Data.Graph.Inductive.Query.MaxFlow

fgl-5.5.2.3: Martin Erwig's Functional Graph Library

Safe Haskell | Safe
Language | Haskell98

Data.Graph.Inductive.Query.MaxFlow

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.

Synopsis

- getRevEdges :: Num b => [Edge] -> [LEdge b]
- augmentGraph :: (DynGraph gr, Num b) => gr a b -> gr a (b, b, b)
- updAdjList :: Num b => Adj (b, b, b) -> Node -> b -> Bool -> Adj (b, b, b)
- updateFlow :: (DynGraph gr, Num 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

# Documentation

getRevEdges :: Num b => [Edge] -> [LEdge b]

                    i                                 0
    For each edge a--->b this function returns edge b--->a .
             i
    Edges a<--->b are ignored
             j

augmentGraph :: (DynGraph gr, Num 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)

updAdjList :: Num b => Adj (b, b, b) -> Node -> b -> Bool -> Adj (b, b, b)

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.

updateFlow :: (DynGraph gr, Num 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.

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.

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

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 graph whose edges are labeled with (y,x) = (current flow, max capacity).

maxFlow :: (DynGraph gr, Num b, Ord b) => gr a b -> Node -> Node -> b

Compute the value of a maximumflow