-
Notifications
You must be signed in to change notification settings - Fork 1
/
Binaryset.ML
481 lines (403 loc) · 16.4 KB
/
Binaryset.ML
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
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
(* SPDX-License-Identifier: SMLNJ *)
(* Binaryset -- modified for Moscow ML
* from SML/NJ library v. 0.2
*
* COPYRIGHT (c) 1993 by AT&T Bell Laboratories.
* See file mosml/copyrght/copyrght.att for details.
*
* This code was adapted from Stephen Adams' binary tree implementation
* of applicative integer sets.
*
* Copyright 1992 Stephen Adams.
*
* This software may be used freely provided that:
* 1. This copyright notice is attached to any copy, derived work,
* or work including all or part of this software.
* 2. Any derived work must contain a prominent notice stating that
* it has been altered from the original.
*
* Name(s): Stephen Adams.
* Department, Institution: Electronics & Computer Science,
* University of Southampton
* Address: Electronics & Computer Science
* University of Southampton
* Southampton SO9 5NH
* Great Britian
* E-mail: [email protected]
*
* Comments:
*
* 1. The implementation is based on Binary search trees of Bounded
* Balance, similar to Nievergelt & Reingold, SIAM J. Computing
* 2(1), March 1973. The main advantage of these trees is that
* they keep the size of the tree in the node, giving a constant
* time size operation.
*
* 2. The bounded balance criterion is simpler than N&R's alpha.
* Simply, one subtree must not have more than `weight' times as
* many elements as the opposite subtree. Rebalancing is
* guaranteed to reinstate the criterion for weight>2.23, but
* the occasional incorrect behaviour for weight=2 is not
* detrimental to performance.
*
* 3. There are two implementations of union. The default,
* hedge_union, is much more complex and usually 20% faster. I
* am not sure that the performance increase warrants the
* complexity (and time it took to write), but I am leaving it
* in for the competition. It is derived from the original
* union by replacing the split_lt(gt) operations with a lazy
* version. The `obvious' version is called old_union.
*
* 4. Most time is spent in T', the rebalancing constructor. If my
* understanding of the output of *<file> in the sml batch
* compiler is correct then the code produced by NJSML 0.75
* (sparc) for the final case is very disappointing. Most
* invocations fall through to this case and most of these cases
* fall to the else part, i.e. the plain contructor,
* T(v,ln+rn+1,l,r). The poor code allocates a 16 word vector
* and saves lots of registers into it. In the common case it
* then retrieves a few of the registers and allocates the 5
* word T node. The values that it retrieves were live in
* registers before the massive save.
*
* Modified to functor to support general ordered values
*)
signature BINARYSET =
sig
type 'item set
exception NotFound
val empty : ('item * 'item -> order) -> 'item set
val singleton : ('item * 'item -> order) -> 'item -> 'item set
val add : 'item set * 'item -> 'item set
val addList : 'item set * 'item list -> 'item set
val retrieve : 'item set * 'item -> 'item
val peek : 'item set * 'item -> 'item option
val isEmpty : 'item set -> bool
val equal : 'item set * 'item set -> bool
val isSubset : 'item set * 'item set -> bool
val member : 'item set * 'item -> bool
val delete : 'item set * 'item -> 'item set
val numItems : 'item set -> int
val union : 'item set * 'item set -> 'item set
val intersection : 'item set * 'item set -> 'item set
val difference : 'item set * 'item set -> 'item set
val listItems : 'item set -> 'item list
val app : ('item -> unit) -> 'item set -> unit
val revapp : ('item -> unit) -> 'item set -> unit
val foldr : ('item * 'b -> 'b) -> 'b -> 'item set -> 'b
val foldl : ('item * 'b -> 'b) -> 'b -> 'item set -> 'b
val find : ('item -> bool) -> 'item set -> 'item option
end
(*
['item set] is the type of sets of ordered elements of type 'item.
The ordering relation on the elements is used in the representation
of the set. The result of combining two sets with different
underlying ordering relations is undefined. The implementation
uses ordered balanced binary trees.
[empty ordr] creates a new empty set with the given ordering
relation.
[singleton ordr i] creates the singleton set containing i, with the
given ordering relation.
[add(s, i)] adds item i to set s.
[addList(s, xs)] adds all items from the list xs to the set s.
[retrieve(s, i)] returns i if it is in s; raises NotFound otherwise.
[peek(s, i)] returns SOME i if i is in s; returns NONE otherwise.
[isEmpty s] returns true if and only if the set is empty.
[equal(s1, s2)] returns true if and only if the two sets have the
same elements.
