def test_feasible( csdfg, bindings, target_mcr ):
# compute pessimistic approximation
pess = transform.single_rate_apx( csdfg, bindings = bindings )
# set path weights
for u, v, data in pess.edges_iter( data = True ):
tokens = data.get( __PARAM_TOKENS__, 0 )
weight = pess.node[ v ].get( __PARAM_WCET__ )
data['weight'] = tokens * target_mcr - weight
# compute longest paths
try:
bfct.find_shortest_paths( pess, None )
return None
except NegativeCycleException as ex:
return ex.cycle
def compute_vertex_keys(graph, variable, q, parameter):
root = next(graph.nodes_iter())
# construct shortest paths tree using token = parameter
for v, w, data in graph.edges_iter(data=True):
tokens = data.get(__param__tokens, 0)
weight = data.get(__param__weight, 0)
data['dist'] = pdistance(weight, tokens)
data['w'] = tokens * parameter - weight
distances = dict()
try:
tree, _, _ = bfct.find_shortest_paths(graph, root, 'w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs(root):
if i == 0:
dv = distances[v]
for w in tree.children(v):
distances[w] = dv + graph.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
raise InfeasibleException(ex.cycle)
# compute vertex keys
for v in graph:
ratio, edge = compute_vertex_key(graph, v, distances)
q[(variable, v)] = (-ratio, edge)
# clean up 'dist' and 'w' parameters
for v, w, data in graph.edges_iter(data=True):
del data['dist']
del data['w']
# return shortest paths tree that is valid for the parameter
return tree
def compute_mcm(g, estimate=None, pweight='weight'):
maxmean, arg_cycle = None, None
for scc in nx.strongly_connected_component_subgraphs(g):
root = next(scc.nodes_iter())
scc_mcm, cycle = compute_mcm_component(scc, root, estimate, pweight)
if scc_mcm is None:
continue
if maxmean is None or scc_mcm > maxmean:
maxmean = scc_mcm
arg_cycle = cycle
forest = Forest()
for scc in nx.strongly_connected_component_subgraphs(g, False):
if scc.number_of_edges() == 0:
continue
for (v, w, scc_data) in scc.edges_iter(data=True):
data = g.get_edge_data(v, w)
# negate weight so that we can construct a longest paths tree for the current solution
scc_data['w'] = maxmean - data.get('weight', 0)
root = w
lpp_tree, _ = bfct.find_shortest_paths(scc, root, arg='w')
forest.add_forest(lpp_tree)
return maxmean, arg_cycle, forest
def test_feasible(csdfg, bindings, target_mcr):
# compute pessimistic approximation
pess = transform.single_rate_apx(csdfg, bindings=bindings)
# set path weights
for u, v, data in pess.edges_iter(data=True):
tokens = data.get(__PARAM_TOKENS__, 0)
weight = pess.node[v].get(__PARAM_WCET__)
data['weight'] = tokens * target_mcr - weight
# compute longest paths
try:
bfct.find_shortest_paths(pess, None)
return None
except NegativeCycleException as ex:
return ex.cycle
def compute_mcm( g, estimate = None, pweight = 'weight' ):
maxmean, arg_cycle = None, None
for scc in nx.strongly_connected_component_subgraphs( g ):
root = next( scc.nodes_iter() )
scc_mcm, cycle = compute_mcm_component( scc, root, estimate, pweight )
if scc_mcm is None:
continue
if maxmean is None or scc_mcm > maxmean:
maxmean = scc_mcm
arg_cycle = cycle
forest = Forest()
for scc in nx.strongly_connected_component_subgraphs( g, False ):
if scc.number_of_edges() == 0:
continue
for ( v, w, scc_data ) in scc.edges_iter( data = True ):
data = g.get_edge_data( v, w )
# negate weight so that we can construct a longest paths tree for the current solution
scc_data['w'] = maxmean - data.get( 'weight', 0 )
root = w
lpp_tree, _ = bfct.find_shortest_paths( scc, root, arg = 'w' )
forest.add_forest( lpp_tree )
return maxmean, arg_cycle, forest
def compute_vertex_keys( graph, variable, q, parameter ):
root = next( graph.nodes_iter() )
# construct shortest paths tree using token = parameter
for v, w, data in graph.edges_iter(data=True):
tokens = data.get(__param__tokens, 0)
weight = data.get(__param__weight, 0)
data['dist'] = pdistance( weight, tokens )
data['w'] = tokens * parameter - weight
distances = dict()
try:
tree, _, _ = bfct.find_shortest_paths(graph, root, 'w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs( root ):
if i == 0:
dv = distances[ v ]
for w in tree.children( v ):
distances[ w ] = dv + graph.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
raise InfeasibleException( ex.cycle )
# compute vertex keys
for v in graph:
ratio, edge = compute_vertex_key( graph, v, distances )
q[(variable, v)] = ( -ratio, edge )
# clean up 'dist' and 'w' parameters
for v, w, data in graph.edges_iter( data = True ):
del data['dist']
del data['w']
# return shortest paths tree that is valid for the parameter
return tree
def path_weight( graph, start, end ):
# compute shortest paths
_, dists, _ = bfct.find_shortest_paths( graph, start )
return dists.get( end, None )
def ensure_liveness(g, factor = 100):
""" Assigns tokens to channels of an SDF graph such that a minimum throughput is attained.
