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Implements the toposort and strongly_connected_components graph algorithms, as a demonstration of how to use the recipe, 'Implementing the observer pattern yet again: this time with coroutines and the with statement'.

See http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/498259

Python, 238 lines
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"""tsort module

Implements the toposort and strongly_connected_components graph
algorithms, as a demonstration of how to use the recipe, 'Implementing
the observer pattern yet again: this time with coroutines and the with
statement'
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/498259)

Requires Python 2.5
Author: Jim Baker (jbaker@zyasoft.com)
"""

from __future__ import with_statement
from observer import consumer, observation
from collections import deque
import unittest


# Colors used by the traversal (DFS) to mark if it has visited all
# vertices leading out of a given vertex.  WHITE is implicit.
# Alternatively use an enumeration as supported by this recipe,
# http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/413486

GRAY = 'gray'   # currently visiting this vertex
BLACK = 'black' # all adjacent vertices visited

def toposort(G):
    """Returns the topological sort of the input graph G.

    The algorithm toposort is described in Cormen, Leiserson, Rivest,
    *Introduction to Algorithms* [CLR].  The code here is explicitly
    modeled on their pseudocode.
    
    David Eppstein's PADS library employs an alternative strategy of
    'shadowing' the Searcher class methods via inheritance
    (http://www.ics.uci.edu/~eppstein/PADS/DFS.py).  His strategy
    makes the preorder, postorder, and backedge events more explicit
    than the implicit usage presented here based on coloring changes.
    """

    ordering = deque()
    seen = set()
    coloring = dict()

    @consumer
    def finishing(order):
        """Returns the vertices in the reverse of their finishing time.

        This function is implemented as a coroutine so that it can be
        decoupled from the actual setting of the vertice's color in
        the DFS. So the only protocol that requires agreement is this
        marking of a coloring.

        It's critical that this function be decorated with the
        @consumer decorator so that it's advanced to waiting on any
        data.
        """

        while True:
            v, old_color, new_color = (yield)
            if new_color == BLACK:
                order.appendleft(v)

    with observation(observe=coloring,
                     notify=[finishing(ordering)]) as coloring:
        for v in DFS(G, coloring):
            # In this code, the real work is in the finishing
            # function.  For now, we will just verify that the vertex
            # is touched at *most* once.
            assert(v not in seen)
            seen.add(v)

    # Now verify that each vertex was touched at *least* once
    assert(seen == set(vertices(G)))
        
    return ordering


def strongly_connected_components(G):
    """Returns the strongly connnected components of G as frozensets.

    SCC is a good test of the toposort algorithm just described. See
    [CLR] for more details. Constructing the components as frozensets
    simplifies their use.

    The following iterator pipeline will return the components in
    order of decreasing size:

    sorted(strongly_connected_components(G), key=len, reverse=True)
    """
    
    ordering = toposort(G)
    R = reverse(G)
    coloring = dict()
    for v in ordering:
        component = frozenset(w for w in DFS_visit(R, coloring, v))
        if component:
            yield component



######################################################################
# What follows is just support code and unit testing.
######################################################################

# Please note, since this is just to illustrate the use of the
# observer pattern, we don't manage reverse edges efficiently.

def edges(G):
    """Returns the edge set of G as pairs (v, w)"""
    for v, adj in G.iteritems():
        for w in adj:
            yield v, w

def vertices(G):
    """Returns the vertex set of G"""
    return iter(G)

def adjacent(G, v):
    """Returns the adjacent vertices to v, if any"""
    for w in G.get(v, ()):
        yield w

def graph_equal(G, H):
    """Tests the equality of graphs"""
    return set(edges(G)) == set(edges(H)) and \
           set(vertices(G)) == set(vertices(H))

def reverse(G):
    """Reverses the edges in a graph.

    Always returns an adjacency list representation, using the GvR
    model. Empty vertices are maintained.
    """

    R = {}
    for v in vertices(G): R[v] = []
    for v, w in edges(G): R[w].append(v)
    return R


def DFS(G, coloring=None, roots=None):
    """Performs a depth-first search of the graph `G`"""
    
    if coloring is None:
        coloring = dict()
    if roots is None:
        roots = vertices(G)
        
    for v in roots:
        for w in DFS_visit(G, coloring, v):
            yield w


def DFS_visit(G, coloring, v):
    """A recursive generator implementation of a depth-first search visitor.

