(* This is an implementation of Patricia trees, following Chris Okasaki's paper at the 1998 ML Workshop in Baltimore.
Both big-endian and little-endian trees are provided. Both sets and maps are implemented on top of Patricia
trees. *)
(*i --------------------------------------------------------------------------------------------------------------- i*)
(*s \mysection{Little-endian vs big-endian trees} *)
(* A tree is little-endian if it expects the key's least significant bits to be tested first during a search. It is
big-endian if it expects the key's most significant bits to be tested first.
Most of the code is independent of this design choice, so it is written as a functor, parameterized by a small
structure which defines endianness. Here is the interface which must be adhered to by such a structure. *)
module Endianness = struct
module type S = sig
(* A mask is an integer with a single one bit (i.e. a power of 2). *)
type mask = int
(* [branching_bit] accepts two distinct integers and returns a mask which identifies the first bit where they
differ. The meaning of ``first'' varies according to the endianness being implemented. *)
val branching_bit: int -> int -> mask
(* [mask i m] returns an integer [i'], where all bits which [m] says are relevant are identical to those in [i],
and all others are set to some unspecified, but fixed value. Which bits are ``relevant'' according to a given
mask varies according to the endianness being implemented. *)
val mask: int -> mask -> int
(* [shorter m1 m2] returns [true] if and only if [m1] describes a shorter prefix than [m2], i.e. if it makes fewer
bits relevant. Which bits are ``relevant'' according to a given mask varies according to the endianness being
implemented. *)
val shorter: mask -> mask -> bool
end
(* Now, let us define [Little] and [Big], two possible [Endiannness] choices. *)
module Little = struct
type mask = int
let lowest_bit x =
x land (-x)
(* Performing a logical ``xor'' of [i0] and [i1] yields a bit field where all differences between [i0] and [i1]
show up as one bits. (There must be at least one, since [i0] and [i1] are distinct.) The ``first'' one is
the lowest bit in this bit field, since we are checking least significant bits first. *)
let branching_bit i0 i1 =
lowest_bit (i0 lxor i1)
(* The ``relevant'' bits in an integer [i] are those which are found (strictly) to the right of the single one bit
in the mask [m]. We keep these bits, and set all others to 0. *)
let mask i m =
i land (m-1)
(* The smaller [m] is, the fewer bits are relevant. *)
let shorter =
(<)
end
module Big = struct
type mask = int
let lowest_bit x =
x land (-x)
let rec highest_bit x =
let m = lowest_bit x in
if x = m then
m
else
highest_bit (x - m)
(* Performing a logical ``xor'' of [i0] and [i1] yields a bit field where all differences between [i0] and [i1]
show up as one bits. (There must be at least one, since [i0] and [i1] are distinct.) The ``first'' one is
the highest bit in this bit field, since we are checking most significant bits first.
In Okasaki's paper, this loop is sped up by computing a conservative initial guess. Indeed, the bit at which
the two prefixes disagree must be somewhere within the shorter prefix, so we can begin searching at the
least-significant valid bit in the shorter prefix. Unfortunately, to allow computing the initial guess, the
main code has to pass in additional parameters, e.g. a mask which describes the length of each prefix. This
``pollutes'' the endianness-independent code. For this reason, this optimization isn't implemented here. *)
let branching_bit i0 i1 =
highest_bit (i0 lxor i1)
(* The ``relevant'' bits in an integer [i] are those which are found (strictly) to the left of the single one bit
in the mask [m]. We keep these bits, and set all others to 0. Okasaki uses a different convention, which allows
big-endian Patricia trees to masquerade as binary search trees. This feature does not seem to be useful here. *)
let mask i m =
i land (lnot (2*m-1))
(* The smaller [m] is, the more bits are relevant. *)
let shorter =
(>)
end
end
(*i --------------------------------------------------------------------------------------------------------------- i*)
(*s \mysection{Patricia-tree-based maps} *)
module Make (X : Endianness.S) = struct
(* Patricia trees are maps whose keys are integers. *)
type key = int
(* A tree is either empty, or a leaf node, containing both the integer key and a piece of data, or a binary node.
