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# README for incremental_cycles library
This library is vendored from
https://gitlab.inria.fr/agueneau/incremental-cycles
## Details on the vendoring process
The vendoring process is a bit involved due to the way the library is
specified upstream. In particular, it assumes a graph interface
`Raw_graph` that we have to copy by hand in Dune [see
`src/dag/dag.ml`], and in particular we have to be careful about not
altering the complexity guarantees.
## Complexity guarantees
The complexity and correctness of the implementation of
`incremental_cycles` has been mechanically-verified using the Coq
theorem prover. Note however, that for the main theorem to hold there
are a few requirements that cannot be captured by ML-level interfaces;
more concretely:
- the current specification for the algorithm requires the
`get_outgoing` function provided by the client to return a list of
all successors, and do so in constant time. This is quite demanding,
as basically requires the client to already have the list at hand.
- the main theorem for `Dag.add` does require that the vertex is not
already in the graph; otherwise the theorem doesn't apply. Thus,
clients must ensure that no duplicate edge is added to the graph.
## Dune-specific modifications
Dune uses incremental_cycles in a way that the no-duplicate-egdes
requirement is not satisfied by construction; thus, before a call to
`Dag.add` edge membership on the graph must be checked.
This is a common operation and thus should be done efficiently, thus
Dune performs the following modifications to `dag.ml`:
- we add a set of children nodes in addition to the current list
- we modify `raw_add_edge` so it updates this set, and `is_child` so
it uses the efficient membership set
The rationale for adding a duplicate children field is to actually
preserve the order the edges were added, this could be important in
other parts of the algo, see comment on `is_child` use at `memo.ml`.
For more details see discussion at https://github.com/ocaml/dune/pull/2959

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(library
(name incremental_cycles))

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include Incremental_cycles_intf
module Make (Raw_graph : Raw_graph) : S
with type graph := Raw_graph.graph
and type vertex := Raw_graph.vertex =
struct
(* This implements the algorithm of incremental cycle detection described in
Section 2 of the following paper:
A New Approach to Incremental Cycle Detection and Related Problems
Bender, M. A., Fineman, J. T., Gilbert, S., & Tarjan, R. E. (2015).
https://dl.acm.org/citation.cfm?id=2756553
*)
(******************************************************************************)
(* The implementation of the algorithm only depends on an abstract graph
structure, here implemented by [Raw_graph].
Here, [raw_graph.ml] provides a concrete implementation, but the proof
only relies on the abstract interface it implements. In the final exported
code (see the export/ directory), [Raw_graph] becomes a functor parameter.
*)
open Raw_graph
(******************************************************************************)
(* Interruptible fold_left.
At each step, the client function decides whether it wants to continue (by
using [Continue new_accumulator]) or stop (by using [Break return_value]).
Ultimately, [interruptible_fold] returns either the last accumulator or the
value returned by [Break], along with a boolean indicating whether it was
interrupted prematurely.
*)
type ('a, 'b) interruptible_fold_step =
| Continue of 'a
| Break of 'b
let rec interruptible_fold f l acc =
match l with
| [] -> Continue acc
| x :: xs ->
let res = f x acc in
match res with
| Continue acc -> interruptible_fold f xs acc
| Break _ -> res
(******************************************************************************)
(* The cycle detection algorithm, implemented as an [add_edge] function on
[graph], which either successfully inserts the edge, or reports a cycle. *)
type visit_backward_result =
| VisitBackwardCompleted
| VisitBackwardInterrupted
| VisitBackwardCyclic
(* Traverse the graph backwards from entries in [stack], looking for [target],
and marking explored vertices with [mark].
If a path to [target] is found, return [VisitBackwardCyclic].
If [fuel] runs out, return [VisitBackwardInterrupted].
Otherwise, report that the search was complete with [VisitBackwardCompleted].