Works by finding cycles of positive weight in the graph's pessimistic approximation
and resolving them by adding sufficient tokens.
Parameters:
----------
g: the SDF graph
factor: indicates the maximum number of time units a single graph iteration may take,
multiplied by the length of a fully sequential iteration.
Lower numbers result in higher throughputs
"""
assert g.is_consistent()
# assert nx.is_strongly_connected( g )
seqperiod = 0
for v, data in g.nodes_iter( data = True ):
wcet = data.get('wcet').sum()
# periods = g.q[v] // data.get('phases')
periods = 1
seqperiod += ( wcet * periods )
target_mcr = Fraction( seqperiod, g.tpi ) / factor
added_tokens = 0
m = 4 * g.number_of_edges()
while m > 0:
m = m - 1
# transform the graph to its single-rate approximation
pess = transform.single_rate_apx( g )
# create graph where each token represents a weight of -factor
lpg = nx.DiGraph()
for u, v, data in pess.edges( data = True ):
wcet = pess.node[v].get('wcet')
tokens = data.get('tokens', 0)
lpg.add_edge( u, v, weight = tokens * target_mcr - wcet )
# find positive cycle
if m == 0:
import pdb; pdb.set_trace()
try:
bfct.find_shortest_paths( lpg, u )
break
except NegativeCycleException as ex:
# find edge with heaviest weight
weight, tokens = 0, 0
max_weight = 0
for u, v in ex.cycle:
weight += pess.node[ v ].get('wcet')
tokens += pess.get_edge_data(u, v).get('tokens', 0)
suv = g.s[ (u, v) ]
if suv > max_weight:
max_weight = suv
arg_max = (u, v)
delta = Fraction( Fraction(weight, target_mcr) - tokens, max_weight ).__ceil__()
edge_data = g.get_edge_data( *arg_max )
edge_data['tokens'] = edge_data.get('tokens', 0) + delta #math.ceil(extra_factor * delta)
added_tokens += delta
# print("Added {} tokens".format( delta ))
else:
assert False, "too many iterations"
return added_tokens
def test_random(n=50, p=0.1, runs=1000, debug=None):
for run in range(runs) if debug is None else [debug]:
g = fast_gnp_random_graph(n, p, seed=run + 1, directed=True)
# add source connected to all nodes
source = 100 * (n // 100) + 200
for v in list(g.nodes()):
g.add_edge(source, v, weight=0, tokens=0)
# add random weights and tokens
wsum = 1
for _, _, data in g.edges_iter(data=True):
data['weight'] = randrange(1, 10)
data['tokens'] = randrange(-1, 8)
wsum += data['weight']
# create shortest path formulation for initial tree
for _, _, data in g.edges_iter(data=True):
data['sp'] = data['tokens'] * wsum - data['weight']
# ensure that the graph admits a feasible solution
its = 0
while True:
try:
its += 1
tree, distances = bfct.find_shortest_paths(g, source, arg='sp')
break
except NegativeCycleException as ex:
toks = sum(
map(lambda vw: g.get_edge_data(*vw).get('tokens'),
ex.cycle))
edge_data = g.get_edge_data(*choice(ex.cycle))
edge_data['tokens'] += (1 - toks)
edge_data[
'sp'] = edge_data['tokens'] * wsum - edge_data['weight']
if debug is not None:
import pdb
pdb.set_trace()
negative_toks = False
for _, _, data in g.edges_iter(data=True):
if data['tokens'] < 0:
negative_toks = True
break
print("Run {}: negative tokens: {}, iterations: {}".format(
run, negative_toks, its))
ratio, cycle = compute_mcr(g, source)
assert ratio is not None, "Deadlocked cycle found"
if not cycle:
# verify that the graph is acyclic
assert is_directed_acyclic_graph(
g), "[run = {}] Graph is not acyclic".format(run)
else:
wsum, tsum = 0, 0
for v, w in cycle:
data = g.get_edge_data(v, w)
wsum += data['weight']
tsum += data['tokens']
assert Fraction(
wsum, tsum
) == ratio, "[run = {}] computed MCR {} does not match ratio of critical cycle {}".format(
run, ratio, Fraction(wsum, tsum))
for v, w, data in g.edges_iter(data=True):
data['weight'] = data['tokens'] * ratio - data['weight']
