/// <summary>
/// Generates all topologies defined on a given set (where set.Count < 6) in O(2^(2^set.Count -2)).
/// </summary>
/// <returns>Set of all topologies that defined on <paramref name="set"/>.</returns>
public static IEnumerable <HashSet <HashSet <T> > > Topologies <T>(
HashSet <T> set, IProgress <double> progress, CancellationToken ct)
{
if (set == null)
{
throw new ArgumentNullException(nameof(set));
}
// if > 6 will case overflow in the long type.
if (set.Count > 5)
{
throw
new Exception("Set elements must be less than 6 elements.");
}
progress.Report(0);
var powerSet = SetUtl.PowerSet(set);
// remove the set and the empty set.
powerSet.RemoveWhere(s => s.Count == 0); // O(2^set.Count)
powerSet.RemoveWhere(s => s.Count == set.Count); // O(2^set.Count)
var n = 1L << powerSet.Count;
// loop to get all n subsets
for (long i = 0; i < n; i++)
{
ct.ThrowIfCancellationRequested();
if (i % 100 == 0)
{
var x = i / (decimal)n;
var p = 100 * x;
progress.Report((double)p);
}
var subset = new HashSet <HashSet <T> >();
// loop though every element in the set and determine with number
// should present in the current subset.
var j = 0;
foreach (var e in powerSet)
{
// if the jth element (bit) in the ith subset (binary number of i) add it.
if (((1L << j++) & i) > 0)
{
subset.Add(e);
}
}
subset.Add(new HashSet <T>());
subset.Add(set);
if (IsTopology(subset, set))
{
yield return(subset);
}
}
progress.Report(0);
}
/// <summary>
/// Generates all topologies defined on a given set (where set.Count < 6) in O(2^(2^set.Count -2)).
/// </summary>
/// Example:
/// - Input: S = {'c', 'b', 'a'} // See unit test.
/// Output Pattern: any pattern also have 000 and 111
/// Single And Double
/// 001
/// 010
/// 011
/// 100
/// 101
/// 110
///
/// Single-double (disjoint)
/// 001 110
/// 010 101
/// 011 100
///
/// Single-double
/// 001 011
/// 001 101
/// 010 011
/// 010 110
/// 100 101
/// 100 110
///
/// Single-single-double
/// 001 010 011
/// 001 100 101
/// 010 100 110
///
/// Single-double-double
/// 001 011 101
/// 010 011 110
/// 100 101 110
///
/// single-single-double-double
/// 001 010 011 101
/// 001 010 011 110
/// 001 011 100 101
/// 001 100 101 110
/// 010 011 100 110
/// 010 100 101 110
///
///
/// Power set
/// 001 010 011 100 101 110
/// - From this pattern the elements of the power set is exist or not
/// so by using a brute force tests we can get all topologies.
/// - Fact:
/// Let T(n) denote the number of distinct topologies on a set with n points.
/// There is no known simple formula to compute T(n) for arbitrary n.
/// - Theorem:
/// The number of subsets of size r (or r-combinations) that can be chosen
/// from a set of n elements, is given by the formula: nCr = !n / r!(n - r)!
/// - Theorem:
/// The number of subsets of all size is 2^n
/// <typeparam name="T">Set elements type.</typeparam>
/// <param name="set">The set that a topologies define.</param>
/// <returns>Set of all topologies that defined on <paramref name="set"/>.</returns>
public static IEnumerable <HashSet <HashSet <T> > > Topologies <T>(HashSet <T> set)
{
if (set == null)
{
throw new ArgumentNullException(nameof(set));
}
// if > 6 will case overflow in the long type.
if (set.Count > 5)
{
throw new Exception("Set elements must be less than 6 elements.");
}
var powerSet = SetUtl.PowerSet(set);
// remove the set and the empty set. for example, for set of 4 element this
// make the complexity decrease from 2^(2^4)= 65,536 to 2^(2^4-2)= 16,384
powerSet.RemoveWhere(s => s.Count == 0); // O(2^set.Count)
powerSet.RemoveWhere(s => s.Count == set.Count); // O(2^set.Count)
var n = 1L << powerSet.Count;
// loop to get all n subsets
for (long i = 0; i < n; i++)
{
var subset = new HashSet <HashSet <T> >();
// loop though every element in the set and determine with number
// should present in the current subset.
var j = 0;
foreach (var e in powerSet)
{
// if the jth element (bit) in the ith subset (binary number of i) add it.
if (((1L << j++) & i) > 0)
{
subset.Add(e);
}
}
subset.Add(new HashSet <T>());
subset.Add(set);
if (IsTopology(subset, set))
{
yield return(subset);
}
}
}
/// <summary>
/// Find the neighborhood for point in the <paramref name="set"/> for a given topology.
/// </summary>
/// Definition: Neighborhood of a point:
/// If X is a topological space and p ∈ X, a neighborhood of p is a subset N of X
/// that includes an open set O containing p, p∈O⊆N.
/// Definition: The collection of all neighborhoods of a point is called the
/// neighborhood system at the point.
public static HashSet <HashSet <T> > NeighbourhoodSystem <T>(
HashSet <T> set, HashSet <HashSet <T> > topology, T point)
{
if (!IsTopology(topology, set))
{
throw new Exception(
"The given topology is not a valid topology on the set.");
}
if (!set.Contains(point))
{
throw new Exception("The set do not contain the point!");
}
var powerSet = SetUtl.PowerSet(set);
// The open sets that contain the point
var enumerable = topology.Where(s => s.Contains(point));
var openSets = enumerable as HashSet <T>[] ?? enumerable.ToArray();
// The smallest open set that contain the point
// Assume that the first element
var o = openSets.First();
foreach (var openSet in openSets)
{
o.IntersectWith(openSet);
}
var neighbourhood = new HashSet <HashSet <T> >();
foreach (var s in powerSet.Where(s => o.IsSubsetOf(s)))
{
neighbourhood.Add(s);
}
return(neighbourhood);
}
