def difference_constraints_with_arbitrary_weight(A, b):
''' An variant to the above difference constraints function
that the weight of the edge from the auxiliary vertex to any
other vertex can be arbitrary value.
'''
row = len(A)
col = len(A[0])
vertices_num = col
vertices = []
edges = []
weights = dict()
distribute = sample(xrange(-10, 10), vertices_num)
for i in range(0, vertices_num + 1):
vertices.append(Vertex(i))
for i in range(1, vertices_num + 1):
edges.append((vertices[0], vertices[i]))
weights[(vertices[0], vertices[i])] = distribute[i - 1]
for i in range(0, row):
u = A[i].index(-1) + 1
v = A[i].index(1) + 1
edges.append((vertices[u], vertices[v]))
weights[(vertices[u], vertices[v])] = b[i]
G = Graph(vertices, edges)
if G.Bellman_Ford(lambda x, y: weights[(x, y)], vertices[0]):
return [v.d for v in vertices[1:]]
else:
return None
def difference_constraints(A, b):
'''A algorithm to solve a system of difference constraints Ax <= b
by solving a single-source shortest-paths problem using Bellman-Ford
algorithm. Each row of the linear-programming matrix A contains
one 1 and one 1, and all other entries of A are 0.
Calling convention: A is a two-dimensional list, b is a list.
Return value: This function returns a list representing x if
there exists feasible solution; otherwise, this function returns None.
'''
row = len(A)
col = len(A[0])
vertices_num = col
vertices = []
edges = []
weights = dict()
for i in range(0, vertices_num + 1):
vertices.append(Vertex(i))
for i in range(1, vertices_num + 1):
edges.append((vertices[0], vertices[i]))
weights[(vertices[0], vertices[i])] = 0
for i in range(0, row):
u = A[i].index(-1) + 1
v = A[i].index(1) + 1
edges.append((vertices[u], vertices[v]))
weights[(vertices[u], vertices[v])] = b[i]
G = Graph(vertices, edges)
if G.Bellman_Ford(lambda x, y: weights[(x, y)], vertices[0]):
return [v.d for v in vertices[1:]]
else:
return None
def testBellmanFord(self):
s = Vertex('s')
t = Vertex('t')
y = Vertex('y')
x = Vertex('x')
z = Vertex('z')
vertices = [s, t, y, x, z]
edges = [(s, t), (s, y), (t, y), (t, x), (t, z), (y, x), (y, z), (x, t), (z, s), (z, x)]
weight = [6, 7, 8, 5, -4, -3, 9, -2, 2, 7]
G = Graph(vertices, edges)
we = dict()
for x,y in zip(edges, weight):
we[x] = y
def w(x, y):
return we[(x, y)]
G.Bellman_Ford(w, z)
def equality_constraints(A, b):
row = len(A)
col = len(A[0])
vertices_num = col
vertices = []
edges = []
weights = dict()
for i in range(0, vertices_num + 1):
vertices.append(Vertex(i))
for i in range(1, vertices_num + 1):
edges.append((vertices[0], vertices[i]))
weights[(vertices[0], vertices[i])] = 0
for i in range(0, row):
u = A[i].index(-1) + 1
v = A[i].index(1) + 1
edges.append((vertices[u], vertices[v]))
weights[(vertices[u], vertices[v])] = b[i]
edges.append((vertices[v], vertices[u]))
weights[(vertices[v], vertices[u])] = -1 * b[i]
G = Graph(vertices, edges)
if G.Bellman_Ford(lambda x, y: weights[(x, y)], vertices[0]):
return [v.d for v in vertices[1:]]
else:
return None
def single_variable_constraints(A, b):
row = len(A)
col = len(A[0])
vertices_num = col
vertices = []
edges = []
weights = dict()
for i in range(0, vertices_num + 1):
vertices.append(Vertex(i))
for i in range(0, row):
for j in range(0, len(A[i])):
if A[i][j] == 1:
edges.append((vertices[0], vertices[j + 1]))
weights[(vertices[0], vertices[j + 1])] = b[i]
break
elif A[i][j] == -1:
edges.append((vertices[j + 1], vertices[0]))
weights[(vertices[j + 1], vertices[0])] = b[i]
break
G = Graph(vertices, edges)
if G.Bellman_Ford(lambda x, y: weights[(x, y)], vertices[0]):
return [v.d for v in vertices[1:]]
else:
return None
