def minimal_steps_number_to(mdp: MDP, T: List[int]) -> List[float]:
"""
Compute the shortest path in term of edges in the underlying graph of the MDP to T (i.e., the minimal number of steps
to reach T in the underlying graph). The function connected_to (a breadth first serach algorithm) is adapted to
number the states instead of mark them.
:param mdp: a MDP.
:param T: a list of target states of the MDP.
:return: a list 'steps' such that, for each state s of the MDP, steps[s] = n where n is the minimal number of steps
to reach T in the underlying graph of the MDP.
"""
steps = [float('inf')] * mdp.number_of_states
for t in T:
steps[t] = 0
next = deque([])
for t in T:
next.extend(mdp.pred(t))
i = 1
while len(next) > 0:
predecessors = next
next = deque([])
while len(predecessors) > 0:
pred = predecessors.pop()
if steps[pred] > i:
steps[pred] = i
next.extend(mdp.pred(pred))
i += 1
return steps
def connected_to(mdp: MDP, T: List[int]) -> List[bool]:
"""
Compute the states connected to T.
For this purpose, a backward breadth-first search algorithm on the underlying graph of the MDP is used.
:param mdp: a MDP.
:param T: a list of target states of the MDP.
:return: a list 'marked' such that, for each state s of the MDP, marked[s] = True if s is connected to T in
the underlying graph of the MDP.
"""
marked = [False] * mdp.number_of_states
for t in T:
marked[t] = True
next = deque([])
for t in T:
next.extend(mdp.pred(t))
while len(next) > 0:
pred = next.pop()
if not marked[pred]:
marked[pred] = True
for predecessor in mdp.pred(pred):
next.appendleft(predecessor)
return marked