pomap / lib / ptset.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 (* * Ptset: Sets of integers implemented as Patricia trees. * Copyright (C) 2000 Jean-Christophe FILLIATRE * * This software is free software; you can redistribute it and/or * modify it under the terms of the GNU Library General Public * License version 2, as published by the Free Software Foundation. * * This software is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * * See the GNU Library General Public License version 2 for more details * (enclosed in the file LGPL). *) (*s Sets of integers implemented as Patricia trees, following Chris Okasaki and Andrew Gill's paper {\em Fast Mergeable Integer Maps} ({\tt\small http://www.cs.columbia.edu/\~{}cdo/papers.html\#ml98maps}). Patricia trees provide faster operations than standard library's module [Set], and especially very fast [union], [subset], [inter] and [diff] operations. *) (*s The idea behind Patricia trees is to build a {\em trie} on the binary digits of the elements, and to compact the representation by branching only one the relevant bits (i.e. the ones for which there is at least on element in each subtree). We implement here {\em little-endian} Patricia trees: bits are processed from least-significant to most-significant. The trie is implemented by the following type [t]. [Empty] stands for the empty trie, and [Leaf k] for the singleton [k]. (Note that [k] is the actual element.) [Branch (m,p,l,r)] represents a branching, where [p] is the prefix (from the root of the trie) and [m] is the branching bit (a power of 2). [l] and [r] contain the subsets for which the branching bit is respectively 0 and 1. Invariant: the trees [l] and [r] are not empty. *) (*i*) type elt = int (*i*) type t = | Empty | Leaf of int | Branch of int * int * t * t (*s Example: the representation of the set $\{1,4,5\}$ is $$\mathtt{Branch~(0,~1,~Leaf~4,~Branch~(1,~4,~Leaf~1,~Leaf~5))}$$ The first branching bit is the bit 0 (and the corresponding prefix is [0b0], not of use here), with $\{4\}$ on the left and $\{1,5\}$ on the right. Then the right subtree branches on bit 2 (and so has a branching value of $2^2 = 4$), with prefix [0b01 = 1]. *) (*s Empty set and singletons. *) let empty = Empty let is_empty = function Empty -> true | _ -> false let singleton k = Leaf k (*s Testing the occurrence of a value is similar to the search in a binary search tree, where the branching bit is used to select the appropriate subtree. *) let zero_bit k m = (k land m) == 0 let rec mem k = function | Empty -> false | Leaf j -> k == j | Branch (_, m, l, r) -> mem k (if zero_bit k m then l else r) (*s The following operation [join] will be used in both insertion and union. Given two non-empty trees [t0] and [t1] with longest common prefixes [p0] and [p1] respectively, which are supposed to disagree, it creates the union of [t0] and [t1]. For this, it computes the first bit [m] where [p0] and [p1] disagree and create a branching node on that bit. Depending on the value of that bit in [p0], [t0] will be the left subtree and [t1] the right one, or the converse. Computing the first branching bit of [p0] and [p1] uses a nice property of twos-complement representation of integers. *) let lowest_bit x = x land (-x) let branching_bit p0 p1 = lowest_bit (p0 lxor p1) let mask p m = p land (m-1) let join (p0,t0,p1,t1) = let m = branching_bit p0 p1 in if zero_bit p0 m then Branch (mask p0 m, m, t0, t1) else Branch (mask p0 m, m, t1, t0) (*s Then the insertion of value [k] in set [t] is easily implemented using [join]. Insertion in a singleton is just the