algebraic-graphs: A library for algebraic graph construction and transformation

[ algebra, algorithms, data-structures, graphs, library, mit ] [ Propose Tags ]

Alga is a library for algebraic construction and manipulation of graphs in Haskell. See this paper for the motivation behind the library, the underlying theory and implementation details.

The top-level module Algebra.Graph defines the core data type Graph, which is a deep embedding of four graph construction primitives empty, vertex, overlay and connect. More conventional graph representations can be found in Algebra.Graph.AdjacencyMap and Algebra.Graph.Relation.

The type classes defined in Algebra.Graph.Class and Algebra.Graph.HigherKinded.Class can be used for polymorphic graph construction and manipulation. Also see Algebra.Graph.Fold that defines the Boehm-Berarducci encoding of algebraic graphs and provides additional flexibility for polymorphic graph manipulation.

This is an experimental library and the API will be unstable until version 1.0.0. Please consider contributing to the on-going discussions on the library API.

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Versions [faq] 0.0.1, 0.0.2, 0.0.3, 0.0.4, 0.0.5, 0.1.0, 0.1.1,, 0.2, 0.3
Dependencies array (>=0.5 && <0.8), base (>=4.9 && <5), containers (>=0.5 && <0.8) [details]
License MIT
Copyright Andrey Mokhov, 2016-2017
Author Andrey Mokhov <>, github: @snowleopard
Maintainer Andrey Mokhov <>, github: @snowleopard
Category Algebra, Algorithms, Data Structures, Graphs
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Source repo head: git clone
Uploaded by snowleopard at Sun Mar 26 23:53:06 UTC 2017
Distributions LTSHaskell:0.3, NixOS:0.3, Stackage:0.3
Downloads 2142 total (226 in the last 30 days)
Rating 2.5 (votes: 4) [estimated by rule of succession]
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Status Hackage Matrix CI
Docs available [build log]
Last success reported on 2017-03-26 [all 1 reports]




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Readme for algebraic-graphs-0.0.4

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Algebraic graphs

Hackage version Linux &amp; OS X status Windows status

Alga is a library for algebraic construction and manipulation of graphs in Haskell. See this paper for the motivation behind the library, the underlying theory and implementation details.

Main idea

Consider the following data type, which is defined in the top-level module Algebra.Graph of the library:

data Graph a = Empty | Vertex a | Overlay (Graph a) (Graph a) | Connect (Graph a) (Graph a)  

We can give the following semantics to the constructors in terms of the pair (V, E) of graph vertices and edges:

  • Empty constructs the empty graph (∅, ∅).
  • Vertex x constructs a graph containing a single vertex, i.e. ({x}, ∅).
  • Overlay x y overlays graphs (Vx, Ex) and (Vy, Ey) constructing (Vx ∪ Vy, Ex ∪ Ey).
  • Connect x y connects graphs (Vx, Ex) and (Vy, Ey) constructing (Vx ∪ Vy, Ex ∪ Ey ∪ Vx × Vy).

Alternatively, we can give an algebraic semantics to the above graph construction primitives by defining the following type class and specifying a set of laws for its instances (see module Algebra.Graph.Class):

class Graph g where
    type Vertex g
    empty   :: g
    vertex  :: Vertex g -> g
    overlay :: g -> g -> g
    connect :: g -> g -> g

The laws of the type class are remarkably similar to those of a semiring, so we use + and * as convenient shortcuts for overlay and connect, respectively:

  • (+, empty) is an idempotent commutative monoid.
  • (*, empty) is a monoid.
  • * distributes over +, that is: x * (y + z) == x * y + x * z and (x + y) * z == x * z + y * z.
  • * can be decomposed: x * y * z == x * y + x * z + y * z.

This algebraic structure corresponds to unlabelled directed graphs: every expression represents a graph, and every graph can be represented by an expression. Other types of graphs (e.g. undirected) can be obtained by modifying the above set of laws. Algebraic graphs provide a convenient, safe and powerful interface for working with graphs in Haskell, and allow the application of equational reasoning for proving the correctness of graph algorithms.

How fast is the library?

Alga can handle graphs comprising millions of vertices and billions of edges in a matter of seconds, which is fast enough for many applications. We believe there is a lot of potential for improving the performance of the library, and this is one of our top priorities. If you come across a performance issue when using the library, please let us know.

Some preliminary benchmarks can be found in doc/benchmarks.

Blog posts

The development of the library has been documented in the series of blog posts: