def __shortest_path_lengths_noinf(g, src):
"""Compute shortest-path distances from src to reachable vertices of g
without python inf notation."""
dist = {} # dist[v] is upper bound from s to v
cloud = {} # map reachable v to its dist[v] value
pq = AdaptableHeapPriorityQueue() # vertex v will have key dist[v]
pq_locator = {} # map from vertex to its pq locator
dist[src] = 0
pq_locator[src] = pq.add(0, src)
while not pq.is_empty():
key, u = pq.remove_min()
cloud[u] = key # its correct dist[u] value
del pq_locator[u] # u is no longer needed
for edge in g.incident_edges(u): # outgoing edges (u, v)
v = edge.opposite(u)
if v not in cloud:
# perform relaxation step on edge (u, v)
new_dist = dist[u] + edge.element()
if v not in dist:
# add an vertex to dist until after an edge reaches it
dist[v] = new_dist
# and add it to the priority queue and save locator for
# future updates
pq_locator[v] = pq.add(new_dist, v)
elif dist[v] > new_dist: # If better path to v exists...
dist[v] = new_dist # update the distance, and...
pq.update(pq_locator[v], new_dist, v) # the pq entry
# Now, dist and cloud are the same, they both only include the reachable
# vertices.
return cloud
def MST_PrimJarnik(g): # O((n+m)logn)
"""Compute a minimum spanning tree of simple connected weighted graph g.
Return a list of edges that comprise the MST (in arbitrary order).
"""
dist = {} # dist[v] is bound on distance to tree
tree = [] # list of edges in spanning tree
pq = AdaptableHeapPriorityQueue() # dist[v] maps to value (v, e=(u, v))
pq_locator = {} # map from vertex to its pq locator
# for each vertex v of the graph, add an entry to the priority queue,
# with the source having distance 0 and all others having infinite dist-
# ance.
for v in g.vertices():
dist[v] = float("inf") if len(dist) > 0 else 0
pq_locator[v] = pq.add(dist[v], (v, None))
while not pq.is_empty():
_, (u, edge) = pq.remove_min() # unpack tuple from pq
del pq_locator[u] # u is no longer in pq
if edge is not None:
tree.append(edge) # add edge to tree
for link in g.incident_edges(u):
v = link.opposite(u)
if v in pq_locator: # thus v not yet in tree
# see if edge (u, v) better connects v to the growing tree
wgt = link.element()
if wgt < dist[v]:
dist[v] = wgt # update the distance and ...
pq.update(pq_locator[v], wgt, (v, link)) # the pq entry
return tree
def __shortest_path_lengths_noinf(g, src):
"""Compute shortest-path distances from src to reachable vertices of g
without python inf notation."""
dist = {} # dist[v] is upper bound from s to v
cloud = {} # map reachable v to its dist[v] value
pq = AdaptableHeapPriorityQueue() # vertex v will have key dist[v]
pq_locator = {} # map from vertex to its pq locator
dist[src] = 0
pq_locator[src] = pq.add(0, src)
while not pq.is_empty():
key, u = pq.remove_min()
cloud[u] = key # its correct dist[u] value
del pq_locator[u] # u is no longer needed
for edge in g.incident_edges(u): # outgoing edges (u, v)
v = edge.opposite(u)
if v not in cloud:
# perform relaxation step on edge (u, v)
new_dist = dist[u] + edge.element()
if v not in dist:
# add an vertex to dist until after an edge reaches it
dist[v] = new_dist
# and add it to the priority queue and save locator for
# future updates
pq_locator[v] = pq.add(new_dist, v)
elif dist[v] > new_dist: # If better path to v exists...
dist[v] = new_dist # update the distance, and...
pq.update(pq_locator[v], new_dist, v) # the pq entry
# Now, dist and cloud are the same, they both only include the reachable
# vertices.
return cloud
def __shortest_path_lengths_inf(g, src):
"""Compute shortest-path distances from src to reachable vertices of g
with python inf notation."""
dist = {} # dist[v] is upper bound from s to v
cloud = {} # map reachable v to its dist[v] value
pq = AdaptableHeapPriorityQueue() # vertex v will have key dist[v]
pq_locator = {} # map from vertex to its pq locator
# for each vertex v of the graph, add an entry to the priority queue, with
# the source having distance 0 and all others having infinite distance
for v in g.vertices():
if v is src:
dist[v] = 0
else:
dist[v] = float('inf') # syntax for positive infinity
pq_locator[v] = pq.add(dist[v], v) # save locator for future updates
while not pq.is_empty():
key, u = pq.remove_min()
cloud[u] = key # its correct dist[u] value
del pq_locator[u] # u is no longer in pq
for e in g.incident_edges(u): # outgoing edges (u,v)
v = e.opposite(u)
if v not in cloud:
# perform relaxation step on edge (u,v)
wgt = e.element()
if dist[u] + wgt < dist[v]: # better path to v?
dist[v] = dist[u] + wgt # update the distance, and
pq.update(pq_locator[v], dist[v], v) # the pq entry
return cloud # only includes reachable vertices
def MST_PrimJarnik(g): # O((n+m)logn)
"""Compute a minimum spanning tree of simple connected weighted graph g.
Return a list of edges that comprise the MST (in arbitrary order).
"""
dist = {} # dist[v] is bound on distance to tree
tree = [] # list of edges in spanning tree
pq = AdaptableHeapPriorityQueue() # dist[v] maps to value (v, e=(u, v))
pq_locator = {} # map from vertex to its pq locator
# for each vertex v of the graph, add an entry to the priority queue,
# with the source having distance 0 and all others having infinite dist-
# ance.
for v in g.vertices():
dist[v] = float("inf") if len(dist) > 0 else 0
pq_locator[v] = pq.add(dist[v], (v, None))
while not pq.is_empty():
_, (u, edge) = pq.remove_min() # unpack tuple from pq
del pq_locator[u] # u is no longer in pq
if edge is not None:
tree.append(edge) # add edge to tree
for link in g.incident_edges(u):
v = link.opposite(u)
if v in pq_locator: # thus v not yet in tree
# see if edge (u, v) better connects v to the growing tree
wgt = link.element()
if wgt < dist[v]:
dist[v] = wgt # update the distance and ...
pq.update(pq_locator[v], wgt, (v, link)) # the pq entry
return tree
def __shortest_path_lengths_inf(g, src):
"""Compute shortest-path distances from src to reachable vertices of g
with python inf notation."""
dist = {} # dist[v] is upper bound from s to v
cloud = {} # map reachable v to its dist[v] value
pq = AdaptableHeapPriorityQueue() # vertex v will have key dist[v]
pq_locator = {} # map from vertex to its pq locator
# for each vertex v of the graph, add an entry to the priority queue, with
# the source having distance 0 and all others having infinite distance
for v in g.vertices():
if v is src:
dist[v] = 0
else:
dist[v] = float('inf') # syntax for positive infinity
pq_locator[v] = pq.add(dist[v], v) # save locator for future updates
while not pq.is_empty():
key, u = pq.remove_min()
cloud[u] = key # its correct dist[u] value
del pq_locator[u] # u is no longer in pq
for e in g.incident_edges(u): # outgoing edges (u,v)
v = e.opposite(u)
if v not in cloud:
# perform relaxation step on edge (u,v)
wgt = e.element()
if dist[u] + wgt < dist[v]: # better path to v?
dist[v] = dist[u] + wgt # update the distance, and
pq.update(pq_locator[v], dist[v], v) # the pq entry
return cloud # only includes reachable vertices