Line data Source code
1 : /*-------------------------------------------------------------------------
2 : *
3 : * array_selfuncs.c
4 : * Functions for selectivity estimation of array operators
5 : *
6 : * Portions Copyright (c) 1996-2023, PostgreSQL Global Development Group
7 : * Portions Copyright (c) 1994, Regents of the University of California
8 : *
9 : *
10 : * IDENTIFICATION
11 : * src/backend/utils/adt/array_selfuncs.c
12 : *
13 : *-------------------------------------------------------------------------
14 : */
15 : #include "postgres.h"
16 :
17 : #include <math.h>
18 :
19 : #include "access/htup_details.h"
20 : #include "catalog/pg_collation.h"
21 : #include "catalog/pg_operator.h"
22 : #include "catalog/pg_statistic.h"
23 : #include "utils/array.h"
24 : #include "utils/builtins.h"
25 : #include "utils/lsyscache.h"
26 : #include "utils/selfuncs.h"
27 : #include "utils/typcache.h"
28 :
29 :
30 : /* Default selectivity constant for "@>" and "<@" operators */
31 : #define DEFAULT_CONTAIN_SEL 0.005
32 :
33 : /* Default selectivity constant for "&&" operator */
34 : #define DEFAULT_OVERLAP_SEL 0.01
35 :
36 : /* Default selectivity for given operator */
37 : #define DEFAULT_SEL(operator) \
38 : ((operator) == OID_ARRAY_OVERLAP_OP ? \
39 : DEFAULT_OVERLAP_SEL : DEFAULT_CONTAIN_SEL)
40 :
41 : static Selectivity calc_arraycontsel(VariableStatData *vardata, Datum constval,
42 : Oid elemtype, Oid operator);
43 : static Selectivity mcelem_array_selec(ArrayType *array,
44 : TypeCacheEntry *typentry,
45 : Datum *mcelem, int nmcelem,
46 : float4 *numbers, int nnumbers,
47 : float4 *hist, int nhist,
48 : Oid operator);
49 : static Selectivity mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
50 : float4 *numbers, int nnumbers,
51 : Datum *array_data, int nitems,
52 : Oid operator, TypeCacheEntry *typentry);
53 : static Selectivity mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
54 : float4 *numbers, int nnumbers,
55 : Datum *array_data, int nitems,
56 : float4 *hist, int nhist,
57 : Oid operator, TypeCacheEntry *typentry);
58 : static float *calc_hist(const float4 *hist, int nhist, int n);
59 : static float *calc_distr(const float *p, int n, int m, float rest);
60 : static int floor_log2(uint32 n);
61 : static bool find_next_mcelem(Datum *mcelem, int nmcelem, Datum value,
62 : int *index, TypeCacheEntry *typentry);
63 : static int element_compare(const void *key1, const void *key2, void *arg);
64 : static int float_compare_desc(const void *key1, const void *key2);
65 :
66 :
67 : /*
68 : * scalararraysel_containment
69 : * Estimate selectivity of ScalarArrayOpExpr via array containment.
70 : *
71 : * If we have const =/<> ANY/ALL (array_var) then we can estimate the
72 : * selectivity as though this were an array containment operator,
73 : * array_var op ARRAY[const].
74 : *
75 : * scalararraysel() has already verified that the ScalarArrayOpExpr's operator
76 : * is the array element type's default equality or inequality operator, and
77 : * has aggressively simplified both inputs to constants.
78 : *
79 : * Returns selectivity (0..1), or -1 if we fail to estimate selectivity.
80 : */
81 : Selectivity
82 14948 : scalararraysel_containment(PlannerInfo *root,
83 : Node *leftop, Node *rightop,
84 : Oid elemtype, bool isEquality, bool useOr,
85 : int varRelid)
86 : {
87 : Selectivity selec;
88 : VariableStatData vardata;
89 : Datum constval;
90 : TypeCacheEntry *typentry;
91 : FmgrInfo *cmpfunc;
92 :
93 : /*
94 : * rightop must be a variable, else punt.
95 : */
96 14948 : examine_variable(root, rightop, varRelid, &vardata);
97 14948 : if (!vardata.rel)
98 : {
99 14600 : ReleaseVariableStats(vardata);
100 14600 : return -1.0;
101 : }
102 :
103 : /*
104 : * leftop must be a constant, else punt.
