#8714: add Bellman-Ford algorithm for shortest paths
----------------------------+-----------------------------------------------
Reporter: mvngu | Owner: jason, ncohen, rlm
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.4
Component: graph theory | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
----------------------------+-----------------------------------------------
Changes (by mvngu):
* cc: wdj (added)
Old description:
> I'm using #698 as a wish list of items to add to the graph theory module
> of Sage. The purpose of this ticket is to implement the
> [http://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm Bellman-
> Ford] algorithm for finding shortest paths in a weighted graph `G` that
> may have negative weights. If `G` doesn't have negative weights,
> Dijkstra's algorithm can be used. However, if `G` has negative weights,
> we fall back on the Bellman-Ford algorithm. The Bellman-Ford algorithm is
> able to handle graphs with negative weights, but not graphs that have
> negative-weight cycles. See also the function
> [http://reference.wolfram.com/mathematica/Combinatorica/ref/BellmanFord.html
> BellmanFord] in Mathematica's
> [http://reference.wolfram.com/mathematica/Combinatorica/guide/CombinatoricaPackage.html
> Combinatorica] package. See this [http://code.google.com/p/graph-theory-
> algorithms-book/ graph theory book] for an algorithmic presentation of
> the Bellman-Ford algorithm.
New description:
I'm using #698 as a wish list of items to add to the graph theory module
of Sage. The purpose of this ticket is to implement the
[http://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm Bellman-Ford]
algorithm for finding shortest paths in a weighted graph `G` that may have
negative weights. If `G` doesn't have negative weights, Dijkstra's
algorithm can be used. However, if `G` has negative weights, we fall back
on the Bellman-Ford algorithm. The Bellman-Ford algorithm is able to
handle graphs with negative weights, but not graphs that have negative-
weight cycles. See also the function
[http://reference.wolfram.com/mathematica/Combinatorica/ref/BellmanFord.html
BellmanFord] in Mathematica's
[http://reference.wolfram.com/mathematica/Combinatorica/guide/CombinatoricaPackage.html
Combinatorica] package. See this [http://code.google.com/p/graph-theory-
algorithms-book/ graph theory book] for an algorithmic presentation of the
Bellman-Ford algorithm.
See also the [/wiki/GraphTheoryRoadmap graph theory roadmap].
--
Comment:
Here is an implementation by David Joyner:
{{{
def bellman_ford(Gamma, s):
"""
Computes the shortest distance from s to all other vertices in Gamma.
If Gamma has a negative weight cycle, then return an error.
INPUT:
- Gamma -- a graph.
- s -- the source vertex.
OUTPUT:
- (d,p) -- pair of dictionaries keyed on the list of vertices,
which store the distance and shortest paths.
REFERENCE:
http://en.wikipedia.org/wiki/Bellman-Ford_algorithm
"""
P = []
dist = {}
predecessor = {}
V = Gamma.vertices()
E = Gamma.edges()
for v in V:
if v == s:
dist[v] = 0
else:
dist[v] = infinity
predecessor[v] = 0
for i in range(1, len(V)):
for e in E:
u = e[0]
v = e[1]
wt = e[2]
if dist[u] + wt < dist[v]:
dist[v] = dist[u] + wt
predecessor[v] = u
# check for negative-weight cycles
for e in E:
u = e[0]
v = e[1]
wt = e[2]
if dist[u] + wt < dist[v]:
raise ValueError("Graph contains a negative-weight cycle")
return dist, predecessor
}}}
And some examples:
{{{
sage: M = matrix([[0,1,4,0], [0,0,1,5], [0,0,0,3], [0,0,0,0]])
sage: G = Graph(M, format="weighted_adjacency_matrix")
sage: bellman_ford(G, G.vertices()[0])
{0: 0, 1: 1, 2: 2, 3: 5}
}}}
and
{{{
sage: M = matrix([[0,1,0,0],[1,0,-4,1],[1,1,0,0],[0,0,1,0]])
sage: G = DiGraph(M, format = "weighted_adjacency_matrix")
sage: bellman_ford(G, G.vertices()[0])
---------------------------------------------------------------------------
...
ValueError: Graph contains a negative-weight cycle
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8714#comment:1>
Sage <http://www.sagemath.org>
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