Each binary node carries two integers. The first one is the longest common prefix of all keys in this
sub-tree. The second integer is the branching bit. It is an integer with a single one bit (i.e. a power of 2),
which describes the bit being tested at this node. *)
type 'a t =
| Empty
| Leaf of int * 'a
| Branch of int * X.mask * 'a t * 'a t
(* The empty map. *)
let empty =
Empty
(* [choose m] returns an arbitrarily chosen binding in [m], if [m]
is nonempty, and raises [Not_found] otherwise. *)
let rec choose = function
| Empty ->
raise Not_found
| Leaf (key, data) ->
key, data
| Branch (_, _, tree0, _) ->
choose tree0
(* [lookup k m] looks up the value associated to the key [k] in the map [m], and raises [Not_found] if no value is
bound to [k].
This implementation takes branches \emph{without} checking whether the key matches the prefix found at the
current node. This means that a query for a non-existent key shall be detected only when finally reaching
a leaf, rather than higher up in the tree. This strategy is better when (most) queries are expected to be
successful. *)
let rec lookup key = function
| Empty ->
raise Not_found
| Leaf (key', data) ->
if key = key' then
data
else
raise Not_found
| Branch (_, mask, tree0, tree1) ->
lookup key (if (key land mask) = 0 then tree0 else tree1)
let find =
lookup
(* [mem k m] tells whether the key [k] appears in the domain of the
map [m]. *)
let mem k m =
try
let _ = lookup k m in
true
with Not_found ->
false
(* The auxiliary function [join] merges two trees in the simple case where their prefixes disagree.
Assume $t_0$ and $t_1$ are non-empty trees, with longest common prefixes $p_0$ and $p_1$, respectively. Further,
suppose that $p_0$ and $p_1$ disagree, that is, neither prefix is contained in the other. Then, no matter how
large $t_0$ and $t_1$ are, we can merge them simply by creating a new [Branch] node that has $t_0$ and $t_1$
as children! *)
let join p0 t0 p1 t1 =
let m = X.branching_bit p0 p1 in
let p = X.mask p0 (* for instance *) m in
if (p0 land m) = 0 then
Branch(p, m, t0, t1)
else
Branch(p, m, t1, t0)
(* The auxiliary function [match_prefix] tells whether a given key has a given prefix. More specifically,
[match_prefix k p m] returns [true] if and only if the key [k] has prefix [p] up to bit [m].
Throughout our implementation of Patricia trees, prefixes are assumed to be in normal form, i.e. their
irrelevant bits are set to some predictable value. Formally, we assume [X.mask p m] equals [p] whenever [p]
is a prefix with [m] relevant bits. This allows implementing [match_prefix] using only one call to
[X.mask]. On the other hand, this requires normalizing prefixes, as done e.g. in [join] above, where
[X.mask p0 m] has to be used instead of [p0]. *)
let match_prefix k p m =
X.mask k m = p
(* [fine_add decide k d m] returns a map whose bindings are all bindings in [m], plus a binding of the key [k] to
the datum [d]. If a binding from [k] to [d0] already exists, then the resulting map contains a binding from
[k] to [decide d0 d]. *)
type 'a decision = 'a -> 'a -> 'a
exception Unchanged
let basic_add decide k d m =
let rec add t =
match t with
| Empty ->
Leaf (k, d)
| Leaf (k0, d0) ->
if k = k0 then
let d' = decide d0 d in
if d' == d0 then
raise Unchanged
else
Leaf (k, d')
else
join k (Leaf (k, d)) k0 t
| Branch (p, m, t0, t1) ->
if match_prefix k p m then
if (k land m) = 0 then Branch (p, m, add t0, t1)
else Branch (p, m, t0, add t1)
else
join k (Leaf (k, d)) p t in
add m
let strict_add k d m =
basic_add (fun _ _ -> raise Unchanged) k d m
let fine_add decide k d m =
try
basic_add decide k d m
with Unchanged ->
