"""
Topological sort.
From Tim Peters, see:
http://mail.python.org/pipermail/python-list/1999-July/006660.html
topsort takes a list of pairs, where each pair (x, y) is taken to
mean that x <= y wrt some abstract partial ordering. The return
value is a list, representing a total ordering that respects all
the input constraints.
E.g.,
topsort( [(1,2), (3,3)] )
Valid topological sorts would be any of (but nothing other than)
[3, 1, 2]
[1, 3, 2]
[1, 2, 3]
... however this variant ensures that 'key' order (first element of
tuple) is preserved so the following will be result returned:
[1, 3, 2]
because those are the permutations of the input elements that
respect the "1 precedes 2" and "3 precedes 3" input constraints.
Note that a constraint of the form (x, x) is really just a trick
to make sure x appears *somewhere* in the output list.
If there's a cycle in the constraints, say
topsort( [(1,2), (2,1)] )
then CycleError is raised, and the exception object supports
many methods to help analyze and break the cycles. This requires
a good deal more code than topsort itself!
"""
[docs]class CycleError(Exception):
[docs] def __init__(self, sofar, numpreds, succs):
Exception.__init__(self, "cycle in constraints", sofar, numpreds, succs)
self.preds = None
# return as much of the total ordering as topsort was able to
# find before it hit a cycle
[docs] def get_partial(self):
return self[1]
# return remaining elt -> count of predecessors map
[docs] def get_pred_counts(self):
return self[2]
# return remaining elt -> list of successors map
[docs] def get_succs(self):
return self[3]
# return remaining elements (== those that don't appear in
# get_partial())
[docs] def get_elements(self):
return self.get_pred_counts().keys()
# Return a list of pairs representing the full state of what's
# remaining (if you pass this list back to topsort, it will raise
# CycleError again, and if you invoke get_pairlist on *that*
# exception object, the result will be isomorphic to *this*
# invocation of get_pairlist).
# The idea is that you can use pick_a_cycle to find a cycle,
# through some means or another pick an (x,y) pair in the cycle
# you no longer want to respect, then remove that pair from the
# output of get_pairlist and try topsort again.
[docs] def get_pairlist(self):
succs = self.get_succs()
answer = []
for x in self.get_elements():
if x in succs:
for y in succs[x]:
answer.append((x, y))
else:
# make sure x appears in topsort's output!
answer.append((x, x))
return answer
# return remaining elt -> list of predecessors map
[docs] def get_preds(self):
if self.preds is not None:
return self.preds
self.preds = preds = {}
remaining_elts = self.get_elements()
for x in remaining_elts:
preds[x] = []
succs = self.get_succs()
for x in remaining_elts:
if x in succs:
for y in succs[x]:
preds[y].append(x)
if __debug__:
for x in remaining_elts:
assert len(preds[x]) > 0
return preds
# return a cycle [x, ..., x] at random
[docs] def pick_a_cycle(self):
remaining_elts = self.get_elements()
# We know that everything in remaining_elts has a predecessor,
# but don't know that everything in it has a successor. So
# crawling forward over succs may hit a dead end. Instead we
# crawl backward over the preds until we hit a duplicate, then
# reverse the path.
preds = self.get_preds()
from random import choice
x = choice(remaining_elts)
answer = []
index = {}
in_answer = index.has_key
while not in_answer(x):
index[x] = len(answer) # index of x in answer
answer.append(x)
x = choice(preds[x])
answer.append(x)
answer = answer[index[x] :]
answer.reverse()
return answer
def _numpreds_and_successors_from_pairlist(pairlist):
numpreds = {} # elt -> # of predecessors
successors = {} # elt -> list of successors
for first, second in pairlist:
# make sure every elt is a key in numpreds
if first not in numpreds:
numpreds[first] = 0
if second not in numpreds:
numpreds[second] = 0
# if they're the same, there's no real dependence
if first == second:
continue
# since first < second, second gains a pred ...
numpreds[second] = numpreds[second] + 1
# ... and first gains a succ
if first in successors:
successors[first].append(second)
else:
successors[first] = [second]
return numpreds, successors
[docs]def topsort(pairlist):
numpreds, successors = _numpreds_and_successors_from_pairlist(pairlist)
# suck up everything without a predecessor
answer = [x for x in numpreds.keys() if numpreds[x] == 0]
# for everything in answer, knock down the pred count on
# its successors; note that answer grows *in* the loop
for x in answer:
assert numpreds[x] == 0
del numpreds[x]
if x in successors:
for y in successors[x]:
numpreds[y] = numpreds[y] - 1
if numpreds[y] == 0:
answer.append(y)
# following "del" isn't needed; just makes
# CycleError details easier to grasp
del successors[x]
if numpreds:
# everything in numpreds has at least one predecessor ->
# there's a cycle
if __debug__:
for x in numpreds.keys():
assert numpreds[x] > 0
raise CycleError(answer, numpreds, successors)
return answer
[docs]def topsort_levels(pairlist):
numpreds, successors = _numpreds_and_successors_from_pairlist(pairlist)
answer = []
while 1:
# Suck up everything without a predecessor.
levparents = [x for x in numpreds.keys() if numpreds[x] == 0]
if not levparents:
break
answer.append(levparents)
for levparent in levparents:
del numpreds[levparent]
if levparent in successors:
for levparentsucc in successors[levparent]:
numpreds[levparentsucc] -= 1
del successors[levparent]
if numpreds:
# Everything in num_parents has at least one child ->
# there's a cycle.
raise CycleError(answer, numpreds, successors)
return answer