105 : */
106 348 : if (!IsA(leftop, Const))
107 : {
108 224 : ReleaseVariableStats(vardata);
109 224 : return -1.0;
110 : }
111 124 : if (((Const *) leftop)->constisnull)
112 : {
113 : /* qual can't succeed if null on left */
114 0 : ReleaseVariableStats(vardata);
115 0 : return (Selectivity) 0.0;
116 : }
117 124 : constval = ((Const *) leftop)->constvalue;
118 :
119 : /* Get element type's default comparison function */
120 124 : typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
121 124 : if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
122 : {
123 0 : ReleaseVariableStats(vardata);
124 0 : return -1.0;
125 : }
126 124 : cmpfunc = &typentry->cmp_proc_finfo;
127 :
128 : /*
129 : * If the operator is <>, swap ANY/ALL, then invert the result later.
130 : */
131 124 : if (!isEquality)
132 86 : useOr = !useOr;
133 :
134 : /* Get array element stats for var, if available */
135 136 : if (HeapTupleIsValid(vardata.statsTuple) &&
136 12 : statistic_proc_security_check(&vardata, cmpfunc->fn_oid))
137 12 : {
138 : Form_pg_statistic stats;
139 : AttStatsSlot sslot;
140 : AttStatsSlot hslot;
141 :
142 12 : stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
143 :
144 : /* MCELEM will be an array of same type as element */
145 12 : if (get_attstatsslot(&sslot, vardata.statsTuple,
146 : STATISTIC_KIND_MCELEM, InvalidOid,
147 : ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
148 : {
149 : /* For ALL case, also get histogram of distinct-element counts */
150 12 : if (useOr ||
151 0 : !get_attstatsslot(&hslot, vardata.statsTuple,
152 : STATISTIC_KIND_DECHIST, InvalidOid,
153 : ATTSTATSSLOT_NUMBERS))
154 12 : memset(&hslot, 0, sizeof(hslot));
155 :
156 : /*
157 : * For = ANY, estimate as var @> ARRAY[const].
158 : *
159 : * For = ALL, estimate as var <@ ARRAY[const].
160 : */
161 12 : if (useOr)
162 12 : selec = mcelem_array_contain_overlap_selec(sslot.values,
163 : sslot.nvalues,
164 : sslot.numbers,
165 : sslot.nnumbers,
166 : &constval, 1,
167 : OID_ARRAY_CONTAINS_OP,
168 : typentry);
169 : else
170 0 : selec = mcelem_array_contained_selec(sslot.values,
171 : sslot.nvalues,
172 : sslot.numbers,
173 : sslot.nnumbers,
174 : &constval, 1,
175 : hslot.numbers,
176 : hslot.nnumbers,
177 : OID_ARRAY_CONTAINED_OP,
178 : typentry);
179 :
180 12 : free_attstatsslot(&hslot);
181 12 : free_attstatsslot(&sslot);
182 : }
183 : else
184 : {
185 : /* No most-common-elements info, so do without */
186 0 : if (useOr)
187 0 : selec = mcelem_array_contain_overlap_selec(NULL, 0,
188 : NULL, 0,
189 : &constval, 1,
190 : OID_ARRAY_CONTAINS_OP,
191 : typentry);
192 : else
193 0 : selec = mcelem_array_contained_selec(NULL, 0,
194 : NULL, 0,
195 : &constval, 1,
196 : NULL, 0,
197 : OID_ARRAY_CONTAINED_OP,
198 : typentry);
199 : }
200 :
201 : /*
202 : * MCE stats count only non-null rows, so adjust for null rows.
203 : */
204 12 : selec *= (1.0 - stats->stanullfrac);
205 : }
206 : else
207 : {
208 : /* No stats at all, so do without */
209 112 : if (useOr)
210 112 : selec = mcelem_array_contain_overlap_selec(NULL, 0,
211 : NULL, 0,
212 : &constval, 1,
213 : OID_ARRAY_CONTAINS_OP,
214 : typentry);
215 : else
216 0 : selec = mcelem_array_contained_selec(NULL, 0,
217 : NULL, 0,
218 : &constval, 1,
219 : NULL, 0,
220 : OID_ARRAY_CONTAINED_OP,
221 : typentry);
222 : /* we assume no nulls here, so no stanullfrac correction */
223 : }
224 :
225 124 : ReleaseVariableStats(vardata);
226 :
227 : /*
228 : * If the operator is <>, invert the results.
