def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
u = rep2vec(vec2rep(U_basis, w), U_basis)
v = rep2vec(vec2rep(V_basis, w), V_basis)
return (u,v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
u = rep2vec(vec2rep(U_basis, w), U_basis)
v = rep2vec(vec2rep(V_basis, w), V_basis)
return (u, v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
U_V_basis = list(U_basis)
U_V_basis.extend(V_basis)
w_coordinates = vec2rep(U_V_basis, w)
w_u_coord = list()
w_v_coord = list()
for n in range(len(U_V_basis)):
if n < len(U_basis):
w_u_coord.append(w_coordinates[n])
else:
w_v_coord.append(w_coordinates[n])
u = rep2vec(list2vec(w_u_coord), U_basis)
v = rep2vec(list2vec(w_v_coord), V_basis)
return (u, v)
def morph(S, B):
"""
Input:
- S: a list of distinct Vecs
- B: a list of linearly independent Vecs all in Span S
Output: a list of pairs of vectors to inject and eject (see problem description)
Example:
>>> # This is how our morph works. Yours may yield different results.
>>> # Note: Make a copy of S to modify instead of modifying S itself.
>>> from vecutil import list2vec
>>> from vec import Vec
>>> S = [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
>>> B = [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
>>> D = {0, 1, 2}
>>> morph(S, B) == [(Vec(D,{0: 1, 1: 1, 2: 0}), Vec(D,{0: 1, 1: 0, 2: 0})), (Vec(D,{0: 0, 1: 1, 2: 1}), Vec(D,{0: 0, 1: 1, 2: 0})), (Vec(D,{0: 1, 1: 0, 2: 1}), Vec(D,{0: 0, 1: 0, 2: 1}))]
True
>>> S == [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
True
>>> B == [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
True
>>> from GF2 import one
>>> D = {0, 1, 2, 3, 4, 5, 6, 7}
>>> S = [Vec(D,{1: one, 2: one, 3: one, 4: one}), Vec(D,{1: one, 3: one}), Vec(D,{0: one, 1: one, 3: one, 5: one, 6: one}), Vec(D,{3: one, 4: one}), Vec(D,{3: one, 5: one, 6: one})]
>>> B = [Vec(D,{2: one, 4: one}), Vec(D,{0: one, 1: one, 2: one, 3: one, 4: one, 5: one, 6: one}), Vec(D,{0: one, 1: one, 2: one, 5: one, 6: one})]
>>> sol = morph(S, B)
>>> sol == [(B[0],S[0]), (B[1],S[2]), (B[2],S[3])] or sol == [(B[0],S[1]), (B[1],S[2]), (B[2],S[3])]
True
>>> # Should work the same regardless of order of S
>>> from random import random
>>> sol = morph(sorted(S, key=lambda x:random()), B)
>>> sol == [(B[0],S[0]), (B[1],S[2]), (B[2],S[3])] or sol == [(B[0],S[1]), (B[1],S[2]), (B[2],S[3])]
True
"""
tmp_S = S[:]
tmp_B = B[:]
A = []
res = []
for i in range(len(tmp_B)):
z = tmp_B[i]
idxes = vec2rep(tmp_S, z)
idxes = [idxes[k] for k in idxes.D]
for i in range(len(idxes)):
if idxes[i] != 0 and tmp_S[i] not in A:
w = tmp_S[i]
break
tmp_S.remove(w)
tmp_S.append(z)
A.append(z)
res.append((z, w))
return res
def morph(S, B):
'''
Input:
- S: a list of distinct Vecs
- B: a list of linearly independent Vecs all in Span S
Output: a list of pairs of vectors to inject and eject (see problem description)
Example:
>>> # This is how our morph works. Yours may yield different results.
>>> # Note: Make a copy of S to modify instead of modifying S itself.
>>> from vecutil import list2vec
>>> from vec import Vec
>>> S = [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
>>> B = [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
>>> D = {0, 1, 2}
>>> morph(S, B) == [(Vec(D,{0: 1, 1: 1, 2: 0}), Vec(D,{0: 1, 1: 0, 2: 0})), (Vec(D,{0: 0, 1: 1, 2: 1}), Vec(D,{0: 0, 1: 1, 2: 0})), (Vec(D,{0: 1, 1: 0, 2: 1}), Vec(D,{0: 0, 1: 0, 2: 1}))]
True
>>> S == [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
True
>>> B == [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
True
>>> from GF2 import one
>>> D = {0, 1, 2, 3, 4, 5, 6, 7}
>>> S = [Vec(D,{1: one, 2: one, 3: one, 4: one}), Vec(D,{1: one, 3: one}), Vec(D,{0: one, 1: one, 3: one, 5: one, 6: one}), Vec(D,{3: one, 4: one}), Vec(D,{3: one, 5: one, 6: one})]
>>> B = [Vec(D,{2: one, 4: one}), Vec(D,{0: one, 1: one, 2: one, 3: one, 4: one, 5: one, 6: one}), Vec(D,{0: one, 1: one, 2: one, 5: one, 6: one})]
>>> sol = morph(S, B)
>>> sol == [(B[0],S[0]), (B[1],S[2]), (B[2],S[3])] or sol == [(B[0],S[1]), (B[1],S[2]), (B[2],S[3])]
True
>>> # Should work the same regardless of order of S
>>> from random import random
>>> sol = morph(sorted(S, key=lambda x:random()), B)
>>> sol == [(B[0],S[0]), (B[1],S[2]), (B[2],S[3])] or sol == [(B[0],S[1]), (B[1],S[2]), (B[2],S[3])]
True
'''
tmp_S = S[:]
tmp_B = B[:]
A = []
res = []
for i in range(len(tmp_B)):
z = tmp_B[i]
idxes = vec2rep(tmp_S, z)
idxes = [idxes[k] for k in idxes.D]
for i in range(len(idxes)):
if idxes[i] != 0 and tmp_S[i] not in A:
w = tmp_S[i]
break
tmp_S.remove(w)
tmp_S.append(z)
A.append(z)
res.append((z, w))
return res
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
direct_sum_basis = U_basis + V_basis
x = vec2rep(direct_sum_basis, w) # len(x) = len(direct_sum_basis)
u = Vec(U_basis[0].D, {})
v = Vec(V_basis[0].D, {})
for i in range(len(U_basis)):
u = u + x[i] * U_basis[i]
for j in range(len(V_basis)):
v = v + x[len(U_basis) + j] * V_basis[j]
return (u, v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
direct_sum_basis = U_basis+V_basis
x = vec2rep(direct_sum_basis, w) # len(x) = len(direct_sum_basis)
u = Vec(U_basis[0].D, {}); v = Vec(V_basis[0].D, {})
for i in range(len(U_basis)):
u = u+x[i]*U_basis[i]
for j in range(len(V_basis)):
v = v+x[len(U_basis)+j]*V_basis[j]
return (u, v)
def exchange(S, A, z):
u = vec2rep(S, z)
for i, v in enumerate(S):
coef = u[i]
if v not in A and coef != 0:
return v