[isSubset(s1, s2)] returns true if and only if s1 is a subset of s2.
[member(s, i)] returns true if and only if i is in s.
[delete(s, i)] removes item i from s. Raises NotFound if i is not in s.
[numItems s] returns the number of items in set s.
[union(s1, s2)] returns the union of s1 and s2.
[intersection(s1, s2)] returns the intersectionof s1 and s2.
[difference(s1, s2)] returns the difference between s1 and s2 (that
is, the set of elements in s1 but not in s2).
[listItems s] returns a list of the items in set s, in increasing
order.
[app f s] applies function f to the elements of s, in increasing
order.
[revapp f s] applies function f to the elements of s, in decreasing
order.
[foldl f e s] applies the folding function f to the entries of the
set in increasing order.
[foldr f e s] applies the folding function f to the entries of the
set in decreasing order.
[find p s] returns SOME i, where i is an item in s which satisfies
p, if one exists; otherwise returns NONE.
*)
structure Binaryset :> BINARYSET =
struct
datatype 'item set = SET of ('item * 'item -> order) * 'item tree
and 'item tree =
E
| T of {elt : 'item,
cnt : int,
left : 'item tree,
right : 'item tree}
fun treeSize E = 0
| treeSize (T{cnt,...}) = cnt
fun numItems (SET(_, t)) = treeSize t
fun isEmpty (SET(_, E)) = true
| isEmpty _ = false
fun mkT(v,n,l,r) = T{elt=v,cnt=n,left=l,right=r}
(* N(v,l,r) = T(v,1+treeSize(l)+treeSize(r),l,r) *)
fun N(v,E,E) = mkT(v,1,E,E)
| N(v,E,r as T{cnt=n,...}) = mkT(v,n+1,E,r)
| N(v,l as T{cnt=n,...}, E) = mkT(v,n+1,l,E)
| N(v,l as T{cnt=n,...}, r as T{cnt=m,...}) = mkT(v,n+m+1,l,r)
fun single_L (a,x,T{elt=b,left=y,right=z,...}) = N(b,N(a,x,y),z)
| single_L _ = raise Match
fun single_R (b,T{elt=a,left=x,right=y,...},z) = N(a,x,N(b,y,z))
| single_R _ = raise Match
fun double_L (a,w,T{elt=c,left=T{elt=b,left=x,right=y,...},right=z,...}) =
N(b,N(a,w,x),N(c,y,z))
| double_L _ = raise Match
fun double_R (c,T{elt=a,left=w,right=T{elt=b,left=x,right=y,...},...},z) =
N(b,N(a,w,x),N(c,y,z))
| double_R _ = raise Match
(*
** val weight = 3
** fun wt i = weight * i
*)
fun wt (i : int) = i + i + i
fun T' (v,E,E) = mkT(v,1,E,E)
| T' (v,E,r as T{left=E,right=E,...}) = mkT(v,2,E,r)
| T' (v,l as T{left=E,right=E,...},E) = mkT(v,2,l,E)
| T' (p as (_,E,T{left=T _,right=E,...})) = double_L p
| T' (p as (_,T{left=E,right=T _,...},E)) = double_R p
(* these cases almost never happen with small weight*)
| T' (p as (_,E,T{left=T{cnt=ln,...},right=T{cnt=rn,...},...})) =
if ln<rn then single_L p else double_L p
| T' (p as (_,T{left=T{cnt=ln,...},right=T{cnt=rn,...},...},E)) =
if ln>rn then single_R p else double_R p
| T' (p as (_,E,T{left=E,...})) = single_L p
| T' (p as (_,T{right=E,...},E)) = single_R p
| T' (p as (v,l as T{elt=lv,cnt=ln,left=ll,right=lr},
r as T{elt=rv,cnt=rn,left=rl,right=rr})) =
if rn >= wt ln (*right is too big*)
then
let val rln = treeSize rl
val rrn = treeSize rr
in
if rln < rrn then single_L p else double_L p
end
else if ln >= wt rn (*left is too big*)
then
let val lln = treeSize ll
val lrn = treeSize lr
in
if lrn < lln then single_R p else double_R p
end
else mkT(v,ln+rn+1,l,r)
fun addt cmpKey t x =
let fun h E = mkT(x,1,E,E)
| h (T{elt=v,left=l,right=r,cnt}) =