try:
bellman_ford(g, source)
except Exception:
print("Exception during run {}".format(run))
def compute_mcm_component(g, root, estimate=None, pweight='weight'):
""" Computes the maximum cycle mean of g.
NOTES:
- The weight on each edge must be non-negative
- The number of tokens on each edge must be non-negative.
- The graph is assumed to be strongly connected.
"""
# initialize:
distances = {}
queue = pq.priority_queue()
# determine lower bound for mcr
init_mcr = 1
for (v, w, data) in g.edges_iter(data=True):
weight = data.get(pweight, 0)
init_mcr = init_mcr + max(0, weight)
data['dist'] = pdistance(weight, 1)
if estimate is not None:
# compute initial tree by computing shortest paths tree
# print("Computing initial tree from estimate {}".format(estimate))
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = estimate - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg='w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs(root):
if i == 0:
dv = distances[v]
for w in tree.children(v):
distances[w] = dv + g.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(estimate, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException(ex.cycle)
else:
# create tree rooted at root
tree = DFSTree(root)
# run Dijkstra on tokens
distances[root] = 0
queue[root] = 0
while len(queue) > 0:
v, _ = queue.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dw = distances.get(w, None)
if (dw is None) or dv + 1 < dw:
distances[w] = dv + 1
queue[w] = distances[w]
tree.append_child(v, w)
# DFS on shortest paths DAG
# (this is the DAG induced by the union of all shortest path trees)
pre, post, index = dict(), dict(), 0
post_order = list()
for v, i in tree.dfs(root):
if i == 0:
pre[v] = index
index += 1
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dw = distances[w]
if dv + 1 == dw and w not in pre:
tree.append_child(v, w)
else:
post[v] = index
index += 1
post_order.append(v)
# scan in topological order
distances.clear()
distances[root] = pdistance(0, 0)
while post_order:
v = post_order.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, None)
newdist = dv + data['dist']
if dw is None or newdist[1] < dw[1]:
distances[w] = newdist
tree.append_child(v, w)
elif (newdist[1] == dw[1] and newdist[0] - dw[0] > 0):
# can't have back edges
assert post[w] < post[v]
distances[w] = newdist
tree.append_child(v, w)
def compute_node_key(node):
maxratio, argmax = None, None
# go over all incoming edges of the node
for u, v, data in g.in_edges_iter(node, data=True):
if u in distances:
delta = distances[u] + data['dist'] - distances[v]
# print("Delta for {} = {}".format((u, v), delta))
if delta[1] > 0:
ratio = Fraction(delta[0], delta[1])
if argmax is None or ratio > maxratio:
maxratio = ratio
argmax = (u, v)
# store the node key for v
if argmax is not None:
queue[node] = (-maxratio, argmax)
elif node in queue:
del queue[node]
return maxratio
# fill priority queue:
# go over all nodes and compute their key
# print("Distances from root {}: {}".format(root, distances))
for v in distances:
compute_node_key(v)
# pivot until cycle is found
path_changes = 0
while len(queue) > 0:
(node, (ratio, (v, w))) = queue.pop()
delta = distances[v] + g.get_edge_data(v, w)['dist'] - distances[w]
for j in tree.pre_order(w):
distances[j] += delta
if v == j:
cycle = list()
if v != w:
child = j
for p in tree.ancestors(child):
cycle.append((p, j))
if p == w:
break
j = p
cycle.reverse()
cycle.append((v, w))
assert __is_cycle(
cycle
), "Found wrong cycle after pivoting with back edge ({}, {})".format(
v, w)
return -ratio, cycle
for (_, k, data) in g.out_edges_iter(j, True):
# update priority of (j, k)
ratio_k = None
if k in queue:
(ratio_k, edge) = queue[k]
