def bruteForce(V, E):
'''
Brute force search through all k=(n-1)-subsets in (E)
'''
validTrees = []
k = len(V) - 1
m = len(E)
numSubsets = combination(m, k)
for i in range(numSubsets):
subset = colexUnrank(i, k, m)
edgeSubset = []
for j in subset:
edgeSubset.append(E[j - 1])
isTree = checkIfTree(V, edgeSubset)
if isTree:
validTrees.append(edgeSubset)
return validTrees
def colexRank(T, k):
'''ranks colex ordering
Algorithm 2.9 in book'''
r = 0
for i in range(1, k + 1):
r += combination(T[i - 1] - 1, k + 1 - i)
return r
def stirling2(m, n):
s = 0
for j in range(1, n + 1):
s += (-1)**(n - j) * combination(n, j) * (j**m)
return s / factorial(n)
def colexUnrank(r, k, n):
'''unranks colex ordering
Algorithm 2.10 in book'''
T = [0] * k
x = n
for i in range(1, k + 1):
c = combination(x, k + 1 - i)
while c > r:
x -= 1
c = combination(x, k + 1 - i)
T[i - 1] = x + 1
r -= combination(x, k + 1 - i)
return T
def lexUnrank(r, k, n):
if k == 1:
return [r]
if r == 1:
return [1] * k
if n == 1:
return [1]
if r > combination(n + k - 1, k):
return 'undefined'
else:
c = 0
for i in range(k):
c += combination(n + k - (i + 2), n - (i + 2))
if r <= c:
return [i + 1] + unrank(r - k, k - 1, n)
def lexRank(T, k, n):
#base cases
if k == 1:
return T[0]
elif n == 1:
return 1
else:
if T[0] != 1:
T = [x - 1 for x in T]
return combination(n + k - 2, k - 1) + rank(T, k, n - 1)
else:
return rank(T[1:], k - 1, n)
def selectPartition(n):
'''
Selects a random partition of a graph on 2*(n) vertices
returns two parts X0, X1
Algorithm 5.7 in book
'''
#selects random partition to be X0
r = np.random.randint(0, combination(2 * n, n))
X0 = lexUnrank(2 * n, r)
#X1 is the complement of X0
X1 = np.array(range(2 * n))
X1 = np.delete(X1, X0)
return ([X0, X1])