors ((n_vertices, n_vertices,) ndarray) – The matrix of predecessors, which can be used to reconstruct the shortest paths. Each entry predecessors[i, j] gives the index of the previous vertex in the path from vertex i to vertex j. If no path exists between vertices i and j, then predecessors[i, j] = -9999.

find_path(start, end, method='bfs', skip_checks=False)

Returns a list with the first path (without cycles) found from the start vertex to the end vertex. It can employ either depth-first search or breadth-first search.

Parameters: start (int) – The vertex from which the path starts. end (int) – The vertex to which the path ends. method ({bfs, dfs}, optional) – The method to be used. skip_checks (bool, optional) – If True, then input arguments won’t pass through checks. Useful for efficiency. path (list) – The path’s vertices. ValueError – Method must be either bfs or dfs.
find_shortest_path(start, end, algorithm='auto', unweighted=False, skip_checks=False)

Returns a list with the shortest path (without cycles) found from start vertex to end vertex.

Parameters:
• start (int) – The vertex from which the path starts.
• end (int) – The vertex to which the path ends.
• algorithm ('str', see below, optional) –

The algorithm to be used. Possible options are:

 ‘dijkstra’ Dijkstra’s algorithm with Fibonacci heaps ‘bellman-ford’ Bellman-Ford algorithm ‘johnson’ Johnson’s algorithm ‘floyd-warshall’ Floyd-Warshall algorithm ‘auto’ Select the best among the above
• unweighted (bool, optional) – If True, then find unweighted distances. That is, rather than finding the path such that the sum of weights is minimized, find the path such that the number of edges is minimized.
• skip_checks (bool, optional) – If True, then input arguments won’t pass through checks. Useful for efficiency.
Returns:

• path (list) – The shortest path’s vertices, including start and end. If there was not path connecting the vertices, then an empty list is returned.
• distance (int or float) – The distance (cost) of the path from start to end.

get_adjacency_list()

Returns the adjacency list of the graph, i.e. a list of length n_vertices that for each vertex has a list of the vertex neighbours. If the graph is directed, the neighbours are children.

Returns: adjacency_list (list of list of length n_vertices) – The adjacency list of the graph.
has_cycles()

Checks if the graph has at least one cycle.

Returns: has_cycles (bool) – True if the graph has cycles.
has_isolated_vertices()

Whether the graph has any isolated vertices, i.e. vertices with no edge connections.

Returns: has_isolated_vertices (bool) – True if the graph has at least one isolated vertex.
classmethod init_from_edges(edges, n_vertices, skip_checks=False)[source]

Initialize graph from edges array.

Parameters: edges ((n_edges, 2, ) ndarray) – The ndarray of edges, i.e. all the pairs of vertices that are connected with an edge. n_vertices (int) – The total number of vertices, assuming that the numbering of vertices starts from 0. edges and n_vertices can be defined in a way to set isolated vertices. skip_checks (bool, optional) – If True, no checks will be performed.

Examples

The following undirected graph

|---0---|
|       |
|       |
1-------2
|       |
|       |
3-------4
|
|
5


can be defined as

from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 1], [1, 0], [0, 2], [2, 0], [1, 2], [2, 1],
[1, 3], [3, 1], [2, 4], [4, 2], [3, 4], [4, 3],
[3, 5], [5, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)


Finally, the following graph with isolated vertices

       0---|
|
|
1       2
|
|
3-------4

5


can be defined as

from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 2], [2, 0], [2, 4], [4, 2], [3, 4], [4, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)

is_edge(vertex_1, vertex_2, skip_checks=False)

Whether there is an edge between the provided vertices.

Parameters: vertex_1 (int) – The first selected vertex. Parent if the graph is directed. vertex_2 (int) – The second selected vertex. Child if the graph is directed. skip_checks (bool, optional) – If False, the given vertices will be checked. is_edge (bool) – True if there is an edge connecting vertex_1 and vertex_2. ValueError – The vertex must be between 0 and {n_vertices-1}.
is_tree()

Checks if the graph is tree.

Returns: is_true (bool) – If the graph is a tree.
isolated_vertices()

Returns the isolated vertices of the graph (if any), i.e. the vertices that have no edge connections.

Returns: isolated_vertices (list) – A list of the isolated vertices. If there aren’t any, it returns an empty list.
minimum_spanning_tree(root_vertex)[source]

Returns the minimum spanning tree of the graph using Kruskal’s algorithm.

Parameters: root_vertex (int) – The vertex that will be set as root in the output MST. mst (Tree) – The computed minimum spanning tree. ValueError – Cannot compute minimum spanning tree of a graph with isolated vertices
n_neighbours(vertex, skip_checks=False)[source]

Returns the number of neighbours of the selected vertex.

Parameters: vertex (int) – The selected vertex. skip_checks (bool, optional) – If False, the given vertex will be checked. n_neighbours (int) – The number of neighbours. ValueError – The vertex must be between 0 and {n_vertices-1}.
n_paths(start, end)

Returns the number of all the paths (without cycles) existing from start vertex to end vertex.

Parameters: start (int) – The vertex from which the paths start. end (int) – The vertex from which the paths end. paths (int) – The paths’ numbers.
neighbours(vertex, skip_checks=False)[source]

Returns the neighbours of the selected vertex.

Parameters: vertex (int) – The selected vertex. skip_checks (bool, optional) – If False, the given vertex will be checked. neighbours (list) – The list of neighbours. ValueError – The vertex must be between 0 and {n_vertices-1}.
n_edges

Returns the number of edges.

Type: int
n_vertices

Returns the number of vertices.

Type: int
vertices

Returns the list of vertices.

Type: list