m
(* [add k d m] returns a map whose bindings are all bindings in [m], plus a binding of the key [k] to the datum
[d]. If a binding already exists for [k], it is overridden. *)
let add k d m =
fine_add (fun _old_binding new_binding -> new_binding) k d m
(* [singleton k d] returns a map whose only binding is from [k] to [d]. *)
let singleton k d =
Leaf (k, d)
(* [is_singleton m] returns [Some (k, d)] if [m] is a singleton map
that maps [k] to [d]. Otherwise, it returns [None]. *)
let is_singleton = function
| Leaf (k, d) ->
Some (k, d)
| Empty
| Branch _ ->
None
(* [is_empty m] returns [true] if and only if the map [m] defines no bindings at all. *)
let is_empty = function
| Empty ->
true
| Leaf _
| Branch _ ->
false
(* [cardinal m] returns [m]'s cardinal, that is, the number of keys it binds, or, in other words, its domain's
cardinal. *)
let rec cardinal = function
| Empty ->
0
| Leaf _ ->
1
| Branch (_, _, t0, t1) ->
cardinal t0 + cardinal t1
(* [remove k m] returns the map [m] deprived from any binding involving [k]. *)
let remove key m =
let rec remove = function
| Empty ->
raise Not_found
| Leaf (key', _) ->
if key = key' then
Empty
else
raise Not_found
| Branch (prefix, mask, tree0, tree1) ->
if (key land mask) = 0 then
match remove tree0 with
| Empty ->
tree1
| tree0 ->
Branch (prefix, mask, tree0, tree1)
else
match remove tree1 with
| Empty ->
tree0
| tree1 ->
Branch (prefix, mask, tree0, tree1) in
try
remove m
with Not_found ->
m
(* [lookup_and_remove k m] looks up the value [v] associated to the key [k] in the map [m], and raises [Not_found]
if no value is bound to [k]. The call returns the value [v], together with the map [m] deprived from the binding
from [k] to [v]. *)
let rec lookup_and_remove key = function
| Empty ->
raise Not_found
| Leaf (key', data) ->
if key = key' then
data, Empty
else
raise Not_found
| Branch (prefix, mask, tree0, tree1) ->
if (key land mask) = 0 then
match lookup_and_remove key tree0 with
| data, Empty ->
data, tree1
| data, tree0 ->
data, Branch (prefix, mask, tree0, tree1)
else
match lookup_and_remove key tree1 with
| data, Empty ->
data, tree0
| data, tree1 ->
data, Branch (prefix, mask, tree0, tree1)
let find_and_remove =
lookup_and_remove
(* [fine_union decide m1 m2] returns the union of the maps [m1] and
[m2]. If a key [k] is bound to [x1] (resp. [x2]) within [m1]
(resp. [m2]), then [decide] is called. It is passed [x1] and
[x2], and must return the value which shall be bound to [k] in
the final map. The operation returns [m2] itself (as opposed to a
copy of it) when its result is equal to [m2]. *)
let reverse decision elem1 elem2 =
decision elem2 elem1
let fine_union decide m1 m2 =
let rec union s t =
match s, t with
| Empty, _ ->
t
| (Leaf _ | Branch _), Empty ->
s
| Leaf(key, value), _ ->
fine_add (reverse decide) key value t
| Branch _, Leaf(key, value) ->
fine_add decide key value s
| Branch(p, m, s0, s1), Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then
(* The trees have the same prefix. Merge their sub-trees. *)
let u0 = union s0 t0
and u1 = union s1 t1 in
if t0 == u0 && t1 == u1 then t
else Branch(p, m, u0, u1)
else if (X.shorter m n) && (match_prefix q p m) then
(* [q] contains [p]. Merge [t] with a sub-tree of [s]. *)
if (q land m) = 0 then
Branch(p, m, union s0 t, s1)
else
Branch(p, m, s0, union s1 t)
else if (X.shorter n m) && (match_prefix p q n) then
(* [p] contains [q]. Merge [s] with a sub-tree of [t]. *)
if (p land n) = 0 then
let u0 = union s t0 in
if t0 == u0 then t
else Branch(q, n, u0, t1)
else
let u1 = union s t1 in
if t1 == u1 then t
else Branch(q, n, t0, u1)
else
(* The prefixes disagree. *)