*)
let rec visit_backward
(g: graph) (target: vertex) (mark: mark)
(fuel: int) (stack: vertex list):
visit_backward_result
=
(* fuel >= 0 *)
match stack with
| [] -> VisitBackwardCompleted
| vertex :: stack ->
let res = interruptible_fold (fun y (stack, fuel) ->
if fuel = 0 then
(* There is no fuel left *)
Break true
else if is_marked g y mark then
(* This vertex has already been visited, skip it *)
Continue (stack, fuel - 1)
else if vertex_eq y target then
(* A path to [target] has been found *)
Break false
else begin
set_mark g y mark;
set_parent g y vertex;
Continue (y :: stack, fuel - 1)
end
) (get_incoming g vertex) (stack, fuel)
in
match res with
| Break timeout ->
if timeout then VisitBackwardInterrupted
else (set_parent g target vertex; VisitBackwardCyclic)
| Continue (stack, fuel) ->
visit_backward g target mark fuel stack
type backward_search_result =
| BackwardForward of int * mark
| BackwardCyclic
| BackwardAcyclic
(* The whole backwards search phase (Step 2 of the algorithm). Explores the
graph backwards starting from [v], and looking for [w].
This function mainly calls [visit_backward] and does some post-processing.
If [w] is found, return [BackwardCyclic].
If [w] is not found and the algorithm should continue with the forward
search phase, return [BackwardForward (new_w_level, visited)], where
[new_w_level] is the level at which [w] needs to be put, and [visited]
is the mark of vertices that have been visited during the search.
If [w] is not found and the algorithm should directly skip to the last step,
return [BackwardAcyclic].
*)
let backward_search
(fuel: int)
(g: graph) (v: vertex) (w: vertex):
backward_search_result
=
let mark = new_mark g in
let v_level = get_level g v in
set_mark g v mark;
match visit_backward g w mark fuel [v] with
| VisitBackwardCyclic -> BackwardCyclic
| VisitBackwardInterrupted ->
(* w_level < v_level + 1 *)
BackwardForward (v_level + 1, mark)
| VisitBackwardCompleted ->
let w_level = get_level g w in
if w_level = v_level then
BackwardAcyclic
else
(* w_level < v_level *)
BackwardForward (v_level, mark)
type forward_search_result =
| ForwardCyclic of vertex * vertex
| ForwardCompleted
(* Traverse the graph forwards. [stack] contains the current working set of
vertices; these are at level [new_level] but their neighbors have not been
yet all visited.
Only follow edges that point to vertices with a smaller level, but update the
incoming edges sets for all vertices encountered.
If a vertex that has been visited during the backward search phase is
encountered, return [ForwardCyclic]. Otherwise, return [ForwardCompleted]. *)
let rec visit_forward
(g: graph) (new_level: int) (visited: mark)
(stack: vertex list):
forward_search_result
=
match stack with
| [] -> ForwardCompleted
| x :: stack ->
let res = interruptible_fold (fun y stack ->
if is_marked g y visited then
(* We found a path to a marked vertex *)
Break y
else begin
let y_level = get_level g y in
set_parent g y x;
if y_level < new_level then begin
set_level g y new_level;
clear_incoming g y;
add_incoming g y x;
Continue (y :: stack)
end else if y_level = new_level then begin
add_incoming g y x;
Continue stack
end else (* y_level > new_level *)
Continue stack
end
) (get_outgoing g x) stack
in
match res with
| Break y -> ForwardCyclic (x, y)
| Continue stack -> visit_forward g new_level visited stack
(* The whole forward search phase (Step 3 of the algorithm). Explores the
graph forwards starting from [w], updating the levels and incoming edges
sets.
This function is a simple wrapper over [visit_forward].
*)
let forward_search
(g: graph) (w: vertex) (new_w_level: int) (visited: mark):
forward_search_result
=
clear_incoming g w;
set_level g w new_w_level;
visit_forward g new_w_level visited [w]
type add_edge_result =
| EdgeAdded
| EdgeCreatesCycle of (unit -> vertex list)
let rec list_of_parents
(g: graph) (x: vertex) (y: vertex) (acc: vertex list):
vertex list
=
if vertex_eq x y then acc
else
let p = get_parent g x in
let acc' = p :: acc in
if vertex_eq p y then acc'
else list_of_parents g p y acc'
(* (z, t) is an edge of the graph such that:
- z has been visited by the forward traversal
- t has been visited by the backward traversal
So the path from w to v is of the form:
w -> ... -> z -> t -> ... -> v
[compute_cycle] returns the list of nodes in that path (including w and v).
*)
let compute_cycle (g: graph) (v: vertex) (w: vertex) (z: vertex) (t: vertex) =
list_of_parents g z w (z :: t :: List.rev (list_of_parents g t v []))
(* The core of the algorithm, wrapping up the previous phases.
This efficiently checks if there is a path from [w] to [v].