case cmpKey(x,v) of
LESS => T'(v, h l, r)
| GREATER => T'(v, l, h r)
| EQUAL => mkT(x,cnt,l,r)
in h t end
fun concat3 cmpKey E v r = addt cmpKey r v
| concat3 cmpKey l v E = addt cmpKey l v
| concat3 cmpKey (l as T{elt=v1,cnt=n1,left=l1,right=r1})
v
(r as T{elt=v2,cnt=n2,left=l2,right=r2}) =
if wt n1 < n2 then T'(v2, concat3 cmpKey l v l2, r2)
else if wt n2 < n1 then T'(v1, l1, concat3 cmpKey r1 v r)
else N(v,l,r)
fun split_lt cmpKey E x = E
| split_lt cmpKey (T{elt=v,left=l,right=r,...}) x =
case cmpKey(v,x) of
GREATER => split_lt cmpKey l x
| LESS => concat3 cmpKey l v (split_lt cmpKey r x)
| _ => l
fun split_gt cmpKey E x = E
| split_gt cmpKey (T{elt=v,left=l,right=r,...}) x =
case cmpKey(v,x) of
LESS => split_gt cmpKey r x
| GREATER => concat3 cmpKey (split_gt cmpKey l x) v r
| _ => r
fun min (T{elt=v,left=E,...}) = v
| min (T{left=l,...}) = min l
| min _ = raise Match
fun delmin (T{left=E,right=r,...}) = r
| delmin (T{elt=v,left=l,right=r,...}) = T'(v,delmin l,r)
| delmin _ = raise Match
fun delete' (E,r) = r
| delete' (l,E) = l
| delete' (l,r) = T'(min r,l,delmin r)
fun concat E s = s
| concat s E = s
| concat (t1 as T{elt=v1,cnt=n1,left=l1,right=r1})
(t2 as T{elt=v2,cnt=n2,left=l2,right=r2}) =
if wt n1 < n2 then T'(v2, concat t1 l2, r2)
else if wt n2 < n1 then T'(v1, l1, concat r1 t2)
else T'(min t2,t1, delmin t2)
fun hedge_union cmpKey s E = s
| hedge_union cmpKey E s = s
| hedge_union cmpKey (T{elt=v,left=l1,right=r1,...})
(s2 as T{elt=v2,left=l2,right=r2,...}) =
let fun trim lo hi E = E
| trim lo hi (s as T{elt=v,left=l,right=r,...}) =
if cmpKey(v,lo) = GREATER
then if cmpKey(v,hi) = LESS then s else trim lo hi l
else trim lo hi r
fun uni_bd s E _ _ = s
| uni_bd E (T{elt=v,left=l,right=r,...}) lo hi =
concat3 cmpKey (split_gt cmpKey l lo) v (split_lt cmpKey r hi)
| uni_bd (T{elt=v,left=l1,right=r1,...})
(s2 as T{elt=v2,left=l2,right=r2,...}) lo hi =
concat3 cmpKey (uni_bd l1 (trim lo v s2) lo v)
v (uni_bd r1 (trim v hi s2) v hi)
(* inv: lo < v < hi *)
(* all the other versions of uni and trim are
* specializations of the above two functions with
* lo=-infinity and/or hi=+infinity
*)
fun trim_lo _ E = E
| trim_lo lo (s as T{elt=v,right=r,...}) =
case cmpKey(v,lo) of
GREATER => s
| _ => trim_lo lo r
fun trim_hi _ E = E
| trim_hi hi (s as T{elt=v,left=l,...}) =
case cmpKey(v,hi) of
LESS => s
| _ => trim_hi hi l
fun uni_hi s E _ = s
| uni_hi E (T{elt=v,left=l,right=r,...}) hi =
concat3 cmpKey l v (split_lt cmpKey r hi)
| uni_hi (T{elt=v,left=l1,right=r1,...})
(s2 as T{elt=v2,left=l2,right=r2,...}) hi =
concat3 cmpKey (uni_hi l1 (trim_hi v s2) v)
v (uni_bd r1 (trim v hi s2) v hi)
fun uni_lo s E _ = s
| uni_lo E (T{elt=v,left=l,right=r,...}) lo =
concat3 cmpKey (split_gt cmpKey l lo) v r
| uni_lo (T{elt=v,left=l1,right=r1,...})
(s2 as T{elt=v2,left=l2,right=r2,...}) lo =
concat3 cmpKey (uni_bd l1 (trim lo v s2) lo v)
v (uni_lo r1 (trim_lo v s2) v)
in
concat3 cmpKey (uni_hi l1 (trim_hi v s2) v)
v (uni_lo r1 (trim_lo v s2) v)
end
(* The old_union version is about 20% slower than
* hedge_union in most cases
*)
fun old_union _ E s2 = s2
| old_union _ s1 E = s1
| old_union cmpKey (T{elt=v,left=l,right=r,...}) s2 =
let val l2 = split_lt cmpKey s2 v
val r2 = split_gt cmpKey s2 v
