def match(G, X, Y): # Maximum bipartite matching
H = tr(G) # The transposed graph
S, T, M = set(X), set(Y), set() # Unmatched left/right + match
while S: # Still unmatched on the left?
s = S.pop() # Get one
Q, P = {s}, {} # Start a traversal from it
while Q: # Discovered, unvisited
u = Q.pop() # Visit one
if u in T: # Finished augmenting path?
T.remove(u) # u is now matched
break # and our traversal is done
forw = (v for v in G[u] if (u, v) not in M) # Possible new edges
back = (v for v in H[u] if (v, u) in M) # Cancellations
for v in chain(forw, back): # Along out- and in-edges
if v in P:
continue # Already visited? Ignore
P[v] = u # Traversal predecessor
Q.add(v) # New node discovered
while u != s: # Augment: Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: # Forward edge?
M.add((u, v)) # New edge
else: # Backward edge?
M.remove((v, u)) # Cancellation
return M # Matching -- a set of edges
def ford_fulkerson(G, s, t, aug=bfs_aug): # Max flow from s to t
H, f = tr(G), defaultdict(int) # Transpose and flow
while True: # While we can improve things
P, c = aug(G, H, s, t, f) # Aug. path and capacity/slack
if c == 0: return f # No augm. path found? Done!
u = t # Start augmentation
while u != s: # Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: f[u,v] += c # Forward edge? Add slack
else: f[v,u] -= c # Backward edge? Cancel slack
def ford_fulkerson(G, s, t, aug=bfs_aug): # Max flow from s to t
H, f = tr(G), defaultdict(int) # Transpose and flow
while True: # While we can improve things
P, c = aug(G, H, s, t, f) # Aug. path and capacity/slack
if c == 0: return f # No augm. path found? Done!
u = t # Start augmentation
while u != s: # Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: f[u,v] += c # Forward edge? Add slack
else: f[v,u] -= c # Backward edge? Cancel slack
def paths(G, s, t): # Edge-disjoint path count
H, M, count = tr(G), set(), 0 # Transpose, matching, result
while True: # Until the function returns
Q, P = {s}, {} # Traversal queue + tree
while Q: # Discovered, unvisited
u = Q.pop() # Get one
if u == t: # Augmenting path!
count += 1 # That means one more path
break # End the traversal
forw = (v for v in G[u] if (u,v) not in M) # Possible new edges
back = (v for v in H[u] if (v,u) in M) # Cancellations
for v in chain(forw, back): # Along out- and in-edges
if v in P: continue # Already visited? Ignore
P[v] = u # Traversal predecessor
Q.add(v) # New node discovered
else: # Didn't reach t?
return count # We're done
while u != s: # Augment: Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: # Forward edge?
M.add((u,v)) # New edge
else: # Backward edge?
M.remove((v,u)) # Cancellation
def paths(G, s, t): # Edge-disjoint path count
H, M, count = tr(G), set(), 0 # Transpose, matching, result
while True: # Until the function returns
Q, P = {s}, {} # Traversal queue + tree
while Q: # Discovered, unvisited
u = Q.pop() # Get one
if u == t: # Augmenting path!
count += 1 # That means one more path
break # End the traversal
forw = (v for v in G[u] if (u,v) not in M) # Possible new edges
back = (v for v in H[u] if (v,u) in M) # Cancellations
for v in chain(forw, back): # Along out- and in-edges
if v in P: continue # Already visited? Ignore
P[v] = u # Traversal predecessor
Q.add(v) # New node discovered
else: # Didn't reach t?
return count # We're done
while u != s: # Augment: Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: # Forward edge?
M.add((u,v)) # New edge
else: # Backward edge?
M.remove((v,u)) # Cancellation
def match(G, X, Y): # Maximum bipartite matching
H = tr(G) # The transposed graph
S, T, M = set(X), set(Y), set() # Unmatched left/right + match
while S: # Still unmatched on the left?
s = S.pop() # Get one
Q, P = {s}, {} # Start a traversal from it
while Q: # Discovered, unvisited
u = Q.pop() # Visit one
if u in T: # Finished augmenting path?
T.remove(u) # u is now matched
break # and our traversal is done
forw = (v for v in G[u] if (u,v) not in M) # Possible new edges
back = (v for v in H[u] if (v,u) in M) # Cancellations
for v in chain(forw, back): # Along out- and in-edges
if v in P: continue # Already visited? Ignore
P[v] = u # Traversal predecessor
Q.add(v) # New node discovered
while u != s: # Augment: Backtrack to s
u, v = P[u], u # Shift one step
if v in G[u]: # Forward edge?
M.add((u,v)) # New edge
else: # Backward edge?
M.remove((v,u)) # Cancellation
return M # Matching -- a set of edges