229 : */
230 124 : if (!isEquality)
231 86 : selec = 1.0 - selec;
232 :
233 124 : CLAMP_PROBABILITY(selec);
234 :
235 124 : return selec;
236 : }
237 :
238 : /*
239 : * arraycontsel -- restriction selectivity for array @>, &&, <@ operators
240 : */
241 : Datum
242 956 : arraycontsel(PG_FUNCTION_ARGS)
243 : {
244 956 : PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
245 956 : Oid operator = PG_GETARG_OID(1);
246 956 : List *args = (List *) PG_GETARG_POINTER(2);
247 956 : int varRelid = PG_GETARG_INT32(3);
248 : VariableStatData vardata;
249 : Node *other;
250 : bool varonleft;
251 : Selectivity selec;
252 : Oid element_typeid;
253 :
254 : /*
255 : * If expression is not (variable op something) or (something op
256 : * variable), then punt and return a default estimate.
257 : */
258 956 : if (!get_restriction_variable(root, args, varRelid,
259 : &vardata, &other, &varonleft))
260 0 : PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
261 :
262 : /*
263 : * Can't do anything useful if the something is not a constant, either.
264 : */
265 956 : if (!IsA(other, Const))
266 : {
267 0 : ReleaseVariableStats(vardata);
268 0 : PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
269 : }
270 :
271 : /*
272 : * The "&&", "@>" and "<@" operators are strict, so we can cope with a
273 : * NULL constant right away.
274 : */
275 956 : if (((Const *) other)->constisnull)
276 : {
277 0 : ReleaseVariableStats(vardata);
278 0 : PG_RETURN_FLOAT8(0.0);
279 : }
280 :
281 : /*
282 : * If var is on the right, commute the operator, so that we can assume the
283 : * var is on the left in what follows.
284 : */
285 956 : if (!varonleft)
286 : {
287 24 : if (operator == OID_ARRAY_CONTAINS_OP)
288 0 : operator = OID_ARRAY_CONTAINED_OP;
289 24 : else if (operator == OID_ARRAY_CONTAINED_OP)
290 24 : operator = OID_ARRAY_CONTAINS_OP;
291 : }
292 :
293 : /*
294 : * OK, there's a Var and a Const we're dealing with here. We need the
295 : * Const to be an array with same element type as column, else we can't do
296 : * anything useful. (Such cases will likely fail at runtime, but here
297 : * we'd rather just return a default estimate.)
298 : */
299 956 : element_typeid = get_base_element_type(((Const *) other)->consttype);
300 1912 : if (element_typeid != InvalidOid &&
301 956 : element_typeid == get_base_element_type(vardata.vartype))
302 : {
303 956 : selec = calc_arraycontsel(&vardata, ((Const *) other)->constvalue,
304 : element_typeid, operator);
305 : }
306 : else
307 : {
308 0 : selec = DEFAULT_SEL(operator);
309 : }
310 :
311 956 : ReleaseVariableStats(vardata);
312 :
313 956 : CLAMP_PROBABILITY(selec);
314 :
315 956 : PG_RETURN_FLOAT8((float8) selec);
316 : }
317 :
318 : /*
319 : * arraycontjoinsel -- join selectivity for array @>, &&, <@ operators
320 : */
321 : Datum
322 0 : arraycontjoinsel(PG_FUNCTION_ARGS)
323 : {
324 : /* For the moment this is just a stub */
325 0 : Oid operator = PG_GETARG_OID(1);
326 :
327 0 : PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
328 : }
329 :
330 : /*
331 : * Calculate selectivity for "arraycolumn @> const", "arraycolumn && const"
332 : * or "arraycolumn <@ const" based on the statistics
333 : *
334 : * This function is mainly responsible for extracting the pg_statistic data
335 : * to be used; we then pass the problem on to mcelem_array_selec().
336 : */
337 : static Selectivity
338 956 : calc_arraycontsel(VariableStatData *vardata, Datum constval,
339 : Oid elemtype, Oid operator)
340 : {
341 : Selectivity selec;
342 : TypeCacheEntry *typentry;
343 : FmgrInfo *cmpfunc;
344 : ArrayType *array;
345 :
346 : /* Get element type's default comparison function */
347 956 : typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
348 956 : if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
349 0 : return DEFAULT_SEL(operator);
350 956 : cmpfunc = &typentry->cmp_proc_finfo;
351 :
352 : /*
353 : * The caller made sure the const is an array with same element type, so
354 : * get it now
355 : */
356 956 : array = DatumGetArrayTypeP(constval);
357 :
358 1424 : if (HeapTupleIsValid(vardata->statsTuple) &&
359 468 : statistic_proc_security_check(vardata, cmpfunc->fn_oid))
360 468 : {
361 : Form_pg_statistic stats;
362 : AttStatsSlot sslot;
363 : AttStatsSlot hslot;
364 :
365 468 : stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
366 :
367 : /* MCELEM will be an array of same type as column */
368 468 : if (get_attstatsslot(&sslot, vardata->statsTuple,
369 : STATISTIC_KIND_MCELEM, InvalidOid,
370 : ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
371 : {
372 : /*
373 : * For "array <@ const" case we also need histogram of distinct
374 : * element counts.
