Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
r""" Convexity properties of graphs
This class gathers the algorithms related to convexity in a graph. It implements the following methods:
.. csv-table:: :class: contentstable :widths: 30, 70 :delim: |
:meth:`ConvexityProperties.hull` | Returns the convex hull of a set of vertices :meth:`ConvexityProperties.hull_number` | Computes the hull number of a graph and a corresponding generating set.
These methods can be used through the :class:`ConvexityProperties` object returned by :meth:`Graph.convexity_properties`.
AUTHORS:
- Nathann Cohen
Methods ------- """
############################################################################## # Copyright (C) 2011 Nathann Cohen <nathann.cohen@gmail.com> # Distributed under the terms of the GNU General Public License (GPL) # The full text of the GPL is available at: # http://www.gnu.org/licenses/ ############################################################################## from __future__ import print_function
include "sage/data_structures/bitset.pxi" from sage.numerical.backends.generic_backend cimport GenericBackend
cdef class ConvexityProperties: r""" This class gathers the algorithms related to convexity in a graph.
**Definitions**
A set `S \subseteq V(G)` of vertices is said to be convex if for all `u,v\in S` the set `S` contains all the vertices located on a shortest path between `u` and `v`. Alternatively, a set `S` is said to be convex if the distances satisfy `\forall u,v\in S, \forall w\in V\backslash S : d_{G}(u,w) + d_{G}(w,v) > d_{G}(u,v)`.
The convex hull `h(S)` of a set `S` of vertices is defined as the smallest convex set containing `S`.
It is a closure operator, as trivially `S\subseteq h(S)` and `h(h(S)) = h(S)`.
**What this class contains**
As operations on convex sets generally involve the computation of distances between vertices, this class' purpose is to cache that information so that computing the convex hulls of several different sets of vertices does not imply recomputing several times the distances between the vertices.
In order to compute the convex hull of a set `S` it is possible to write the following algorithm.
*For any pair `u,v` of elements in the set `S`, and for any vertex `w`* *outside of it, add `w` to `S` if `d_{G}(u,w) + d_{G}(w,v) = d_{G}(u,v)`.* *When no vertex can be added anymore, the set `S` is convex*
The distances are not actually that relevant. The same algorithm can be implemented by remembering for each pair `u, v` of vertices the list of elements `w` satisfying the condition, and this is precisely what this class remembers, encoded as bitsets to make storage and union operations more efficient.
.. NOTE::
* This class is useful if you compute the convex hulls of many sets in the same graph, or if you want to compute the hull number itself as it involves many calls to :meth:`hull`
* Using this class on non-connected graphs is a waste of space and efficiency ! If your graph is disconnected, the best for you is to deal independently with each connected component, whatever you are doing.
**Possible improvements**
When computing a convex set, all the pairs of elements belonging to the set `S` are enumerated several times.
* There should be a smart way to avoid enumerating pairs of vertices which have already been tested. The cost of each of them is not very high, so keeping track of those which have been tested already may be too expensive to gain any efficiency.
* The ordering in which they are visited is currently purely lexicographic, while there is a Poset structure to exploit. In particular, when two vertices `u, v` are far apart and generate a set `h(\{u,v\})` of vertices, all the pairs of vertices `u', v'\in h(\{u,v\})` satisfy `h(\{u',v'\}) \subseteq h(\{u,v\})`, and so it is useless to test the pair `u', v'` when both `u` and `v` where present.
* The information cached is for any pair `u,v` of vertices the list of elements `z` with `d_{G}(u,w) + d_{G}(w,v) = d_{G}(u,v)`. This is not in general equal to `h(\{u,v\})` !
Nothing says these recommandations will actually lead to any actual improvements. There are just some ideas remembered while writing this code. Trying to optimize may well lead to lost in efficiency on many instances.
EXAMPLES::
sage: from sage.graphs.convexity_properties import ConvexityProperties sage: g = graphs.PetersenGraph() sage: CP = ConvexityProperties(g) sage: CP.hull([1,3]) [1, 2, 3] sage: CP.hull_number() 3
TESTS::
sage: ConvexityProperties(digraphs.Circuit(5)) Traceback (most recent call last): ... ValueError: This is currenly implemented for Graphs only.Only minor updates are needed if you want to makeit support DiGraphs too. """
def __init__(self, G): r""" Constructor
EXAMPLES::
sage: from sage.graphs.convexity_properties import ConvexityProperties sage: g = graphs.PetersenGraph() sage: ConvexityProperties(g) <sage.graphs.convexity_properties.ConvexityProperties object at ...> """ "it support DiGraphs too.")
