132 lines
4.3 KiB
Python
132 lines
4.3 KiB
Python
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# -*- coding: utf-8 -*-
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# Copyright (C) 2011-2012 by
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# Nicholas Mancuso <nick.mancuso@gmail.com>
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# All rights reserved.
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# BSD license.
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"""Functions for finding node and edge dominating sets.
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A `dominating set`_ for an undirected graph *G* with vertex set *V*
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and edge set *E* is a subset *D* of *V* such that every vertex not in
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*D* is adjacent to at least one member of *D*. An `edge dominating set`_
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is a subset *F* of *E* such that every edge not in *F* is
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incident to an endpoint of at least one edge in *F*.
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.. _dominating set: https://en.wikipedia.org/wiki/Dominating_set
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.. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set
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"""
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from ..matching import maximal_matching
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from ...utils import not_implemented_for
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__all__ = ["min_weighted_dominating_set",
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"min_edge_dominating_set"]
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__author__ = """Nicholas Mancuso (nick.mancuso@gmail.com)"""
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# TODO Why doesn't this algorithm work for directed graphs?
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@not_implemented_for('directed')
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def min_weighted_dominating_set(G, weight=None):
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r"""Returns a dominating set that approximates the minimum weight node
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dominating set.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph.
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weight : string
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The node attribute storing the weight of an node. If provided,
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the node attribute with this key must be a number for each
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node. If not provided, each node is assumed to have weight one.
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Returns
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-------
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min_weight_dominating_set : set
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A set of nodes, the sum of whose weights is no more than `(\log
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w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of
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each node in the graph and `w(V^*)` denotes the sum of the
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weights of each node in the minimum weight dominating set.
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Notes
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-----
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This algorithm computes an approximate minimum weighted dominating
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set for the graph `G`. The returned solution has weight `(\log
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w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each
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node in the graph and `w(V^*)` denotes the sum of the weights of
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each node in the minimum weight dominating set for the graph.
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This implementation of the algorithm runs in $O(m)$ time, where $m$
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is the number of edges in the graph.
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References
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----------
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.. [1] Vazirani, Vijay V.
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*Approximation Algorithms*.
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Springer Science & Business Media, 2001.
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"""
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# The unique dominating set for the null graph is the empty set.
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if len(G) == 0:
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return set()
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# This is the dominating set that will eventually be returned.
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dom_set = set()
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def _cost(node_and_neighborhood):
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"""Returns the cost-effectiveness of greedily choosing the given
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node.
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`node_and_neighborhood` is a two-tuple comprising a node and its
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closed neighborhood.
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"""
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v, neighborhood = node_and_neighborhood
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return G.nodes[v].get(weight, 1) / len(neighborhood - dom_set)
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# This is a set of all vertices not already covered by the
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# dominating set.
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vertices = set(G)
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# This is a dictionary mapping each node to the closed neighborhood
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# of that node.
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neighborhoods = {v: {v} | set(G[v]) for v in G}
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# Continue until all vertices are adjacent to some node in the
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# dominating set.
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while vertices:
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# Find the most cost-effective node to add, along with its
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# closed neighborhood.
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dom_node, min_set = min(neighborhoods.items(), key=_cost)
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# Add the node to the dominating set and reduce the remaining
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# set of nodes to cover.
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dom_set.add(dom_node)
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del neighborhoods[dom_node]
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vertices -= min_set
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return dom_set
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def min_edge_dominating_set(G):
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r"""Returns minimum cardinality edge dominating set.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph
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Returns
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-------
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min_edge_dominating_set : set
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Returns a set of dominating edges whose size is no more than 2 * OPT.
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Notes
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-----
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The algorithm computes an approximate solution to the edge dominating set
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problem. The result is no more than 2 * OPT in terms of size of the set.
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Runtime of the algorithm is $O(|E|)$.
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"""
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if not G:
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raise ValueError("Expected non-empty NetworkX graph!")
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return maximal_matching(G)
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