containers-0.4.0.0: Assorted concrete container types

Portability | portable
Stability | experimental
Maintainer | libraries@haskell.org

Data.Graph

Contents

- External interface
- Graphs
  - Building graphs
  - Graph properties
- Algorithms

Description

A version of the graph algorithms described in:

Lazy Depth-First Search and Linear Graph Algorithms in Haskell, by David King and John Launchbury.

Synopsis

- stronglyConnComp :: Ord key => [(node, key, [key])] -> [SCC node]
- stronglyConnCompR :: Ord key => [(node, key, [key])] -> [SCC (node, key, [key])]
- data SCC vertex
  - = AcyclicSCC vertex
  - | CyclicSCC [vertex]
- flattenSCC :: SCC vertex -> [vertex]
- flattenSCCs :: [SCC a] -> [a]
- type Graph = Table [Vertex]
- type Table a = Array Vertex a
- type Bounds = (Vertex, Vertex)
- type Edge = (Vertex, Vertex)
- type Vertex = Int
- graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)
- graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))
- buildG :: Bounds -> [Edge] -> Graph
- transposeG :: Graph -> Graph
- vertices :: Graph -> [Vertex]
- edges :: Graph -> [Edge]
- outdegree :: Graph -> Table Int
- indegree :: Graph -> Table Int
- dfs :: Graph -> [Vertex] -> Forest Vertex
- dff :: Graph -> Forest Vertex
- topSort :: Graph -> [Vertex]
- components :: Graph -> Forest Vertex
- scc :: Graph -> Forest Vertex
- bcc :: Graph -> Forest [Vertex]
- reachable :: Graph -> Vertex -> [Vertex]
- path :: Graph -> Vertex -> Vertex -> Bool
- module Data.Tree

# External interface

stronglyConnComp

Arguments

:: Ord key |
=> [(node, key, [key])] | The graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored.
-> [SCC node] |

The strongly connected components of a directed graph, topologically sorted.

stronglyConnCompR

Arguments

:: Ord key |
=> [(node, key, [key])] | The graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored.
-> [SCC (node, key, [key])] | Topologically sorted

The strongly connected components of a directed graph, topologically sorted. The function is the same as `stronglyConnComp`, except that all the information about each node retained. This interface is used when you expect to apply `SCC` to (some of) the result of `SCC`, so you don't want to lose the dependency information.

data SCC vertex

Strongly connected component.

Constructors

AcyclicSCC vertex | A single vertex that is not in any cycle.
CyclicSCC [vertex] | A maximal set of mutually reachable vertices.

flattenSCC :: SCC vertex -> [vertex]

The vertices of a strongly connected component.

flattenSCCs :: [SCC a] -> [a]

The vertices of a list of strongly connected components.

# Graphs

type Graph = Table [Vertex]

Adjacency list representation of a graph, mapping each vertex to its list of successors.

type Table a = Array Vertex a

Table indexed by a contiguous set of vertices.

type Bounds = (Vertex, Vertex)

The bounds of a `Table`.

type Edge = (Vertex, Vertex)

An edge from the first vertex to the second.

type Vertex = Int

Abstract representation of vertices.

## Building graphs

graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)

Build a graph from a list of nodes uniquely identified by keys, with a list of keys of nodes this node should have edges to. The out-list may contain keys that don't correspond to nodes of the graph; they are ignored.

graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))

Identical to `graphFromEdges`, except that the return value does not include the function which maps keys to vertices. This version of `graphFromEdges` is for backwards compatibility.

buildG :: Bounds -> [Edge] -> Graph

Build a graph from a list of edges.

transposeG :: Graph -> Graph

The graph obtained by reversing all edges.

## Graph properties

vertices :: Graph -> [Vertex]

All vertices of a graph.

edges :: Graph -> [Edge]

All edges of a graph.

outdegree :: Graph -> Table Int

A table of the count of edges from each node.

indegree :: Graph -> Table Int

A table of the count of edges into each node.

# Algorithms

dfs :: Graph -> [Vertex] -> Forest Vertex

A spanning forest of the part of the graph reachable from the listed vertices, obtained from a depth-first search of the graph starting at each of the listed vertices in order.

dff :: Graph -> Forest Vertex

A spanning forest of the graph, obtained from a depth-first search of the graph starting from each vertex in an unspecified order.

topSort :: Graph -> [Vertex]

A topological sort of the graph. The order is partially specified by the condition that a vertex i precedes j whenever j is reachable from i but not vice versa.

components :: Graph -> Forest Vertex

The connected components of a graph. Two vertices are connected if there is a path between them, traversing edges in either direction.

scc :: Graph -> Forest Vertex

The strongly connected components of a graph.

bcc :: Graph -> Forest [Vertex]

The biconnected components of a graph. An undirected graph is biconnected if the deletion of any vertex leaves it connected.

reachable :: Graph -> Vertex -> [Vertex]

A list of vertices reachable from a given vertex.

path :: Graph -> Vertex -> Vertex -> Bool

Is the second vertex reachable from the first?

module Data.Tree