1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** An imperative implementation of partitions via Union-Find *)
(** Paths are compressed imperatively at each lookup of a
canonical representative. Each union also modifies in-place
the partition structure.
Nota: For the moment we use Pervasive's comparison for
choosing the smallest object as representative. This could
be made more generic.
*)
module type PartitionSig = sig
(** The type of elements in the partition *)
type elt
(** A set structure over elements *)
type set
(** The type of partitions *)
type t
(** Initialise an empty partition *)
val create : unit -> t
(** Add (in place) an element in the partition, or do nothing
if the element is already in the partition. *)
val add : elt -> t -> unit
(** Find the canonical representative of an element.
Raise [not_found] if the element isn't known yet. *)
val find : elt -> t -> elt
(** Merge (in place) the equivalence classes of two elements.
This will add the elements in the partition if necessary. *)
val union : elt -> elt -> t -> unit
(** Merge (in place) the equivalence classes of many elements. *)
val union_set : set -> t -> unit
(** Listing the different components of the partition *)
val partition : t -> set list
end
module type SetS =
sig
type t
type elt
val singleton : elt -> t
val union : t -> t -> t
val choose : t -> elt
val iter : (elt -> unit) -> t -> unit
end
module type MapS =
sig
type key
type +'a t
val empty : 'a t
val find : key -> 'a t -> 'a
val add : key -> 'a -> 'a t -> 'a t
val mem : key -> 'a t -> bool
val fold : (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b
end
module Make (S:SetS)(M:MapS with type key = S.elt) = struct
type elt = S.elt
type set = S.t
type node =
| Canon of set
| Equiv of elt
type t = node ref M.t ref
let create () = ref (M.empty : node ref M.t)
let fresh x p =
let node = ref (Canon (S.singleton x)) in
p := M.add x node !p;
x, node
let rec lookup x p =
let node = M.find x !p in
match !node with
| Canon _ -> x, node
| Equiv y ->
let ((z,_) as res) = lookup y p in
if not (z == y) then node := Equiv z;
res
let add x p = if not (M.mem x !p) then ignore (fresh x p)
let find x p = fst (lookup x p)
let canonical x p = try lookup x p with Not_found -> fresh x p
let union x y p =
let ((x,_) as xcan) = canonical x p in
let ((y,_) as ycan) = canonical y p in
if x = y then ()
else
let xcan, ycan = if x < y then xcan, ycan else ycan, xcan in
let x,xnode = xcan and y,ynode = ycan in
match !xnode, !ynode with
| Canon lx, Canon ly ->
xnode := Canon (S.union lx ly);
ynode := Equiv x;
| _ -> assert false
let union_set s p =
try
let x = S.choose s in
S.iter (fun y -> union x y p) s
with Not_found -> ()
let partition p =
List.rev (M.fold
(fun x node acc -> match !node with
| Equiv _ -> acc
| Canon lx -> lx::acc)
!p [])
end
|