join p s q t in
union m1 m2
(* [union m1 m2] returns the union of the maps [m1] and
[m2]. Bindings in [m2] take precedence over those in [m1]. *)
let union m1 m2 =
fine_union (fun _d d' -> d') m1 m2
(* [iter f m] invokes [f k x], in turn, for each binding from key [k] to element [x] in the map [m]. Keys are
presented to [f] according to some unspecified, but fixed, order. *)
let rec iter f = function
| Empty ->
()
| Leaf (key, data) ->
f key data
| Branch (_, _, tree0, tree1) ->
iter f tree0;
iter f tree1
(* [fold f m seed] invokes [f k d accu], in turn, for each binding from key [k] to datum [d] in the map
[m]. Keys are presented to [f] in increasing order according to the map's ordering. The initial value of
[accu] is [seed]; then, at each new call, its value is the value returned by the previous invocation of [f]. The
value returned by [fold] is the final value of [accu]. *)
let rec fold f m accu =
match m with
| Empty ->
accu
| Leaf (key, data) ->
f key data accu
| Branch (_, _, tree0, tree1) ->
fold f tree1 (fold f tree0 accu)
(* [fold_rev] performs exactly the same job as [fold], but presents keys to [f] in the opposite order. *)
let rec fold_rev f m accu =
match m with
| Empty ->
accu
| Leaf (key, data) ->
f key data accu
| Branch (_, _, tree0, tree1) ->
fold_rev f tree0 (fold_rev f tree1 accu)
(* It is valid to evaluate [iter2 f m1 m2] if and only if [m1] and [m2] have the same domain. Doing so invokes
[f k x1 x2], in turn, for each key [k] bound to [x1] in [m1] and to [x2] in [m2]. Bindings are presented to [f]
according to some unspecified, but fixed, order. *)
let rec iter2 f t1 t2 =
match t1, t2 with
| Empty, Empty ->
()
| Leaf (key1, data1), Leaf (key2, data2) ->
assert (key1 = key2);
f key1 (* for instance *) data1 data2
| Branch (p1, m1, left1, right1), Branch (p2, m2, left2, right2) ->
assert (p1 = p2);
assert (m1 = m2);
iter2 f left1 left2;
iter2 f right1 right2
| _, _ ->
assert false
(* [map f m] returns the map obtained by composing the map [m] with the function [f]; that is, the map
$k\mapsto f(m(k))$. *)
let rec map f = function
| Empty ->
Empty
| Leaf (key, data) ->
Leaf(key, f data)
| Branch (p, m, tree0, tree1) ->
Branch (p, m, map f tree0, map f tree1)
(* [endo_map] is similar to [map], but attempts to physically share its result with its input. This saves
memory when [f] is the identity function. *)
let rec endo_map f tree =
match tree with
| Empty ->
tree
| Leaf (key, data) ->
let data' = f data in
if data == data' then
tree
else
Leaf(key, data')
| Branch (p, m, tree0, tree1) ->
let tree0' = endo_map f tree0 in
let tree1' = endo_map f tree1 in
if (tree0' == tree0) && (tree1' == tree1) then
tree
else
Branch (p, m, tree0', tree1')
(* [filter f m] returns a copy of the map [m] where only the bindings
that satisfy [f] have been retained. *)
let filter f m =
fold (fun key data accu ->
if f key data then
add key data accu
else
accu
) m empty
(* [iterator m] returns a stateful iterator over the map [m]. *)
(* TEMPORARY performance could be improved, see JCF's paper *)
let iterator m =
let remainder = ref [ m ] in
let rec next () =
match !remainder with
| [] ->
None
| Empty :: parent ->
remainder := parent;
next()
| (Leaf (key, data)) :: parent ->
remainder := parent;
Some (key, data)
| (Branch(_, _, s0, s1)) :: parent ->
remainder := s0 :: s1 :: parent;
next () in
next
(* If [dcompare] is an ordering over data, then [compare dcompare]
is an ordering over maps. *)
exception Got of int
let compare dcompare m1 m2 =
let iterator2 = iterator m2 in
try
iter (fun key1 data1 ->
match iterator2() with
| None ->
raise (Got 1)
| Some (key2, data2) ->