375 : */
376 468 : if (operator != OID_ARRAY_CONTAINED_OP ||
377 72 : !get_attstatsslot(&hslot, vardata->statsTuple,
378 : STATISTIC_KIND_DECHIST, InvalidOid,
379 : ATTSTATSSLOT_NUMBERS))
380 396 : memset(&hslot, 0, sizeof(hslot));
381 :
382 : /* Use the most-common-elements slot for the array Var. */
383 468 : selec = mcelem_array_selec(array, typentry,
384 : sslot.values, sslot.nvalues,
385 : sslot.numbers, sslot.nnumbers,
386 : hslot.numbers, hslot.nnumbers,
387 : operator);
388 :
389 468 : free_attstatsslot(&hslot);
390 468 : free_attstatsslot(&sslot);
391 : }
392 : else
393 : {
394 : /* No most-common-elements info, so do without */
395 0 : selec = mcelem_array_selec(array, typentry,
396 : NULL, 0, NULL, 0, NULL, 0,
397 : operator);
398 : }
399 :
400 : /*
401 : * MCE stats count only non-null rows, so adjust for null rows.
402 : */
403 468 : selec *= (1.0 - stats->stanullfrac);
404 : }
405 : else
406 : {
407 : /* No stats at all, so do without */
408 488 : selec = mcelem_array_selec(array, typentry,
409 : NULL, 0, NULL, 0, NULL, 0,
410 : operator);
411 : /* we assume no nulls here, so no stanullfrac correction */
412 : }
413 :
414 : /* If constant was toasted, release the copy we made */
415 956 : if (PointerGetDatum(array) != constval)
416 0 : pfree(array);
417 :
418 956 : return selec;
419 : }
420 :
421 : /*
422 : * Array selectivity estimation based on most common elements statistics
423 : *
424 : * This function just deconstructs and sorts the array constant's contents,
425 : * and then passes the problem on to mcelem_array_contain_overlap_selec or
426 : * mcelem_array_contained_selec depending on the operator.
427 : */
428 : static Selectivity
429 956 : mcelem_array_selec(ArrayType *array, TypeCacheEntry *typentry,
430 : Datum *mcelem, int nmcelem,
431 : float4 *numbers, int nnumbers,
432 : float4 *hist, int nhist,
433 : Oid operator)
434 : {
435 : Selectivity selec;
436 : int num_elems;
437 : Datum *elem_values;
438 : bool *elem_nulls;
439 : bool null_present;
440 : int nonnull_nitems;
441 : int i;
442 :
443 : /*
444 : * Prepare constant array data for sorting. Sorting lets us find unique
445 : * elements and efficiently merge with the MCELEM array.
446 : */
447 956 : deconstruct_array(array,
448 : typentry->type_id,
449 956 : typentry->typlen,
450 956 : typentry->typbyval,
451 956 : typentry->typalign,
452 : &elem_values, &elem_nulls, &num_elems);
453 :
454 : /* Collapse out any null elements */
455 956 : nonnull_nitems = 0;
456 956 : null_present = false;
457 1864 : for (i = 0; i < num_elems; i++)
458 : {
459 908 : if (elem_nulls[i])
460 36 : null_present = true;
461 : else
462 872 : elem_values[nonnull_nitems++] = elem_values[i];
463 : }
464 :
465 : /*
466 : * Query "column @> '{anything, null}'" matches nothing. For the other
467 : * two operators, presence of a null in the constant can be ignored.
