def test_prim(self):
w, edges = prim(self.dasgupta_graph)
self.assertEqual(w, self.dasgupta_weight)
self.assertEqual(edges, self.dasgupta_edges)
w, pred = prim(self.dasgupta_graph, 'a', edge_list=False)
self.assertEqual(w, self.dasgupta_weight)
self.assertEqual(pred, self.dasgupta_dict)
def test_prim(self):
w, edges = prim(self.dasgupta_graph)
self.assertEqual(w, self.dasgupta_weight)
self.assertEqual(edges, self.dasgupta_edges)
w, pred = prim(self.dasgupta_graph, 'a', edge_list=False)
self.assertEqual(w, self.dasgupta_weight)
self.assertEqual(pred, self.dasgupta_dict)
def _improve(graph, edges, terminals):
r"""Attempts to improve the quality of a tree.
Given a graph, the edges of a subtree, and a list of terminal vertices,
this method attempts to improve the quality of the tree.
Parameters
----------
graph : rosemary.graphs.graphs.Graph
Parent graph.
edges : list
A list of the edges of a tree in `graph`.
terminals : list
A list of the terminal vertices.
Returns
-------
(weight, edges) : tuple
The number `weight` is the weight of the new tree. The list `edges`
contains the edges of the new tree.
Notes
-----
This method implements the improvement procedure outlined in [1]. Given
a candidate tree with edges in `edges`, we construct the subgraph of
`graph` induced by the vertices of the tree. Next, we construct a
minimum spanning tree of this induced subgraph. Finally, we delete all
non-terminal leaves from this new tree.
References
----------
.. [1] S. Voss, "Steiner's Problem in Graphs: Heuristic Methods",
Discrete Applied Mathematics 40, 1992, 45-72.
"""
tree_vertices = set()
for (u, v) in edges:
tree_vertices.add(u)
tree_vertices.add(v)
induced = graph.induced_subgraph(tree_vertices)
__, mst_edges = prim(induced)
improved = _delete_nonterminal_leaves(mst_edges, terminals)
weight = sum(graph[u][v] for (u, v) in improved)
return weight, improved
def distance_network_heuristic(graph, terminals):
r"""Returns an approximate minimal Steiner tree connecting `terminals`
in `graph`.
Given a connected, undirected graph `graph` with positive edge weights
and a subset of the vertices `terminals`, this method finds a subgraph
with near-minimal total edge cost among all connected subgraphs that
contain these vertices.
Parameters
----------
graph : rosemary.graphs.graphs.Graph
terminals : list or set or tuple
A collection of vertices of `graph`.
Returns
-------
(weight, edges) : tuple
The number `weight` is the weight of the Steiner tree. The list
`edges` is a list of the edges in the Steiner tree.
Notes
-----
The Steiner problem in graphs asks to find the minimal tree connecting
the terminal vertices. This problem is NP-complete and has proven to be
extremely difficult even for small problem instances. Given this, it is
of practical importance to have heuristics for finding low-cost Steiner
trees.
This method uses the heuristic algorithm given in [1], which produces a
tree spanning the terminals with total cost <= 2*(1 - 1/t)*MST, where t
is the number of terminal vertices and MST is the weight of a minimal
Steiner tree. We also apply the improvement procedure given in [3].
The implementation given runs in time O(t*|E|*log(|V|)), where |E| is
the number of edges of `graph` and |V| is the number of vertices.
References
----------
.. [1] L. Kou, G. Markowsky, L. Berman, "A Fast Algorithm for Steiner
Trees", Acta Informatica, Volume 15, Issue 2, June 1981, 141-145.
.. [2] P. Winter, "Steiner Problem in Networks: A Survey", Networks,
Volume 17, Issue 2, Summer 1987, 129-167.
.. [3] S. Voss, "Steiner's Problem in Graphs: Heuristic Methods",
Discrete Applied Mathematics 40, 1992, 45-72.
.. [4] H.J. Proemel, A. Steger, "The Steiner Tree Problem - A Tour
through Graphs, Algorithms, and Complexity", Vieweg, 2002.
Examples
--------
>>> G = Graph()
>>> G.add_edges([('u1', 'u2', 1), ('u1', 'v1', 2), ('u1', 'v2', 2),
('u2', 'v3', 4), ('u2', 'v4', 3), ('u2', 'v5', 5),
('u3', 'v1', 2), ('u3', 'v3', 8), ('u4', 'v2', 8),
('u4', 'v5', 8), ('v1', 'v2', 8), ('v1', 'v3', 9),
('v2', 'v5', 5), ('v3', 'v4', 8)])
>>> distance_network_heuristic(G, ['v1', 'v2', 'v3', 'v4', 'v5'])
(17, [('u1', 'u2'), ('u1', 'v1'), ('u1', 'v2'), ('u2', 'v3'),
('u2', 'v4'), ('v2', 'v5')])
"""
# Create the distance network induced by the terminal set.
distance_network = Graph()
# shortest_prev[u] holds the predecessor dict for the shortest path
# tree rooted at u.
shortest_prev = {}
shortest_dist = {}
for u in terminals:
u_dist, u_prev = dijkstra(graph, u)
shortest_dist[u] = u_dist
shortest_prev[u] = u_prev
# For each pair (u, v) of terminal vertices, add an edge with weight
# equal to the length of the shortest u, v path.
distance_network.add_edges([(u, v, shortest_dist[u][v]) for (u, v) in itertools.combinations(terminals, 2)])
# Determine the minimum spanning tree of the distance network.
_, mst_edges = prim(distance_network, edge_list=True)
subnetwork = Graph()
# Construct a subnetwork of the graph by replacing each edge in the
# minimum spanning tree by the corresponding minimum cost path.
for (u, v) in mst_edges:
a, b = shortest_prev[u][v], v
while a is not None:
subnetwork.add_edge(a, b, graph[a][b])
a, b = shortest_prev[u][a], a
# Determine the minimum spanning tree of the subnetwork.
_, subnetwork_mst_edges = prim(subnetwork, edge_list=True)
tree_weight, tree_edges = _improve(graph, subnetwork_mst_edges, terminals)
return (tree_weight, tree_edges)
