# Module WeakTopological

module WeakTopological: sig .. end
Weak topological ordering of the vertices of a graph, as described by François Bourdoncle.

Weak topological ordering is an extension of topological ordering for potentially cyclic graphs.

A hierarchical ordering of a set is a well-parenthesized permutation of its elements with no consecutive (. The elements between two parentheses are called a component, and the first elements of a component is called the head. A weak topological ordering of a graph is a hierarchical ordering of its vertices such that for every edge u -> v of the graph, either u comes (strictly) before v, or v is the head of a component containing u.

One may notice that :

• For an acyclic graph, every topological ordering is also a weak topological ordering.
• For any graph with the vertices v1, ..., vN, the following trivial weak topological ordering is valid : (v1 (v2 (... (vN))...)).
Weak topological ordering are useful for fixpoint computation (see ChaoticIteration). This module aims at computing weak topological orderings which improve the precision and the convergence speed of these analyses.
Author(s): Thibault Suzanne
See also Efficient chaotic iteration strategies with widenings , François Bourdoncle, Formal Methods in Programming and their Applications, Springer Berlin Heidelberg, 1993

module type G = sig .. end
Minimal graph signature for the algorithm
type 'a element =
 | Vertex of 'a | Component of 'a * 'a t
The type of the elements of a weak topological ordering over a set of 'a.

• Vertex v represents a single vertex.
• Component (head, cs) is a component of the wto, that is a sequence of elements between two parentheses. head is the head of the component, that is the first element, which is guaranteed to be a vertex by definition. cs is the rest of the component.

type 'a t
The type of a sequence of outermost elements in a weak topological ordering. This is also the type of a weak topological ordering over a set of 'a.
val fold_left : ('a -> 'b element -> 'a) -> 'a -> 'b t -> 'a
Folding over the elements of a weak topological ordering. They are given to the accumulating function according to their order.

Note that as elements present in an ordering of type t can contain values of type t itself due to the Component variant, this function should be used by defining a recursive function f, which will call fold_left with f used to define the first parameter.

module Make:
functor (G : G) -> sig .. end