# Algebraic graphs

**Alga** is a library for algebraic construction and manipulation of graphs in Haskell. See
this Haskell Symposium paper and the
corresponding talk for the motivation
behind the library, the underlying theory and implementation details. There is also a
Haskell eXchange talk,
and a tutorial by Alexandre Moine.

## 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.

To represent *non-empty graphs*, we can drop the `Empty`

constructor – see module
Algebra.Graph.NonEmpty.

To represent *edge-labelled graphs*, we can switch to the following data type, as
explained in my Haskell eXchange 2018 talk:

```
data Graph e a = Empty
| Vertex a
| Connect e (Graph e a) (Graph e a)
```

Here `e`

is the type of edge labels. If `e`

is a monoid `(<+>, zero)`

then graph overlay can be recovered
as `Connect zero`

, and `<+>`

corresponds to *parallel composition* of edge labels.

## 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 here.

## Blog posts

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

## Algebraic graphs in other languages

Algebraic graphs were implemented in a few other languages, including
Agda,
F#,
Scala and
TypeScript.