identity or a call to [join], depending on the value of [k]. When inserting in a branching tree, we first check if the value to insert [k] matches the prefix [p]: if not, [join] will take care of creating the above branching; if so, we just insert [k] in the appropriate subtree, depending of the branching bit. *) let match_prefix k p m = (mask k m) == p let add k t = let rec ins = function | Empty -> Leaf k | Leaf j as t -> if j == k then t else join (k, Leaf k, j, t) | Branch (p,m,t0,t1) as t -> if match_prefix k p m then if zero_bit k m then Branch (p, m, ins t0, t1) else Branch (p, m, t0, ins t1) else join (k, Leaf k, p, t) in ins t (*s The code to remove an element is basically similar to the code of insertion. But since we have to maintain the invariant that both subtrees of a [Branch] node are non-empty, we use here the smart constructor'' [branch] instead of [Branch]. *) let branch = function | (_,_,Empty,t) -> t | (_,_,t,Empty) -> t | (p,m,t0,t1) -> Branch (p,m,t0,t1) let remove k t = let rec rmv = function | Empty -> Empty | Leaf j as t -> if k == j then Empty else t | Branch (p,m,t0,t1) as t -> if match_prefix k p m then if zero_bit k m then branch (p, m, rmv t0, t1) else branch (p, m, t0, rmv t1) else t in rmv t (*s One nice property of Patricia trees is to support a fast union operation (and also fast subset, difference and intersection operations). When merging two branching trees we examine the following four cases: (1) the trees have exactly the same prefix; (2/3) one prefix contains the other one; and (4) the prefixes disagree. In cases (1), (2) and (3) the recursion is immediate; in case (4) the function [join] creates the appropriate branching. *) let rec merge = function | Empty, t -> t | t, Empty -> t | Leaf k, t -> add k t | t, Leaf k -> add k t | (Branch (p,m,s0,s1) as s), (Branch (q,n,t0,t1) as t) -> if m == n && match_prefix q p m then (* The trees have the same prefix. Merge the subtrees. *) Branch (p, m, merge (s0,t0), merge (s1,t1)) else if m < n && match_prefix q p m then (* [q] contains [p]. Merge [t] with a subtree of [s]. *) if zero_bit q m then Branch (p, m, merge (s0,t), s1) else Branch (p, m, s0, merge (s1,t)) else if m > n && match_prefix p q n then (* [p] contains [q]. Merge [s] with a subtree of [t]. *) if zero_bit p n then Branch (q, n, merge (s,t0), t1) else Branch (q, n, t0, merge (s,t1)) else (* The prefixes disagree. *) join (p, s, q, t) let union s t = merge (s,t) (*s When checking if [s1] is a subset of [s2] only two of the above four cases are relevant: when the prefixes are the same and when the prefix of [s1] contains the one of [s2], and then the recursion is obvious. In the other two cases, the result is [false]. *) let rec subset s1 s2 = match (s1,s2) with | Empty, _ -> true | _, Empty -> false | Leaf k1, _ -> mem k1 s2 | Branch _, Leaf _ -> false | Branch (p1,m1,l1,r1), Branch (p2,m2,l2,r2) -> if m1 == m2 && p1 == p2 then subset l1 l2 && subset r1 r2 else if m1 > m2 && match_prefix p1 p2 m2 then if zero_bit p1 m2 then subset l1 l2 && subset r1 l2 else subset l1 r2 && subset r1 r2 else false let rec intersect s1 s2 = match (s1,s2) with | Empty, _ -> false | _, Empty -> false | Leaf k1, _ -> mem k1 s2 | _, Leaf k2 -> mem k2 s1 | Branch (p1,m1,l1,r1), Branch (p2,m2,l2,r2) -> if m1 == m2 && p1 == p2 then intersect l1 l2 || intersect r1 r2 else if m1 < m2 && match_prefix p2 p1 m1 then intersect (if zero_bit p2 m1 then l1 else r1) s2 else if m1 > m2 && match_prefix p1 p2 m2 then intersect s1 (if zero_bit p1 m2 then l2 else r2) else false (*s To compute the intersection and the difference of two