(* This signature defines a few operations over maps of keys to nonempty sets of items. Keys and items can have distinct types, hence the name [Heterogeneous]. These maps can be used to represent directed bipartite graphs whose source vertices are keys and whose target vertices are items. Each key is mapped to the set of its successors. *) module type Heterogeneous = sig (* These are the types of keys, items, and sets of items. *) type key type item type itemset (* This is the type of maps of keys to sets of items. *) type t (* [find x m] is the item set associated with key [x] in map [m], if such an association is defined; it is the empty set otherwise. *) val find: key -> t -> itemset (* [add x is m] extends [m] with a binding of [x] to the item set [is], if [is] is nonempty. If [is] is empty, it removes [x] from [m]. *) val add: key -> itemset -> t -> t (* [update x f m] is [add x (f (find x m)) m]. *) val update: key -> (itemset -> itemset) -> t -> t (* [mkedge x i m] extends [m] with a binding of [x] to the union of the set [m x] and the singleton [i], where [m x] is taken to be empty if undefined. In terms of graphs, [mkedge x i m] extends the graph [m] with an edge of [x] to [i]. *) val mkedge: key -> item -> t -> t (* [rmedge x i m] extends [m] with a binding of [x] to the difference of the set [m x] and the singleton [i], where the binding is considered undefined if that difference is empty. In terms of graphs, [rmedge x i m] removes an edge of [x] to [i] to the graph [m]. *) val rmedge: key -> item -> t -> t (* [iter] and [fold] iterate over all edges in the graph. *) val iter: (key * item -> unit) -> t -> unit val fold: (key * item -> 'a -> 'a) -> t -> 'a -> 'a (* [pick m p] returns an arbitrary edge that satisfies predicate [p], if the graph contains one. *) val pick: t -> (key * item -> bool) -> (key * item) option end (* This functor offers an implementation of [Heterogeneous] out of standard implementations of sets and maps. *) module MakeHetero (Set : sig type elt type t val empty: t val is_empty: t -> bool val add: elt -> t -> t val remove: elt -> t -> t val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a end) (Map : sig type key type 'a t val add: key -> 'a -> 'a t -> 'a t val find: key -> 'a t -> 'a val remove: key -> 'a t -> 'a t val fold: (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b end) = struct type key = Map.key type item = Set.elt type itemset = Set.t type t = Set.t Map.t let find x m = try Map.find x m with Not_found -> Set.empty let add x is m = if Set.is_empty is then Map.remove x m else Map.add x is m let update x f m = add x (f (find x m)) m let mkedge x i m = update x (Set.add i) m let rmedge x i m = update x (Set.remove i) m let fold f m accu = Map.fold (fun source targets accu -> Set.fold (fun target accu -> f (source, target) accu ) targets accu ) m accu let iter f m = fold (fun edge () -> f edge) m () exception Picked of (key * item) let pick m p = try iter (fun edge -> if p edge then raise (Picked edge) ) m; None with Picked edge -> Some edge end (* This signature defines a few common operations over maps of keys to sets of keys -- that is, keys and items have the same type, hence the name [Homogeneous]. These maps can be used to represent general directed graphs. *) module type Homogeneous = sig include Heterogeneous (* [key] and [item] intended to be equal *) (* [mkbiedge x1 x2 m] is [mkedge x1 x2 (mkedge x2 x1 m)]. *) val mkbiedge: key -> key -> t -> t (* [rmbiedge x1 x2 m] is [rmedge x1 x2 (rmedge x2 x1 m)]. *) val rmbiedge: key -> key -> t -> t (* [reverse m] is the reverse of graph [m]. *) val reverse: t -> t (* [restrict m] is the graph obtained by keeping only the vertices that satisfy predicate [p]. *) val restrict: (key -> bool) -> t -> t end module MakeHomo (Set : sig type elt type t val empty: t val is_empty: t -> bool val add: elt -> t -> t val remove: elt -> t -> t val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a val filter: (elt -> bool) -> t -> t end) (Map : sig type key = Set.elt type 'a t val empty: 'a t val add: key -> 'a -> 'a t -> 'a t val find: key -> 'a t -> 'a val remove: key -> 'a t -> 'a t val fold: (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b end) = struct include MakeHetero(Set)(Map) let symmetric transform x1 x2 m = transform x1 x2 (transform x2 x1 m) let mkbiedge = symmetric mkedge let rmbiedge = symmetric rmedge let reverse m = Map.fold (fun source targets predecessors -> Set.fold (fun target predecessors -> (* We have a direct edge from [source] to [target]. Thus, we record the existence of a reverse edge from [target] to [source]. *) mkedge target source predecessors ) targets predecessors ) m Map.empty let restrict p m = Map.fold (fun source targets m -> if p source then let targets = Set.filter p targets in if Set.is_empty targets then m else Map.add source targets m else m ) m Map.empty end