468 : */
469 956 : if (null_present && operator == OID_ARRAY_CONTAINS_OP)
470 : {
471 12 : pfree(elem_values);
472 12 : pfree(elem_nulls);
473 12 : return (Selectivity) 0.0;
474 : }
475 :
476 : /* Sort extracted elements using their default comparison function. */
477 944 : qsort_arg(elem_values, nonnull_nitems, sizeof(Datum),
478 : element_compare, typentry);
479 :
480 : /* Separate cases according to operator */
481 944 : if (operator == OID_ARRAY_CONTAINS_OP || operator == OID_ARRAY_OVERLAP_OP)
482 872 : selec = mcelem_array_contain_overlap_selec(mcelem, nmcelem,
483 : numbers, nnumbers,
484 : elem_values, nonnull_nitems,
485 : operator, typentry);
486 72 : else if (operator == OID_ARRAY_CONTAINED_OP)
487 72 : selec = mcelem_array_contained_selec(mcelem, nmcelem,
488 : numbers, nnumbers,
489 : elem_values, nonnull_nitems,
490 : hist, nhist,
491 : operator, typentry);
492 : else
493 : {
494 0 : elog(ERROR, "arraycontsel called for unrecognized operator %u",
495 : operator);
496 : selec = 0.0; /* keep compiler quiet */
497 : }
498 :
499 944 : pfree(elem_values);
500 944 : pfree(elem_nulls);
501 944 : return selec;
502 : }
503 :
504 : /*
505 : * Estimate selectivity of "column @> const" and "column && const" based on
506 : * most common element statistics. This estimation assumes element
507 : * occurrences are independent.
508 : *
509 : * mcelem (of length nmcelem) and numbers (of length nnumbers) are from
510 : * the array column's MCELEM statistics slot, or are NULL/0 if stats are
511 : * not available. array_data (of length nitems) is the constant's elements.
512 : *
513 : * Both the mcelem and array_data arrays are assumed presorted according
514 : * to the element type's cmpfunc. Null elements are not present.
515 : *
516 : * TODO: this estimate probably could be improved by using the distinct
517 : * elements count histogram. For example, excepting the special case of
518 : * "column @> '{}'", we can multiply the calculated selectivity by the
519 : * fraction of nonempty arrays in the column.
520 : */
521 : static Selectivity
522 996 : mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
523 : float4 *numbers, int nnumbers,
524 : Datum *array_data, int nitems,
525 : Oid operator, TypeCacheEntry *typentry)
526 : {
527 : Selectivity selec,
528 : elem_selec;
529 : int mcelem_index,
530 : i;
531 : bool use_bsearch;
532 : float4 minfreq;
533 :
534 : /*
535 : * There should be three more Numbers than Values, because the last three
536 : * cells should hold minimal and maximal frequency among the non-null
537 : * elements, and then the frequency of null elements. Ignore the Numbers
538 : * if not right.
539 : */
540 996 : if (nnumbers != nmcelem + 3)
541 : {
542 600 : numbers = NULL;
543 600 : nnumbers = 0;
544 : }
545 :
546 996 : if (numbers)
547 : {
548 : /* Grab the lowest observed frequency */
549 396 : minfreq = numbers[nmcelem];
550 : }
551 : else
552 : {
553 : /* Without statistics make some default assumptions */
554 600 : minfreq = 2 * (float4) DEFAULT_CONTAIN_SEL;
555 : }
556 :
557 : /* Decide whether it is faster to use binary search or not. */
558 996 : if (nitems * floor_log2((uint32) nmcelem) < nmcelem + nitems)
559 798 : use_bsearch = true;
560 : else
561 198 : use_bsearch = false;
562 :
563 996 : if (operator == OID_ARRAY_CONTAINS_OP)
564 : {
565 : /*
566 : * Initial selectivity for "column @> const" query is 1.0, and it will
567 : * be decreased with each element of constant array.
568 : */
569 852 : selec = 1.0;
570 : }
571 : else
572 : {
573 : /*
574 : * Initial selectivity for "column && const" query is 0.0, and it will
575 : * be increased with each element of constant array.
576 : */
577 144 : selec = 0.0;
578 : }
579 :
580 : /* Scan mcelem and array in parallel. */
581 996 : mcelem_index = 0;
582 1836 : for (i = 0; i < nitems; i++)
583 : {
584 840 : bool match = false;
585 :
586 : /* Ignore any duplicates in the array data. */
587 960 : if (i > 0 &&
588 120 : element_compare(&array_data[i - 1], &array_data[i], typentry) == 0)
589 0 : continue;
590 :
591 : /* Find the smallest MCELEM >= this array item. */
592 840 : if (use_bsearch)
593 : {
594 840 : match = find_next_mcelem(mcelem, nmcelem, array_data[i],
595 : &mcelem_index, typentry);
596 : }
597 : else
598 : {
599 0 : while (mcelem_index < nmcelem)
600 : {
601 0 : int cmp = element_compare(&mcelem[mcelem_index],
602 0 : &array_data[i],
603 : typentry);
604 :
605 0 : if (cmp < 0)
606 0 : mcelem_index++;
607 : else
608 : {
609 0 : if (cmp == 0)
610 0 : match = true; /* mcelem is found */
611 0 : break;
612 : }
613 : }
614 : }
615 :
616 840 : if (match && numbers)
617 : {
618 : /* MCELEM matches the array item; use its frequency. */
619 408 : elem_selec = numbers[mcelem_index];
620 408 : mcelem_index++;
621 : }
622 : else
623 : {
624 : /*
625 : * The element is not in MCELEM. Punt, but assume that the
626 : * selectivity cannot be more than minfreq / 2.