sets, we still examine the same four cases as in [merge]. The recursion is then obvious. *) let rec inter s1 s2 = match (s1,s2) with | Empty, _ -> Empty | _, Empty -> Empty | Leaf k1, _ -> if mem k1 s2 then s1 else Empty | _, Leaf k2 -> if mem k2 s1 then s2 else Empty | Branch (p1,m1,l1,r1), Branch (p2,m2,l2,r2) -> if m1 == m2 && p1 == p2 then merge (inter l1 l2, inter r1 r2) else if m1 < m2 && match_prefix p2 p1 m1 then inter (if zero_bit p2 m1 then l1 else r1) s2 else if m1 > m2 && match_prefix p1 p2 m2 then inter s1 (if zero_bit p1 m2 then l2 else r2) else Empty let rec diff s1 s2 = match (s1,s2) with | Empty, _ -> Empty | _, Empty -> s1 | Leaf k1, _ -> if mem k1 s2 then Empty else s1 | _, Leaf k2 -> remove k2 s1 | Branch (p1,m1,l1,r1), Branch (p2,m2,l2,r2) -> if m1 == m2 && p1 == p2 then merge (diff l1 l2, diff r1 r2) else if m1 < m2 && match_prefix p2 p1 m1 then if zero_bit p2 m1 then merge (diff l1 s2, r1) else merge (l1, diff r1 s2) else if m1 > m2 && match_prefix p1 p2 m2 then if zero_bit p1 m2 then diff s1 l2 else diff s1 r2 else s1 (*s All the following operations ([cardinal], [iter], [fold], [for_all], [exists], [filter], [partition], [choose], [elements]) are implemented as for any other kind of binary trees. *) let rec cardinal = function | Empty -> 0 | Leaf _ -> 1 | Branch (_,_,t0,t1) -> cardinal t0 + cardinal t1 let rec iter f = function | Empty -> () | Leaf k -> f k | Branch (_,_,t0,t1) -> iter f t0; iter f t1 let rec fold f s accu = match s with | Empty -> accu | Leaf k -> f k accu | Branch (_,_,t0,t1) -> fold f t0 (fold f t1 accu) let rec for_all p = function | Empty -> true | Leaf k -> p k | Branch (_,_,t0,t1) -> for_all p t0 && for_all p t1 let rec exists p = function | Empty -> false | Leaf k -> p k | Branch (_,_,t0,t1) -> exists p t0 || exists p t1 let filter p s = let rec filt acc = function | Empty -> acc | Leaf k -> if p k then add k acc else acc | Branch (_,_,t0,t1) -> filt (filt acc t0) t1 in filt Empty s let partition p s = let rec part (t,f as acc) = function | Empty -> acc | Leaf k -> if p k then (add k t, f) else (t, add k f) | Branch (_,_,t0,t1) -> part (part acc t0) t1 in part (Empty, Empty) s let rec choose = function | Empty -> raise Not_found | Leaf k -> k | Branch (_, _,t0,_) -> choose t0 (* we know that [t0] is non-empty *) let split x s = let coll k (l, b, r) = if k < x then add k l, b, r else if k > x then l, b, add k r else l, true, r in fold coll s (Empty, false, Empty) let elements s = let rec elements_aux acc = function | Empty -> acc | Leaf k -> k :: acc | Branch (_,_,l,r) -> elements_aux (elements_aux acc l) r in elements_aux [] s (*s There is no way to give an efficient implementation of [min_elt] and [max_elt], as with binary search trees. The following implementation is a traversal of all elements, barely more efficient than [fold min t (choose t)] (resp. [fold max t (choose t)]). Note that we use the fact that there is no constructor [Empty] under [Branch] and therefore always a minimal (resp. maximal) element there. *) let rec min_elt = function | Empty -> raise Not_found | Leaf k -> k | Branch (_,_,s,t) -> min (min_elt s) (min_elt t) let rec max_elt = function | Empty -> raise Not_found | Leaf k -> k | Branch (_,_,s,t) -> max (max_elt s) (max_elt t) let find e t = if mem e t then e else raise Not_found (*s Another nice property of Patricia trees is to be independent of the order of insertion. As a consequence, two Patricia trees have the same elements if and only if they are structurally equal. *) let equal = (=) let compare = compare 
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