in
concat3 cmpKey (old_union cmpKey l l2) v (old_union cmpKey r r2)
end
exception NotFound
fun empty cmpKey = SET(cmpKey, E)
fun singleton cmpKey x = SET(cmpKey, T{elt=x,cnt=1,left=E,right=E})
fun addList (SET(cmpKey, t), l) =
SET(cmpKey, List.foldl (fn (i,s) => addt cmpKey s i) t l)
fun add (SET(cmpKey, t), x) = SET(cmpKey, addt cmpKey t x)
fun peekt cmpKey t x =
let fun pk E = NONE
| pk (T{elt=v,left=l,right=r,...}) =
case cmpKey(x,v) of
LESS => pk l
| GREATER => pk r
| _ => SOME v
in pk t end;
fun membert cmpKey t x =
case peekt cmpKey t x of NONE => false | _ => true
fun peek (SET(cmpKey, t), x) = peekt cmpKey t x;
fun member arg = case peek arg of NONE => false | _ => true
local
(* true if every item in t is in t' *)
fun treeIn cmpKey (t,t') =
let fun isIn E = true
| isIn (T{elt,left=E,right=E,...}) =
membert cmpKey t' elt
| isIn (T{elt,left,right=E,...}) =
membert cmpKey t' elt andalso isIn left
| isIn (T{elt,left=E,right,...}) =
membert cmpKey t' elt andalso isIn right
| isIn (T{elt,left,right,...}) =
membert cmpKey t' elt andalso isIn left andalso isIn right
in isIn t end
in
fun isSubset (SET(_, E),_) = true
| isSubset (_,SET(_, E)) = false
| isSubset (SET(cmpKey, t as T{cnt=n,...}),
SET(_, t' as T{cnt=n',...})) =
(n<=n') andalso treeIn cmpKey (t,t')
fun equal (SET(_,E), SET(_, E)) = true
| equal (SET(cmpKey, t as T{cnt=n,...}),
SET(_, t' as T{cnt=n',...})) =
(n=n') andalso treeIn cmpKey (t,t')
| equal _ = false
end
fun retrieve arg =
case peek arg of NONE => raise NotFound | SOME v => v
fun delete (SET(cmpKey, t), x) =
let fun delt E = raise NotFound
| delt (t as T{elt=v,left=l,right=r,...}) =
case cmpKey(x,v) of
LESS => T'(v, delt l, r)
| GREATER => T'(v, l, delt r)
| _ => delete'(l,r)
in SET(cmpKey, delt t) end;
fun union (SET(cmpKey, t1), SET(_, t2)) =
SET(cmpKey, hedge_union cmpKey t1 t2)
fun intersection (SET(cmpKey, t1), SET(_, t2)) =
let fun intert E _ = E
| intert _ E = E
| intert t (T{elt=v,left=l,right=r,...}) =
let val l2 = split_lt cmpKey t v
val r2 = split_gt cmpKey t v
in
case peekt cmpKey t v of
NONE => concat (intert l2 l) (intert r2 r)
| _ => concat3 cmpKey (intert l2 l) v (intert r2 r)
end
in SET(cmpKey, intert t1 t2) end
fun difference (SET(cmpKey, t1), SET(_, t2)) =
let fun difft E s = E
| difft s E = s
| difft s (T{elt=v,left=l,right=r,...}) =
let val l2 = split_lt cmpKey s v
val r2 = split_gt cmpKey s v
in
concat (difft l2 l) (difft r2 r)
end
in SET(cmpKey, difft t1 t2) end
fun foldr f b (SET(_, t)) =
let fun foldf E b = b
| foldf (T{elt,left,right,...}) b =
foldf left (f(elt, foldf right b))
in foldf t b end
fun foldl f b (SET(_, t)) =
let fun foldf E b = b
| foldf (T{elt,left,right,...}) b =
foldf right (f(elt, foldf left b))
in foldf t b end
fun listItems set = foldr (op::) [] set
fun revapp f (SET(_, t)) =
let fun apply E = ()
| apply (T{elt,left,right,...}) =
(apply right; ignore (f elt); apply left)
in apply t end
fun app f (SET(_, t)) =
let fun apply E = ()
| apply (T{elt,left,right,...}) =
(apply left; ignore (f elt); apply right)
in apply t end
fun find p (SET(_, t)) =
let fun findt E = NONE
| findt (T{elt,left,right,...}) =
if p elt then SOME elt
else case findt left of
NONE => findt right
| a => a
in findt t end
end;