If there is none, then it adds the edge [(v, w)] to the graph. *)
let add_edge_or_detect_cycle (g: graph) (v: vertex) (w: vertex) =
let succeed () =
raw_add_edge g v w;
if get_level g v = get_level g w then
add_incoming g w v;
EdgeAdded
in
if vertex_eq v w then
EdgeCreatesCycle (fun () -> [v])
else if get_level g w > get_level g v then
(* There cannot be a path from [w] to [v], as levels form a
pseudo-lexicographic ordering: edges always go to equal or increasing
levels. *)
succeed ()
else match backward_search (get_level g v) g v w with
| BackwardCyclic ->
EdgeCreatesCycle (fun () -> w :: List.rev (list_of_parents g w v []))
| BackwardAcyclic -> succeed ()
| BackwardForward (new_level, visited) ->
match forward_search g w new_level visited with
| ForwardCyclic (z, t) ->
EdgeCreatesCycle (fun () -> compute_cycle g v w z t)
| ForwardCompleted -> succeed ()
let add_vertex (g: graph) (v: vertex) =
raw_add_vertex g v
end

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(** An incremental cycle detection algorithm for directed graphs. *)
(** {1 Functorial interface} *)
(** Signature for the graph data structure on which the algorithm operates.
It is the input signature of the [Make] functor.
This corresponds to a standard imperative directed graph structure.
Additionally, extra meta-data is associated to each node, to hold internal
data of the cycle detection algorithm. The meta-data is written and accessed
by the algorithm through the corresponding [set_*] and [get_*] functions; it
must not be modified otherwise.
The standard graph operations provided by [Raw_graph] are:
- adding a new vertex;
- adding a new edge between two existing vertices;
- returning the list of successors of a vertex;
- testing for equality of vertices.
The extra meta-data that [Raw_graph] must provide is:
- Vertices can be marked, and it must be possible to generate fresh marks.
Intuitively, a mark can be implemented as an integer, and generating a
fresh mark as incrementing some mark counter.
- Each vertex has an associated integer "level", which can be read and set.
- Each vertex has an associated list of "incoming" vertices.
- Each vertex has an associated "parent", which can be read and set to an
other vertex of the graph.
No particular assumption should be made by the implementor of [Raw_graph]
about the contents of these fields.
*)
module type Raw_graph = Incremental_cycles_intf.Raw_graph
(** Output signature of the functor [Incremental_cycles.Make]. *)
module type S = Incremental_cycles_intf.S
(** The algorithm is provided as a functor parameterized over the directed graph
implementation [Raw_graph].
NB: The algorithm does not allocate or maintain (long-lived) data itself: it
only mutates the graph by calling the operations provided by [Raw_graph].
*)
module Make (Raw_graph : Raw_graph) : S
with type graph := Raw_graph.graph
and type vertex := Raw_graph.vertex

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module type Raw_graph = sig
(** {1 Types} *)
(** The graph data structure that is modified by the operations below. *)
type graph
(** The type of vertices of the graph. *)
type vertex
(** The type of marks (each vertex has an associated mark). *)
type mark
(** {1 Standard graph operations} *)
(** NB: One must {e not} call [raw_add_edge] and [raw_add_vertex] manually, as
it would break the internal invariants of the cycle detection algorithm.
One must use instead the safe wrappers [add_vertex] and
[add_edge_or_detect_cycle] that are provided as output of the
[Incremental_cycles.Make] functor. *)
(** [vertex_eq v1 v2] tests whether vertices [v1] and [v2] are equal. *)
val vertex_eq : vertex -> vertex -> bool
(** [get_outgoing g v] returns the list of successors of [v] in the graph. *)
val get_outgoing : graph -> vertex -> vertex list
(** [raw_add_edge g v w] inserts a new (directed) arc between vertices [v] and
[w].
[v] and [w] must have been previously added to the graph using
[raw_add_vertex], and the arc [v]->[w] must not already be in the graph. *)
val raw_add_edge : graph -> vertex -> vertex -> unit
(** [raw_add_vertex g v] inserts a new vertex [v] into the graph.
- The mark of a new vertex must be some "default mark" which is different
from all marks that can be returned by [new_mark].
- The level (returned by [get_level]) of a new vertex must be [1].
- The incoming vertices (returned by [get_incoming]) of a new vertex
must be [\[\]] (the empty list).