delta_k = distances[j] + data['dist'] - distances[k]
if delta_k[1] > 0:
r = -Fraction(delta_k[0], delta_k[1])
if ratio_k is None or r < ratio_k:
queue[k] = (r, (j, k))
# recompute vertex key of j
compute_node_key(j)
tree.append_child(v, w)
else:
# no cycle found, any period is admissible
# Note that this implies that the graph is acyclic
return None, None
def compute_mcr_component(g, root, estimate=None):
""" Computes the maximum cycle ratio of g.
NOTES:
- The weight on each edge must be non-negative
- The number of tokens on each edge must be non-negative.
- The graph is assumed to be strongly connected.
"""
# initialize:
distances = {}
queue = pq.priority_queue()
# print("Computing MCR for graph {}".format( g.edges( data = True )))
# determine lower bound for mcr
# while doing so, determine vertices with outgoing tokenless
init_mcr = 1
negative_tokens_or_weights = False
for (v, w, data) in g.edges_iter(data=True):
tokens = data.get('tokens', 0)
weight = data.get('weight', 0)
init_mcr = init_mcr + max(0, weight)
data['dist'] = pdistance(weight, tokens)
if tokens < 0 or weight < 0:
negative_tokens_or_weights = True
if estimate is not None:
# compute initial tree by computing shortest paths tree
# print("Computing initial tree from estimate {}".format(estimate))
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = dist[1] * estimate - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg='w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs(root):
if i == 0:
dv = distances[v]
for w in tree.children(v):
distances[w] = dv + g.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(estimate, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException(ex.cycle)
elif True:
# print("Computing initial tree from estimate {}".format(init_mcr))
# compute initial tree by computing shortest paths tree
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = dist[1] * init_mcr - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg='w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs(root):
if i == 0:
dv = distances[v]
for w in tree.children(v):
distances[w] = dv + g.get_edge_data(v, w).get('dist')
if False:
# verify distances
for v in distances:
distv = distances[v]
distv = distv[0] - distv[1] * init_mcr
for _, w, data in g.out_edges_iter(v, True):
distw = distances[w]
distw = distw[0] - distw[1] * init_mcr
edge_dist = data['dist']
edge_dist = edge_dist[0] - edge_dist[1] * init_mcr
assert distv + edge_dist <= distw
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(init_mcr, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException(ex.cycle)
else:
# create tree rooted at root
tree = DFSTree(root)
# run Dijkstra on tokens
distances[root] = 0
queue[root] = 0
while len(queue) > 0:
v, _ = queue.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, dv + dist[1] + 1)
if dv + dist[1] - dw < 0:
distances[w] = dv + dist[1]
queue[w] = distances[w]
tree.append_child(v, w)
# DFS on shortest paths DAG
# (this is the DAG induced by the union of all shortest path trees)
pre, post, index = dict(), dict(), 0
post_order = list()
for v, i in tree.dfs(root):
if i == 0:
pre[v] = index
index += 1
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
tokens = data['dist'][1]
dw = distances[w]
if dv + tokens == dw and w not in pre:
tree.append_child(v, w)
else:
post[v] = index
index += 1
post_order.append(v)
# scan in topological order
distances.clear()
distances[root] = pdistance(0, 0)
while post_order:
v = post_order.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, None)
newdist = dv + data['dist']
if dw is None or newdist[1] < dw[1]:
distances[w] = newdist
tree.append_child(v, w)
elif (newdist[1] == dw[1] and newdist[0] - dw[0] > 0):
