 | fgl-5.4.1.1: Martin Erwig's Functional Graph Library | Contents | Index |
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| Data.Graph.Inductive.Query.MaxFlow |
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| Description |
Maximum Flow algorithm
We are given a flow network G=(V,E) with source s and sink t where each
edge (u,v) in E has a nonnegative capacity c(u,v)>=0, and we wish to
find a flow of maximum value from s to t.
A flow in G=(V,E) is a real-valued function f:VxV->R that satisfies:
For all u,v in V, f(u,v)<=c(u,v)
For all u,v in V, f(u,v)=-f(v,u)
For all u in V-{s,t}, Sum{f(u,v):v in V } = 0
The value of a flow f is defined as |f|=Sum {f(s,v)|v in V}, i.e.,
the total net flow out of the source.
In this module we implement the Edmonds-Karp algorithm, which is the
Ford-Fulkerson method but using the shortest path from s to t as the
augmenting path along which the flow is incremented.
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| Synopsis |
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| getRevEdges :: (Num b, Ord b) => [(Node, Node)] -> [(Node, Node, b)] | | | augmentGraph :: (DynGraph gr, Num b, Ord b) => gr a b -> gr a (b, b, b) | | | updAdjList :: (Num b, Ord b) => [((b, b, b), Node)] -> Node -> b -> Bool -> [((b, b, b), Node)] | | | updateFlow :: (DynGraph gr, Num b, Ord b) => Path -> b -> gr a (b, b, b) -> gr a (b, b, b) | | | mfmg :: (DynGraph gr, Num b, Ord b) => gr a (b, b, b) -> Node -> Node -> gr a (b, b, b) | | | mf :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> gr a (b, b, b) | | | maxFlowgraph :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> gr a (b, b) | | | maxFlow :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> b |
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| Documentation |
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| getRevEdges :: (Num b, Ord b) => [(Node, Node)] -> [(Node, Node, b)] |
i 0
For each edge a--->b this function returns edge b--->a .
i
Edges a<--->b are ignored
j
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| augmentGraph :: (DynGraph gr, Num b, Ord b) => gr a b -> gr a (b, b, b) |
i 0
For each edge a--->b insert into graph the edge a<---b . Then change the
i (i,0,i)
label of every edge from a---->b to a------->b
where label (x,y,z)=(Max Capacity, Current flow, Residual capacity)
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| updAdjList :: (Num b, Ord b) => [((b, b, b), Node)] -> Node -> b -> Bool -> [((b, b, b), Node)] |
| Given a successor or predecessor list for node u and given node v, find
the label corresponding to edge (u,v) and update the flow and residual
capacity of that edge's label. Then return the updated list.
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| updateFlow :: (DynGraph gr, Num b, Ord b) => Path -> b -> gr a (b, b, b) -> gr a (b, b, b) |
| Update flow and residual capacity along augmenting path from s to t in
graph G. For a path [u,v,w,...] find the node u in G and its successor and
predecessor list, then update the corresponding edges (u,v) and (v,u) on
those lists by using the minimum residual capacity of the path.
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| mfmg :: (DynGraph gr, Num b, Ord b) => gr a (b, b, b) -> Node -> Node -> gr a (b, b, b) |
| Compute the flow from s to t on a graph whose edges are labeled with
(x,y,z)=(max capacity,current flow,residual capacity) and all edges
are of the form a<---->b. First compute the residual graph, that is,
delete those edges whose residual capacity is zero. Then compute the
shortest augmenting path from s to t, and finally update the flow and
residual capacity along that path by using the minimum capacity of
that path. Repeat this process until no shortest path from s to t exist.
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| mf :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> gr a (b, b, b) |
| Compute the flow from s to t on a graph whose edges are labeled with
x, which is the max capacity and where not all edges need to be of the
form a<---->b. Return the flow as a grap whose edges are labeled with
(x,y,z)=(max capacity,current flow,residual capacity) and all edges
are of the form a<---->b
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| maxFlowgraph :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> gr a (b, b) |
| Compute the maximum flow from s to t on a graph whose edges are labeled
with x, which is the max capacity and where not all edges need to be of
the form a<---->b. Return the flow as a grap whose edges are labeled with
(y,x) = (current flow, max capacity).
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| maxFlow :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> b |
| Compute the value of a maximumflow
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