let c = Pervasives.compare key1 key2 in
if c <> 0 then
raise (Got c)
else
let c = dcompare data1 data2 in
if c <> 0 then
raise (Got c)
) m1;
match iterator2() with
| None ->
0
| Some _ ->
-1
with Got c ->
c
(*i --------------------------------------------------------------------------------------------------------------- i*)
(*s \mysection{Patricia-tree-based sets} *)
(* To enhance code sharing, it would be possible to implement maps as sets of pairs, or (vice-versa) to implement
sets as maps to the unit element. However, both possibilities introduce some space and time inefficiency. To
avoid it, we define each structure separately. *)
module Domain = struct
type element = int
type t =
| Empty
| Leaf of int
| Branch of int * X.mask * t * t
(* The empty set. *)
let empty =
Empty
(* [is_empty s] returns [true] if and only if the set [s] is empty. *)
let is_empty = function
| Empty ->
true
| Leaf _
| Branch _ ->
false
(* [singleton x] returns a set whose only element is [x]. *)
let singleton x =
Leaf x
(* [is_singleton s] tests whether [s] is a singleton set. *)
let is_singleton = function
| Leaf _ ->
true
| Empty
| Branch _ ->
false
(* [choose s] returns an arbitrarily chosen element of [s], if [s]
is nonempty, and raises [Not_found] otherwise. *)
let rec choose = function
| Empty ->
raise Not_found
| Leaf x ->
x
| Branch (_, _, tree0, _) ->
choose tree0
(* [cardinal s] returns [s]'s cardinal. *)
let rec cardinal = function
| Empty ->
0
| Leaf _ ->
1
| Branch (_, _, t0, t1) ->
cardinal t0 + cardinal t1
(* [mem x s] returns [true] if and only if [x] appears in the set [s]. *)
let rec mem x = function
| Empty ->
false
| Leaf x' ->
x = x'
| Branch (_, mask, tree0, tree1) ->
mem x (if (x land mask) = 0 then tree0 else tree1)
(* The auxiliary function [join] merges two trees in the simple case where their prefixes disagree. *)
let join p0 t0 p1 t1 =
let m = X.branching_bit p0 p1 in
let p = X.mask p0 (* for instance *) m in
if (p0 land m) = 0 then
Branch(p, m, t0, t1)
else
Branch(p, m, t1, t0)
(* [add x s] returns a set whose elements are all elements of [s], plus [x]. *)
exception Unchanged
let rec strict_add x t =
match t with
| Empty ->
Leaf x
| Leaf x0 ->
if x = x0 then
raise Unchanged
else
join x (Leaf x) x0 t
| Branch (p, m, t0, t1) ->
if match_prefix x p m then
if (x land m) = 0 then Branch (p, m, strict_add x t0, t1)
else Branch (p, m, t0, strict_add x t1)
else
join x (Leaf x) p t
let add x s =
try
strict_add x s
with Unchanged ->
s
(* [remove x s] returns a set whose elements are all elements of [s], except [x]. *)
let remove x s =
let rec strict_remove = function
| Empty ->
raise Not_found
| Leaf x' ->
if x = x' then
Empty
else
raise Not_found
| Branch (prefix, mask, tree0, tree1) ->
if (x land mask) = 0 then
match strict_remove tree0 with
| Empty ->
tree1
| tree0 ->
Branch (prefix, mask, tree0, tree1)
else
match strict_remove tree1 with
| Empty ->
tree0
| tree1 ->
Branch (prefix, mask, tree0, tree1) in
try
strict_remove s
with Not_found ->
s
(* [union s1 s2] returns the union of the sets [s1] and [s2]. *)
let rec union s t =
match s, t with
| Empty, _ ->
t
| _, Empty ->
s
| Leaf x, _ ->
add x t
| _, Leaf x ->
add x s
| Branch(p, m, s0, s1), Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then
(* The trees have the same prefix. Merge their sub-trees. *)
let u0 = union s0 t0
and u1 = union s1 t1 in
if t0 == u0 && t1 == u1 then t
else Branch(p, m, u0, u1)
else if (X.shorter m n) && (match_prefix q p m) then
(* [q] contains [p]. Merge [t] with a sub-tree of [s]. *)
if (q land m) = 0 then
Branch(p, m, union s0 t, s1)
else
Branch(p, m, s0, union s1 t)