# check that (v, w) is not a back edge!
if post[w] >= post[v]:
# back edge, MCR not defined
prev = v
cycle = list()
if v != w:
for p in tree.ancestors(v):
cycle.append((p, prev))
if p == w: break
prev = p
cycle.reverse()
cycle.append((v, w))
assert __is_cycle(cycle), "Not a cycle: {}".format(
cycle)
# print(cycle)
# print("Positive cycle found: {}".format( cycle ))
raise InfeasibleException(cycle)
distances[w] = newdist
tree.append_child(v, w)
def compute_node_key(node):
maxratio, argmax = None, None
# go over all incoming edges of the node
for (u, v, data) in g.in_edges_iter(nbunch=[node], data=True):
if u in distances:
delta = distances[u] + data['dist'] - distances[v]
# print("Delta for {} = {}".format((u, v), delta))
if delta[1] > 0:
ratio = Fraction(delta[0], delta[1])
if argmax is None or ratio > maxratio:
maxratio = ratio
argmax = (u, v)
# store the node key for v
if argmax is not None:
queue[node] = (-maxratio, argmax)
# print("Vertex key for {} = {}, attained by edge {}".format(node, maxratio, argmax))
elif node in queue:
del queue[node]
return maxratio
# fill priority queue:
# go over all nodes and compute their key
# print("Distances from root {}: {}".format(root, distances))
for v in distances:
compute_node_key(v)
# pivot until cycle is found
path_changes = 0
pivots = 0
while len(queue) > 0:
(node, (ratio, (v, w))) = queue.pop()
pivots += 1
delta = distances[v] + g.get_edge_data(v, w)['dist'] - distances[w]
# print("Pivoting with edge ({}, {}), delta = {}".format(v, w, delta))
for j in tree.pre_order(w):
# update parametric distance to j
distances[j] += delta
path_changes += 1
if v == j:
cycle = list()
if v != w:
child = j
for p in tree.ancestors(child):
cycle.append((p, j))
if p == w:
break
j = p
cycle.reverse()
cycle.append((v, w))
assert __is_cycle(
cycle
), "Found wrong cycle after pivoting with back edge ({}, {})".format(
v, w)
return -ratio, cycle
for (_, k, data) in g.out_edges_iter(j, True):
# update priority of (j, k)
ratio_k = None
if k in queue:
(ratio_k, edge) = queue[k]
delta_k = distances[j] + data['dist'] - distances[k]
# commented out line below to allow a negative MCR
if delta_k[1] > 0:
r = -Fraction(delta_k[0], delta_k[1])
if ratio_k is None or r < ratio_k:
queue[k] = (r, (j, k))
# print("New arc key for ({}, {}) = {}, delta = {} + {} - {} = {}".format(j, k, -r, distances[j], data['dist'], distances[k], delta_k))
# recompute vertex key of j
compute_node_key(j)
tree.append_child(v, w)
else:
# no cycle found, any period is admissible
# Note that this implies that the graph is acyclic
return None, None
def path_weight(graph, start, end):
# compute shortest paths
_, dists, _ = bfct.find_shortest_paths(graph, start)
return dists.get(end, None)
def test_random(n = 50, p = 0.1, runs = 1000, debug = None):
for run in range(runs) if debug is None else [debug]:
g = fast_gnp_random_graph(n, p, seed = run + 1, directed = True)
# add source connected to all nodes
source = 100 * (n // 100) + 200
for v in list(g.nodes()): g.add_edge(source, v, weight = 0, tokens = 0)
# add random weights and tokens
wsum = 1
for _, _, data in g.edges_iter( data = True ):
data['weight'] = randrange(1, 10)
data['tokens'] = randrange(-1, 8)
wsum += data['weight']
# create shortest path formulation for initial tree
for _, _, data in g.edges_iter( data = True ):
data['sp'] = data['tokens'] * wsum - data['weight']
# ensure that the graph admits a feasible solution
its = 0
while True:
try:
its += 1
tree, distances = bfct.find_shortest_paths(g, source, arg = 'sp')
break
except NegativeCycleException as ex:
toks = sum(map(lambda vw : g.get_edge_data(*vw).get('tokens'), ex.cycle))
edge_data = g.get_edge_data(*choice(ex.cycle))
edge_data['tokens'] += (1 - toks)
edge_data['sp'] = edge_data['tokens'] * wsum - edge_data['weight']
if debug is not None:
import pdb; pdb.set_trace()
negative_toks = False
for _, _, data in g.edges_iter( data = True ):
if data['tokens'] < 0:
negative_toks = True; break
print("Run {}: negative tokens: {}, iterations: {}".format(run, negative_toks, its))
ratio, cycle = compute_mcr(g, source)
assert ratio is not None, "Deadlocked cycle found"