# Cached number of vertices
cdef int j,k
# Temporary variables cdef dict d_i cdef dict d_j cdef int d_ij
# Remembering integers instead of the labels, and building dictionaries # in both directions.
# Computation of distances between all pairs. Costly.
# _cache_hull_pairs[u*n + v] is a bitset whose 1 bits are the vertices located on a shortest path from vertex u to v # # Note that u < v
# Filling the cache # # The p_bitset variable iterates over the successive elements of the cache # # For any pair i,j of vertices (i<j), we built the bitset of all the # elements k which are on a shortest path from i to j
# Caching the distances from i to the other vertices
# Caching the distances from j to the other vertices
# Caching the distance between i and j
# Initializing the new bitset
# Filling it
# Next bitset !
def __destruct__(self): r""" Destructor
EXAMPLES::
sage: from sage.graphs.convexity_properties import ConvexityProperties sage: g = graphs.PetersenGraph() sage: ConvexityProperties(g) <sage.graphs.convexity_properties.ConvexityProperties object at ...>
""" cdef bitset_t * p_bitset = self._cache_hull_pairs cdef int i
for 0 <= i < ((self._n*(self._n-1))>>1): bitset_free(p_bitset[0]) p_bitset = p_bitset + 1
sig_free(self._cache_hull_pairs)
cdef list _vertices_to_integers(self, vertices): r""" Converts a list of vertices to a list of integers with the cached data. """ cdef list answer = [] for v in v: answer.append(self._dict_vertices_to_integers[v]) return answer
cdef list _integers_to_vertices(self, integers): r""" Converts a list of integers to a list of vertices with the cached data. """
cdef _bitset_convex_hull(self, bitset_t hull): r""" Computes the convex hull of a list of vertices given as a bitset.
(this method returns nothing and modifies the input) """ cdef int count cdef int tmp_count cdef int i,j
cdef bitset_t * p_bitset
# Current size of the set
# Iterating over all the elements in the cache
# For any vertex i
# If i is not in the current set, we skip it !
# If it is, we iterate over all the elements j
# If both i and j are inside, we add all the (cached) # vertices on a shortest ij-path
# Next bitset !
# If we added nothing new during the previous loop, our set is # convex !
# Otherwise, update and back to the loop
cpdef hull(self, list vertices): r""" Returns the convex hull of a set of vertices.
INPUT:
* ``vertices`` -- A list of vertices.
EXAMPLES::
sage: from sage.graphs.convexity_properties import ConvexityProperties sage: g = graphs.PetersenGraph() sage: CP = ConvexityProperties(g) sage: CP.hull([1,3]) [1, 2, 3] """ cdef bitset_t bs
#cdef list answer = bitset_list(bs)
cdef _greedy_increase(self, bitset_t bs): r""" Given a bitset whose hull is not the whole set, greedily add vertices and stop before its hull is the whole set.
NOTE:
* Counting the bits at each turn is not the best way... """ cdef bitset_t tmp
cpdef hull_number(self, value_only = True, verbose = False): r""" Computes the hull number and a corresponding generating set.
The hull number `hn(G)` of a graph `G` is the cardinality of a smallest set of vertices `S` such that `h(S)=V(G)`.
INPUT:
* ``value_only`` (boolean) -- whether to return only the hull number (default) or a minimum set whose convex hull is the whole graph.
* ``verbose`` (boolean) -- whether to display information on the LP.
**COMPLEXITY:**
This problem is NP-Hard [CHZ02]_, but seems to be of the "nice" kind. Update this comment if you fall on hard instances `:-)`
**ALGORITHM:**
This is solved by linear programming.
As the function `h(S)` associating to each set `S` its convex hull is a closure operator, it is clear that any set `S_G` of vertices such that `h(S_G)=V(G)` must satisfy `S_G \not \subseteq C` for any *proper* convex set `C \subsetneq V(G)`. The following formulation is hence correct
.. MATH::
\text{Minimize :}& \sum_{v\in G}b_v\\ \text{Such that :}&\\ &\forall C\subsetneq V(G)\text{ a proper convex set }\\ &\sum_{v\in V(G)\backslash C} b_v \geq 1
Of course, the number of convex sets -- and so the number of constraints -- can be huge, and hard to enumerate, so at first an incomplete formulation is solved (it is missing some constraints). If the answer returned by the LP solver is a set `S` generating the whole graph, then it is optimal and so is returned. Otherwise, the constraint corresponding to the set `h(S)` can be added to the LP, which makes the answer `S` infeasible, and another solution computed.
This being said, simply adding the constraint corresponding to `h(S)` is a bit slow, as these sets can be large (and the corresponding constraint a bit weak). To improve it a bit, before being added, the set `h(S)` is "greedily enriched" to a set `S'` with vertices for as long as `h(S')\neq V(G)`. This way, we obtain a set `S'` with `h(S)\subseteq h(S')\subsetneq V(G)`, and the constraint corresponding to `h(S')` -- which is stronger than the one corresponding to `h(S)` -- is added.
This can actually be seen as a hitting set problem on the complement of convex sets.
EXAMPLES:
The Hull number of Petersen's graph::
sage: from sage.graphs.convexity_properties import ConvexityProperties sage: g = graphs.PetersenGraph() sage: CP = ConvexityProperties(g) sage: CP.hull_number() 3 sage: generating_set = CP.hull_number(value_only = False) sage: CP.hull(generating_set) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
REFERENCE:
.. [CHZ02] \F. Harary, E. Loukakis, C. Tsouros The geodetic number of a graph Mathematical and computer modelling vol. 17 n11 pp.89--95, 1993 """
cdef int i cdef list constraint # temporary variable to add constraints to the LP
if value_only: return self._n else: return self._list_integers_to_vertices
# Minimization
# We have exactly n binary variables, all of them with a coefficient of # 1 in the objective function
# We know that at least 2 vertices are required to cover the whole graph
# The set of vertices generated by the current LP solution cdef bitset_t current_hull
# Which is at first empty
# Greedily increase it to obtain a better constraint
print("Adding a constraint corresponding to convex set ", end="") print(bitset_list(current_hull))
# Building the corresponding constraint
# Computing the current solution's convex hull
# Are we done ?
|