*)
val raw_add_vertex : graph -> vertex -> unit
(** {1 Operations on graph meta-data} *)
(** [new_mark g] generates a fresh mark.
More specifically, this mark must be different from all the marks
previously returned by [new_mark g] (on the same graph [g]). It must also
be different from all the marks currently associated to vertices of the
graph [g]. *)
val new_mark : graph -> mark
(** [is_marked g v m] tests whether the vertex [v] has mark [m].
NB: [is_marked g v m] can only hold if [set_mark g v m] has been called
previously. *)
val is_marked : graph -> vertex -> mark -> bool
(** [set_mark g v m] sets the mark of vertex [v] to be [m]. *)
val set_mark : graph -> vertex -> mark -> unit
(** [get_level g v] returns the level of node [v].
It is either the value previously set by [set_level], or the default value
for a newly created vertex (i.e. [1], see [raw_add_vertex]). *)
val get_level : graph -> vertex -> int
(** [set_level g v l] sets the level of node [v] to be [l]. *)
val set_level : graph -> vertex -> int -> unit
(** [get_incoming g v] returns the list of "incoming" vertices of node [v].
It corresponds to either the default value for a newly created vertex
(i.e. the empty list, see [raw_add_vertex]), or the result of previous
calls to [clear_incoming] and [add_incoming]. *)
val get_incoming : graph -> vertex -> vertex list
(** [clear_incoming g v] sets the list of "incoming" vertices of [v] to be the
empty list. *)
val clear_incoming : graph -> vertex -> unit
(** [add_incoming g v w] adds [w] to the list of "incoming" vertices of [v]. *)
val add_incoming : graph -> vertex -> vertex -> unit
(** [get_parent g v] returns the "parent" of node [v], as set by [set_parent].
{e Note: there is no default value for [get_parent]}. It is fine for
[get_parent g v] to fail if it is called while [set_parent g v w] has not
be called beforehand. *)
val get_parent : graph -> vertex -> vertex
(** [set_parent g v w] sets the "parent" of node [v] to be [w]. *)
val set_parent : graph -> vertex -> vertex -> unit
(** {1 Asymptotic complexity} *)
(** All the operations provided by [Raw_graph] must run in constant time. *)
end
module type S = sig
(** The [graph] type from [Raw_graph].
NB: one must always start from an empty [graph], and add vertices and edges
using the functions [add_edge_or_detect_cycle] and [add_vertex] provided
below. It is {e not} safe to use these functions on a graph manually
constructed using the internal operations of [Raw_graph]. *)
type graph
(** The [vertex] type from [Raw_graph]. *)
type vertex
(** {1 Operations} *)
(** The result of [add_edge_or_detect_cycle]. *)
type add_edge_result =
| EdgeAdded
| EdgeCreatesCycle of (unit -> vertex list)
(** [add_edge_or_detect_cycle g v w] adds the edge [v]->[w] to the graph [g],
provided doing so does not make the graph cyclic.
This assumes that [v] and [w] have previously been added to the graph
using [add_vertex], and that the edge [v]->[w] is not already in the graph.
- If adding the edge does not make the graph cyclic, the function returns
[EdgeAdded], and updates the graph [g].
- If adding the edge would make the graph cyclic (i.e. there is currently
a path from [w] to [v]), the function returns [EdgeCreatesCycle
compute_cycle]. Then, [compute_cycle ()] can be called to get the list of
vertices that form a path from [w] to [v] (in linear time).
In the [EdgeCreatesCycle] case, the edge [v]->[w] is not inserted in the
graph, but the internal invariants of the graph do not hold anymore. It is
{e not} safe to call again [add_edge_or_detect_cycle] or [add_vertex] on
the graph.
*)
val add_edge_or_detect_cycle :
graph -> vertex -> vertex ->
add_edge_result
(** [add_vertex g v] adds the vertex [v] to the graph [g]. *)
val add_vertex : graph -> vertex -> unit
(** {1 Asymptotic complexity} *)
(** Inserting [n] vertices and [m] edges (using [add_edge_or_detect_cycle] and
[add_vertex]) has complexity [O(m * min(m^1/2, n^2/3) + n)].
Roughly speaking, in a sparse enough graph (for which this algorithm is
optimized), this means that each edge insertion has amortized complexity
[O(sqrt(m))] (as opposed to [O(m)] for a naive algorithm).
*)
end