if not cycle:
# verify that the graph is acyclic
assert is_directed_acyclic_graph(g), "[run = {}] Graph is not acyclic".format(run)
else:
wsum, tsum = 0, 0
for v, w in cycle:
data = g.get_edge_data(v, w)
wsum += data['weight']
tsum += data['tokens']
assert Fraction( wsum, tsum ) == ratio, "[run = {}] computed MCR {} does not match ratio of critical cycle {}".format(run, ratio, Fraction(wsum, tsum))
for v, w, data in g.edges_iter( data = True ):
data['weight'] = data['tokens'] * ratio - data['weight']
try:
bellman_ford(g, source)
except Exception:
print("Exception during run {}".format(run))
def compute_mcm_component( g, root, estimate = None, pweight = 'weight' ):
""" Computes the maximum cycle mean of g.
NOTES:
- The weight on each edge must be non-negative
- The number of tokens on each edge must be non-negative.
- The graph is assumed to be strongly connected.
"""
# initialize:
distances = {}
queue = pq.priority_queue()
# determine lower bound for mcr
init_mcr = 1
for (v, w, data) in g.edges_iter(data=True):
weight = data.get(pweight, 0)
init_mcr = init_mcr + max(0, weight)
data['dist'] = pdistance(weight, 1)
if estimate is not None:
# compute initial tree by computing shortest paths tree
# print("Computing initial tree from estimate {}".format(estimate))
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = estimate - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg = 'w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs( root ):
if i == 0:
dv = distances[ v ]
for w in tree.children( v ):
distances[ w ] = dv + g.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(estimate, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException( ex.cycle )
else:
# create tree rooted at root
tree = DFSTree( root )
# run Dijkstra on tokens
distances[root] = 0
queue[ root ] = 0
while len(queue) > 0:
v, _ = queue.pop()
dv = distances[ v ]
for _, w, data in g.out_edges_iter(v, True):
dw = distances.get(w, None)
if (dw is None) or dv + 1 < dw:
distances[w] = dv + 1
queue[ w ] = distances[w]
tree.append_child( v, w )
# DFS on shortest paths DAG
# (this is the DAG induced by the union of all shortest path trees)
pre, post, index = dict(), dict(), 0
post_order = list()
for v, i in tree.dfs( root ):
if i == 0:
pre[v] = index
index += 1
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dw = distances[w]
if dv + 1 == dw and w not in pre:
tree.append_child( v, w )
else:
post[v] = index
index += 1
post_order.append(v)
# scan in topological order
distances.clear()
distances[root] = pdistance(0, 0)
while post_order:
v = post_order.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, None)
newdist = dv + data['dist']
if dw is None or newdist[1] < dw[1]:
distances[w] = newdist
tree.append_child( v, w )
elif (newdist[1] == dw[1] and newdist[0] - dw[0] > 0):
# can't have back edges
assert post[w] < post[v]
distances[w] = newdist
tree.append_child( v, w )
def compute_node_key(node):
maxratio, argmax = None, None
# go over all incoming edges of the node
for u, v, data in g.in_edges_iter( node, data = True ):
if u in distances:
delta = distances[u] + data['dist'] - distances[v]
# print("Delta for {} = {}".format((u, v), delta))
if delta[1] > 0:
ratio = Fraction(delta[0], delta[1])
if argmax is None or ratio > maxratio:
maxratio = ratio
argmax = (u, v)
# store the node key for v
if argmax is not None:
queue[node] = (-maxratio, argmax)
elif node in queue:
del queue[node]
return maxratio
# fill priority queue:
# go over all nodes and compute their key
# print("Distances from root {}: {}".format(root, distances))
for v in distances:
compute_node_key(v)
# pivot until cycle is found
path_changes = 0
while len(queue) > 0:
(node, (ratio, (v, w))) = queue.pop()
delta = distances[v] + g.get_edge_data(v, w)['dist'] - distances[w]
for j in tree.pre_order(w):
distances[j] += delta
if v == j:
cycle = list()
if v != w:
child = j
for p in tree.ancestors(child):
cycle.append( (p, j) )
if p == w:
break
j = p
cycle.reverse()
cycle.append( (v, w) )
assert __is_cycle( cycle ), "Found wrong cycle after pivoting with back edge ({}, {})".format( v, w )
return -ratio, cycle
