I would like to find a minimum spanning tree (MST) on a weighted directed graph. I have been trying to use Chu-Liu/Edmond's algorithm, which I have implemented in Python (code below). A simple, clear description of the algorithm can be found here. I have two questions.
Is Edmond's algorithm guaranteed to converge on a solution?
I am concerned that removing a cycle will add another cycle. If this happens, the algorithm will continue trying to remove cycles forever.
I seem to have found an example where this happens. The input graph is shown below (in the code). The algorithm never finishes because it switches between cycles [1,2] and [1,3], and [5,4] and [5,6]. The edge added to the graph to resolve the cycle [5,4] creates cycle [5,6] and vice versa, and similarly for [1,2] and [1,3].
I should note that I am not certain that my implementation is correct.
To resolve this issue, I introduced an ad hoc patch. When an edge is removed to remove a cycle, I permanently remove that edge from the underlying graph G on which we are searching for an MST. Consequently, that edge cannot be added again and this should prevent the algorithm from getting stuck. With this change, am I guaranteed to find an MST?
I suspect that one can find a pathological case where this step will lead to a result that is not an MST, but I have not been able to think of one. It seems to work on all the simple test cases that I have tried.
Code:
import sys
# --------------------------------------------------------------------------------- #
def _reverse(graph):
r = {}
for src in graph:
for (dst,c) in graph[src].items():
if dst in r:
r[dst][src] = c
else:
r[dst] = { src : c }
return r
# Finds all cycles in graph using Tarjan's algorithm
def strongly_connected_components(graph):
"""
Tarjan's Algorithm (named for its discoverer, Robert Tarjan) is a graph theory algorithm
for finding the strongly connected components of a graph.
Based on: http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
"""
index_counter = [0]
stack = []
lowlinks = {}
index = {}
result = []
def strongconnect(node):
# set the depth index for this node to the smallest unused index
index[node] = index_counter[0]
lowlinks[node] = index_counter[0]
index_counter[0] += 1
stack.append(node)
# Consider successors of `node`
try:
successors = graph[node]
except:
successors = []
for successor in successors:
if successor not in lowlinks:
# Successor has not yet been visited; recurse on it
strongconnect(successor)
lowlinks[node] = min(lowlinks[node],lowlinks[successor])
elif successor in stack:
# the successor is in the stack and hence in the current strongly connected component (SCC)
lowlinks[node] = min(lowlinks[node],index[successor])
# If `node` is a root node, pop the stack and generate an SCC
if lowlinks[node] == index[node]:
connected_component = []
while True:
successor = stack.pop()
connected_component.append(successor)
if successor == node: break
component = tuple(connected_component)
# storing the result
result.append(component)
for node in graph:
if node not in lowlinks:
strongconnect(node)
return result
def _mergeCycles(cycle,G,RG,g,rg):
allInEdges = [] # all edges entering cycle from outside cycle
minInternal = None
minInternalWeight = sys.maxint
# Find minimal internal edge weight
for n in cycle:
for e in RG[n]:
if e in cycle:
if minInternal is None or RG[n][e] < minInternalWeight:
minInternal = (n,e)
minInternalWeight = RG[n][e]
continue
else:
allInEdges.append((n,e)) # edge enters cycle
# Find the incoming edge with minimum modified cost
# modified cost c(i,k) = c(i,j) - (c(x_j, j) - min{j}(c(x_j, j)))
minExternal = None
minModifiedWeight = 0
for j,i in allInEdges: # j is vertex in cycle, i is candidate vertex outside cycle
xj, weight_xj_j = rg[j].popitem() # xj is vertex in cycle that currently goes to j
rg[j][xj] = weight_xj_j # put item back in dictionary
w = RG[j][i] - (weight_xj_j - minInternalWeight) # c(i,k) = c(i,j) - (c(x_j, j) - min{j}(c(x_j, j)))
if minExternal is None or w <= minModifiedWeight:
minExternal = (j,i)
minModifiedWeight = w
w = RG[minExternal[0]][minExternal[1]] # weight of edge entering cycle
xj,_ = rg[minExternal[0]].popitem() # xj is vertex in cycle that currently goes to j
rem = (minExternal[0], xj) # edge to remove
rg[minExternal[0]].clear() # popitem() should delete the one edge into j, but we ensure that
# Remove offending edge from RG
# RG[minExternal[0]].pop(xj, None) #highly experimental. throw away the offending edge, so we never get it again
if rem[1] in g:
if rem[0] in g[rem[1]]:
del g[rem[1]][rem[0]]
if minExternal[1] in g:
g[minExternal[1]][minExternal[0]] = w
else:
g[minExternal[1]] = { minExternal[0] : w }
rg = _reverse(g)
# --------------------------------------------------------------------------------- #
def mst(root,G):
""" The Chu-Liu/Edmond's algorithm
arguments:
root - the root of the MST
G - the graph in which the MST lies
returns: a graph representation of the MST
Graph representation is the same as the one found at:
http://code.activestate.com/recipes/119466/
Explanation is copied verbatim here:
The input graph G is assumed to have the following
representation: A vertex can be any object that can
be used as an index into a dictionary. G is a
dictionary, indexed by vertices. For any vertex v,
G[v] is itself a dictionary, indexed by the neighbors
of v. For any edge v->w, G[v][w] is the length of
the edge.
"""
RG = _reverse(G)
g = {}
for n in RG:
if len(RG[n]) == 0:
continue
minimum = sys.maxint
s,d = None,None
for e in RG[n]:
if RG[n][e] < minimum:
minimum = RG[n][e]
s,d = n,e
if d in g:
g[d][s] = RG[s][d]
else:
g[d] = { s : RG[s][d] }
cycles = [list(c) for c in strongly_connected_components(g)]
cycles_exist = True
while cycles_exist:
cycles_exist = False
cycles = [list(c) for c in strongly_connected_components(g)]
rg = _reverse(g)
for cycle in cycles:
if root in cycle:
continue
if len(cycle) == 1:
continue
_mergeCycles(cycle, G, RG, g, rg)
cycles_exist = True
return g
# --------------------------------------------------------------------------------- #
if __name__ == "__main__":
# an example of an input that works
root = 0
g = {0: {1: 23, 2: 22, 3: 22}, 1: {2: 1, 3: 1}, 3: {1: 1, 2: 0}}
# an example of an input that causes infinite cycle
root = 0
g = {0: {1: 17, 2: 16, 3: 19, 4: 16, 5: 16, 6: 18}, 1: {2: 3, 3: 3, 4: 11, 5: 10, 6: 12}, 2: {1: 3, 3: 4, 4: 8, 5: 8, 6: 11}, 3: {1: 3, 2: 4, 4: 12, 5: 11, 6: 14}, 4: {1: 11, 2: 8, 3: 12, 5: 6, 6: 10}, 5: {1: 10, 2: 8, 3: 11, 4: 6, 6: 4}, 6: {1: 12, 2: 11, 3: 14, 4: 10, 5: 4}}
h = mst(int(root),g)
print h
for s in h:
for t in h[s]:
print "%d-%d" % (s,t)