    Please note that as a recursive function, it may exhaust Python's
    stack. Consider using NetworkX or PADS instead. Results are
    produced incrementally, a nice benefit of using a generator.
    """
    
    if v in coloring:
        return
        
    yield v
    coloring[v] = GRAY

    for w in adjacent(G, v):
        if w not in coloring:
            for descendant in DFS_visit(G, coloring, w):
                yield descendant
    coloring[v] = BLACK




def pairwise(iterable):
    from itertools import tee, izip
    "s -> (s0,s1), (s1,s2), (s2, s3), ..."
    a, b = tee(iterable)
    try:
        b.next()
    except StopIteration:
        pass
    return izip(a, b)

class TsortTestCase(unittest.TestCase):

    # from itertools recipes

    def testTopoSort(self):
        def verify_partial_ordering(G, ordering):
            """Verifies that `ordering` is a partial ordering of graph `G`."""

            ordering = tuple(ordering)
            assert(set(ordering) == set(vertices(G)))
            for v, w in pairwise(ordering):
                # need to ensure that v > w, from a graph theoretic perspective
                self.assert_(v not in adjacent(G,w))

        # Prof. Bumstead's dependency graph from CLR
        Bumstead = {
            'undershorts':['pants', 'shoes'],
            'socks': ['shoes'],
            'watch': [],
            'pants': ['belt'],
            'shirt': ['belt', 'tie'],
            'belt': ['jacket'],
            'jacket': [],
            'shoes': [],
            'tie':  [],
            }

        verify_partial_ordering(Bumstead, toposort(Bumstead))
        verify_partial_ordering(reverse(Bumstead), toposort(reverse(Bumstead)))

    def testSCC(self):
        def verify_strongly_connected(precomputed, computed):
            # turn components into a set of frozensets, simplifies the comparison
            self.assertEquals(set(frozenset(component) for component in precomputed),
                              set(frozenset(component) for component in computed))

        # from Eppstein's tests
        G1 = { 0:[1], 1:[2,3], 2:[4,5], 3:[4,5], 4:[6], 5:[], 6:[] }
        C1 = [[0],[1],[2],[3],[4],[5],[6]]

        G2 = { 0:[1], 1:[2,3,4], 2:[0,3], 3:[4], 4:[3] }
        C2 = [[0,1,2],[3,4]]

        C1_computed = sorted(strongly_connected_components(G1), key=len, reverse=True)
        C2_computed = sorted(strongly_connected_components(G2), key=len, reverse=True)

        verify_strongly_connected(C1, C1_computed)
        verify_strongly_connected(C2, C2_computed)

if __name__ == "__main__":
    unittest.main()

Graphs use the model introduced by GvR in this essay, http://www.python.org/doc/essays/graphs/

Here's then how to use it:

from tsort import toposort, strongly_connected_components

Bumstead = { 'undershorts':['pants', 'shoes'], 'socks': ['shoes'], 'watch': [], 'pants': ['belt'], 'shirt': ['belt', 'tie'], 'belt': ['jacket'], 'jacket': [], 'shoes': [], 'tie': [], }

print toposort(Bumstead) print sorted(strongly_connected_components({ 0:[1], 1:[2,3,4], 2:[0,3], 3:[4], 4:[3] }), key=len, reverse=True)

1 comment

Christopher Dunn 14 years, 11 months ago  # | flag

Interesting example. This example is very interesting, but it does not demonstrate why a co-routine is useful. In this case, a straight-forward, partially evaluated closure would have been simpler.

from functools import partial

def record_if_BLACK(order, v, old_color, new_color):
    if new_color == BLACK:
        order.appendleft(v)

with observation(observe=coloring,
                 notify=[partial(record_if_BLACK,
                                 ordering)]) as coloring:
    ....

(Of course, 'observation' would have to be modified to call 'notify' directly, rather than notify.send().)

In fact, we could turn the whole toposort into an iterator (in order of finishing time).

def yield_if_BLACK(v, old_color, new_color):
    if new_color == BLACK:
        yield v

with observation(observe=coloring,
                 notify=[yield_if_BLACK]) as coloring:
    ....

Now, we would get the results from toposort as they are found (but not in reverse order, naturally). However, only one 'consumer' can yield to the caller of toposort.

While the direct function call is much clearer, the co-routine is potentially quite powerful since it can maintain state, not only between calls, but before and after the entire algorithm. I do like the co-routine very much. Python looks more like Ruby all the time.