for (_, k, data) in g.out_edges_iter(j, True):
# update priority of (j, k)
ratio_k = None
if k in queue:
(ratio_k, edge) = queue[k]
delta_k = distances[j] + data['dist'] - distances[k]
if delta_k[1] > 0:
r = -Fraction(delta_k[0], delta_k[1])
if ratio_k is None or r < ratio_k:
queue[k] = (r, (j, k))
# recompute vertex key of j
compute_node_key(j)
tree.append_child( v, w )
else:
# no cycle found, any period is admissible
# Note that this implies that the graph is acyclic
return None, None
def compute_mcr_component( g, root, estimate = None ):
""" Computes the maximum cycle ratio of g.
NOTES:
- The weight on each edge must be non-negative
- The number of tokens on each edge must be non-negative.
- The graph is assumed to be strongly connected.
"""
# initialize:
distances = {}
queue = pq.priority_queue()
# print("Computing MCR for graph {}".format( g.edges( data = True )))
# determine lower bound for mcr
# while doing so, determine vertices with outgoing tokenless
init_mcr = 1
negative_tokens_or_weights = False
for (v, w, data) in g.edges_iter(data=True):
tokens = data.get('tokens', 0)
weight = data.get('weight', 0)
init_mcr = init_mcr + max(0, weight)
data['dist'] = pdistance(weight, tokens)
if tokens < 0 or weight < 0:
negative_tokens_or_weights = True
if estimate is not None:
# compute initial tree by computing shortest paths tree
# print("Computing initial tree from estimate {}".format(estimate))
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = dist[1] * estimate - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg = 'w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs( root ):
if i == 0:
dv = distances[ v ]
for w in tree.children( v ):
distances[ w ] = dv + g.get_edge_data(v, w).get('dist')
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(estimate, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException( ex.cycle )
elif True:
# print("Computing initial tree from estimate {}".format(init_mcr))
# compute initial tree by computing shortest paths tree
for (v, w, data) in g.edges_iter(data=True):
dist = data['dist']
data['w'] = dist[1] * init_mcr - dist[0]
try:
tree, _ = bfct.find_shortest_paths(g, root, arg = 'w')
distances[root] = pdistance(0, 0)
for v, i in tree.dfs( root ):
if i == 0:
dv = distances[ v ]
for w in tree.children( v ):
distances[ w ] = dv + g.get_edge_data(v, w).get('dist')
if False:
# verify distances
for v in distances:
distv = distances[v]
distv = distv[0] - distv[1] * init_mcr
for _, w, data in g.out_edges_iter( v, True ):
distw = distances[w]
distw = distw[0] - distw[1] * init_mcr
edge_dist = data['dist']
edge_dist = edge_dist[0] - edge_dist[1] * init_mcr
assert distv + edge_dist <= distw
except NegativeCycleException as ex:
# print("Infeasible solution for estimate {}, due to cycle {}".format(init_mcr, [(u, v, data) for u, v, data in g.edges_iter(data = True) if (u, v) in ex.cycle]))
raise InfeasibleException( ex.cycle )
else:
# create tree rooted at root
tree = DFSTree( root )
# run Dijkstra on tokens
distances[root] = 0
queue[ root ] = 0
while len(queue) > 0:
v, _ = queue.pop()
dv = distances[ v ]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, dv + dist[1] + 1)
if dv + dist[1] - dw < 0:
distances[w] = dv + dist[1]
queue[ w ] = distances[w]
tree.append_child( v, w )
# DFS on shortest paths DAG
# (this is the DAG induced by the union of all shortest path trees)
pre, post, index = dict(), dict(), 0
post_order = list()
for v, i in tree.dfs( root ):
if i == 0:
pre[v] = index
index += 1
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
tokens = data['dist'][1]
dw = distances[w]
if dv + tokens == dw and w not in pre:
tree.append_child( v, w )
else:
post[v] = index
index += 1
post_order.append(v)
# scan in topological order
distances.clear()
distances[root] = pdistance(0, 0)
while post_order:
v = post_order.pop()
dv = distances[v]
for _, w, data in g.out_edges_iter(v, True):
dist = data['dist']
dw = distances.get(w, None)
newdist = dv + data['dist']
if dw is None or newdist[1] < dw[1]:
distances[w] = newdist
tree.append_child( v, w )
elif (newdist[1] == dw[1] and newdist[0] - dw[0] > 0):
