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
'''
S0 = S.copy()
A = set([])
pairs = []
for z in B:
w = exchange(set(S0), A, z)
# S0 = (S0 | {z}) - {w}
S0.append(z)
S0.remove(w)
# A = A | {z}
A = A | {z}
pairs.append((z,w))
return pairs