else if (X.shorter n m) && (match_prefix p q n) then
(* [p] contains [q]. Merge [s] with a sub-tree of [t]. *)
if (p land n) = 0 then
let u0 = union s t0 in
if t0 == u0 then t
else Branch(q, n, u0, t1)
else
let u1 = union s t1 in
if t1 == u1 then t
else Branch(q, n, t0, u1)
else
(* The prefixes disagree. *)
join p s q t
(* [build] is a ``smart constructor''. It builds a [Branch] node with the specified arguments, but ensures
that the newly created node does not have an [Empty] child. *)
let build p m t0 t1 =
match t0, t1 with
| Empty, Empty ->
Empty
| Empty, _ ->
t1
| _, Empty ->
t0
| _, _ ->
Branch(p, m, t0, t1)
(* [inter s t] returns the set intersection of [s] and [t], that is, $s\cap t$. *)
let rec inter s t =
match s, t with
| Empty, _
| _, Empty ->
Empty
| (Leaf x as s), t
| t, (Leaf x as s) ->
if mem x t then s else Empty
| Branch(p, m, s0, s1), Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then
(* The trees have the same prefix. Compute the intersections of their sub-trees. *)
build p m (inter s0 t0) (inter s1 t1)
else if (X.shorter m n) && (match_prefix q p m) then
(* [q] contains [p]. Intersect [t] with a sub-tree of [s]. *)
inter (if (q land m) = 0 then s0 else s1) t
else if (X.shorter n m) && (match_prefix p q n) then
(* [p] contains [q]. Intersect [s] with a sub-tree of [t]. *)
inter s (if (p land n) = 0 then t0 else t1)
else
(* The prefixes disagree. *)
Empty
(* [disjoint s1 s2] returns [true] if and only if the sets [s1] and [s2] are disjoint, i.e. iff their intersection
is empty. It is a specialized version of [inter], which uses less space. *)
exception NotDisjoint
let disjoint s t =
let rec inter s t =
match s, t with
| Empty, _
| _, Empty ->
()
| Leaf x, _ ->
if mem x t then
raise NotDisjoint
| _, Leaf x ->
if mem x s then
raise NotDisjoint
| Branch(p, m, s0, s1), Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then begin
inter s0 t0;
inter s1 t1
end
else if (X.shorter m n) && (match_prefix q p m) then
inter (if (q land m) = 0 then s0 else s1) t
else if (X.shorter n m) && (match_prefix p q n) then
inter s (if (p land n) = 0 then t0 else t1)
else
() in
try
inter s t;
true
with NotDisjoint ->
false
(* [iter f s] invokes [f x], in turn, for each element [x] of the set [s]. Elements are presented to [f] according
to some unspecified, but fixed, order. *)
let rec iter f = function
| Empty ->
()
| Leaf x ->
f x
| Branch (_, _, tree0, tree1) ->
iter f tree0;
iter f tree1
(* [fold f s seed] invokes [f x accu], in turn, for each element [x] of the set [s]. Elements are presented to [f]
according to some unspecified, but fixed, order. The initial value of [accu] is [seed]; then, at each new call,
its value is the value returned by the previous invocation of [f]. The value returned by [fold] is the final
value of [accu]. In other words, if $s = \{ x_1, x_2, \ldots, x_n \}$, where $x_1 < x_2 < \ldots < x_n$, then
[fold f s seed] computes $([f]\,x_n\,\ldots\,([f]\,x_2\,([f]\,x_1\,[seed]))\ldots)$. *)
let rec fold f s accu =
match s with
| Empty ->
accu
| Leaf x ->
f x accu
| Branch (_, _, s0, s1) ->
fold f s1 (fold f s0 accu)
(* [elements s] is a list of all elements in the set [s]. *)
let elements s =
fold (fun tl hd -> tl :: hd) s []
(* [iterator s] returns a stateful iterator over the set [s]. That is, if $s = \{ x_1, x_2, \ldots, x_n \}$, where
$x_1 < x_2 < \ldots < x_n$, then [iterator s] is a function which, when invoked for the $k^{\text{th}}$ time,
returns [Some]$x_k$, if $k\leq n$, and [None] otherwise. Such a function can be useful when one wishes to
iterate over a set's elements, without being restricted by the call stack's discipline.