# check that (v, w) is not a back edge!
if post[w] >= post[v]:
# back edge, MCR not defined
prev = v
cycle = list()
if v != w:
for p in tree.ancestors(v):
cycle.append( (p, prev) )
if p == w: break
prev = p
cycle.reverse()
cycle.append( (v, w) )
assert __is_cycle( cycle ), "Not a cycle: {}".format( cycle )
# print(cycle)
# print("Positive cycle found: {}".format( cycle ))
raise InfeasibleException(cycle)
distances[w] = newdist
tree.append_child( v, w )
def compute_node_key(node):
maxratio, argmax = None, None
# go over all incoming edges of the node
for (u, v, data) in g.in_edges_iter(nbunch=[node], data=True):
if u in distances:
delta = distances[u] + data['dist'] - distances[v]
# print("Delta for {} = {}".format((u, v), delta))
if delta[1] > 0:
ratio = Fraction(delta[0], delta[1])
if argmax is None or ratio > maxratio:
maxratio = ratio
argmax = (u, v)
# store the node key for v
if argmax is not None:
queue[node] = (-maxratio, argmax)
# print("Vertex key for {} = {}, attained by edge {}".format(node, maxratio, argmax))
elif node in queue:
del queue[node]
return maxratio
# fill priority queue:
# go over all nodes and compute their key
# print("Distances from root {}: {}".format(root, distances))
for v in distances:
compute_node_key(v)
# pivot until cycle is found
path_changes = 0
pivots = 0
while len(queue) > 0:
(node, (ratio, (v, w))) = queue.pop()
pivots += 1
delta = distances[v] + g.get_edge_data(v, w)['dist'] - distances[w]
# print("Pivoting with edge ({}, {}), delta = {}".format(v, w, delta))
for j in tree.pre_order(w):
# update parametric distance to j
distances[j] += delta
path_changes += 1
if v == j:
cycle = list()
if v != w:
child = j
for p in tree.ancestors(child):
cycle.append( (p, j) )
if p == w:
break
j = p
cycle.reverse()
cycle.append( (v, w) )
assert __is_cycle( cycle ), "Found wrong cycle after pivoting with back edge ({}, {})".format( v, w )
return -ratio, cycle
for (_, k, data) in g.out_edges_iter(j, True):
# update priority of (j, k)
ratio_k = None
if k in queue:
(ratio_k, edge) = queue[k]
delta_k = distances[j] + data['dist'] - distances[k]
# commented out line below to allow a negative MCR
if delta_k[1] > 0:
r = -Fraction(delta_k[0], delta_k[1])
if ratio_k is None or r < ratio_k:
queue[k] = (r, (j, k))
# print("New arc key for ({}, {}) = {}, delta = {} + {} - {} = {}".format(j, k, -r, distances[j], data['dist'], distances[k], delta_k))
# recompute vertex key of j
compute_node_key(j)
tree.append_child( v, w )
else:
# no cycle found, any period is admissible
# Note that this implies that the graph is acyclic
return None, None