For more comments about this algorithm, please see module [Baltree], which defines a similar one. *)
let iterator s =
let remainder = ref [ s ] in
let rec next () =
match !remainder with
| [] ->
None
| Empty :: parent ->
remainder := parent;
next()
| (Leaf x) :: parent ->
remainder := parent;
Some x
| (Branch(_, _, s0, s1)) :: parent ->
remainder := s0 :: s1 :: parent;
next () in
next
(* [compare] is an ordering over sets. *)
exception Got of int
let compare s1 s2 =
let iterator2 = iterator s2 in
try
iter (fun x1 ->
match iterator2() with
| None ->
raise (Got 1)
| Some x2 ->
let c = Pervasives.compare x1 x2 in
if c <> 0 then
raise (Got c)
) s1;
match iterator2() with
| None ->
0
| Some _ ->
-1
with Got c ->
c
(* [equal] implements equality over sets. *)
let equal s1 s2 =
compare s1 s2 = 0
(* [subset] implements the subset predicate over sets. In other words, [subset s t] returns [true] if and only if
$s\subseteq t$. It is a specialized version of [diff]. *)
exception NotSubset
let subset s t =
let rec diff s t =
match s, t with
| Empty, _ ->
()
| _, Empty
| Branch _, Leaf _ ->
raise NotSubset
| Leaf x, _ ->
if not (mem x t) then
raise NotSubset
| Branch(p, m, s0, s1), Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then begin
diff s0 t0;
diff s1 t1
end
else if (X.shorter n m) && (match_prefix p q n) then
diff s (if (p land n) = 0 then t0 else t1)
else
(* Either [q] contains [p], which means at least one of [s]'s sub-trees is not contained within [t],
or the prefixes disagree. In either case, the subset relationship cannot possibly hold. *)
raise NotSubset in
try
diff s t;
true
with NotSubset ->
false
end
(*i --------------------------------------------------------------------------------------------------------------- i*)
(*s \mysection{Relating sets and maps} *)
(* Back to the world of maps. Let us now describe the relationship which exists between maps and their domains. *)
(* [domain m] returns [m]'s domain. *)
let rec domain = function
| Empty ->
Domain.Empty
| Leaf (k, _) ->
Domain.Leaf k
| Branch (p, m, t0, t1) ->
Domain.Branch (p, m, domain t0, domain t1)
(* [lift f s] returns the map $k\mapsto f(k)$, where $k$ ranges over a set of keys [s]. *)
let rec lift f = function
| Domain.Empty ->
Empty
| Domain.Leaf k ->
Leaf (k, f k)
| Domain.Branch (p, m, t0, t1) ->
Branch(p, m, lift f t0, lift f t1)
(* [build] is a ``smart constructor''. It builds a [Branch] node with the specified arguments, but ensures
that the newly created node does not have an [Empty] child. *)
let build p m t0 t1 =
match t0, t1 with
| Empty, Empty ->
Empty
| Empty, _ ->
t1
| _, Empty ->
t0
| _, _ ->
Branch(p, m, t0, t1)
(* [corestrict m d] performs a co-restriction of the map [m] to the domain [d]. That is, it returns the map
$k\mapsto m(k)$, where $k$ ranges over all keys bound in [m] but \emph{not} present in [d]. Its code resembles
[diff]'s. *)
let rec corestrict s t =
match s, t with
| Empty, _
| _, Domain.Empty ->
s
| Leaf (k, _), _ ->
if Domain.mem k t then Empty else s
| _, Domain.Leaf k ->
remove k s
| Branch(p, m, s0, s1), Domain.Branch(q, n, t0, t1) ->
if (p = q) && (m = n) then
build p m (corestrict s0 t0) (corestrict s1 t1)
else if (X.shorter m n) && (match_prefix q p m) then
if (q land m) = 0 then
build p m (corestrict s0 t) s1
else
build p m s0 (corestrict s1 t)
else if (X.shorter n m) && (match_prefix p q n) then
corestrict s (if (p land n) = 0 then t0 else t1)
else
s
end
(*i --------------------------------------------------------------------------------------------------------------- i*)
(*s \mysection{Instantiating the functor} *)
module Little = Make(Endianness.Little)
module Big = Make(Endianness.Big)