627 : */
628 432 : elem_selec = Min(DEFAULT_CONTAIN_SEL, minfreq / 2);
629 : }
630 :
631 : /*
632 : * Update overall selectivity using the current element's selectivity
633 : * and an assumption of element occurrence independence.
634 : */
635 840 : if (operator == OID_ARRAY_CONTAINS_OP)
636 696 : selec *= elem_selec;
637 : else
638 144 : selec = selec + elem_selec - selec * elem_selec;
639 :
640 : /* Clamp intermediate results to stay sane despite roundoff error */
641 840 : CLAMP_PROBABILITY(selec);
642 : }
643 :
644 996 : return selec;
645 : }
646 :
647 : /*
648 : * Estimate selectivity of "column <@ const" based on most common element
649 : * statistics.
650 : *
651 : * mcelem (of length nmcelem) and numbers (of length nnumbers) are from
652 : * the array column's MCELEM statistics slot, or are NULL/0 if stats are
653 : * not available. array_data (of length nitems) is the constant's elements.
654 : * hist (of length nhist) is from the array column's DECHIST statistics slot,
655 : * or is NULL/0 if those stats are not available.
656 : *
657 : * Both the mcelem and array_data arrays are assumed presorted according
658 : * to the element type's cmpfunc. Null elements are not present.
659 : *
660 : * Independent element occurrence would imply a particular distribution of
661 : * distinct element counts among matching rows. Real data usually falsifies
662 : * that assumption. For example, in a set of 11-element integer arrays having
663 : * elements in the range [0..10], element occurrences are typically not
664 : * independent. If they were, a sufficiently-large set would include all
665 : * distinct element counts 0 through 11. We correct for this using the
666 : * histogram of distinct element counts.
667 : *
668 : * In the "column @> const" and "column && const" cases, we usually have a
669 : * "const" with low number of elements (otherwise we have selectivity close
670 : * to 0 or 1 respectively). That's why the effect of dependence related
671 : * to distinct element count distribution is negligible there. In the
672 : * "column <@ const" case, number of elements is usually high (otherwise we
673 : * have selectivity close to 0). That's why we should do a correction with
674 : * the array distinct element count distribution here.
675 : *
676 : * Using the histogram of distinct element counts produces a different
677 : * distribution law than independent occurrences of elements. This
678 : * distribution law can be described as follows:
679 : *
680 : * P(o1, o2, ..., on) = f1^o1 * (1 - f1)^(1 - o1) * f2^o2 *
681 : * (1 - f2)^(1 - o2) * ... * fn^on * (1 - fn)^(1 - on) * hist[m] / ind[m]
682 : *
683 : * where:
684 : * o1, o2, ..., on - occurrences of elements 1, 2, ..., n
685 : * (1 - occurrence, 0 - no occurrence) in row
686 : * f1, f2, ..., fn - frequencies of elements 1, 2, ..., n
687 : * (scalar values in [0..1]) according to collected statistics
688 : * m = o1 + o2 + ... + on = total number of distinct elements in row
689 : * hist[m] - histogram data for occurrence of m elements.
690 : * ind[m] - probability of m occurrences from n events assuming their
691 : * probabilities to be equal to frequencies of array elements.
692 : *
693 : * ind[m] = sum(f1^o1 * (1 - f1)^(1 - o1) * f2^o2 * (1 - f2)^(1 - o2) *
694 : * ... * fn^on * (1 - fn)^(1 - on), o1, o2, ..., on) | o1 + o2 + .. on = m
695 : */
696 : static Selectivity
697 72 : mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
698 : float4 *numbers, int nnumbers,
699 : Datum *array_data, int nitems,
700 : float4 *hist, int nhist,
701 : Oid operator, TypeCacheEntry *typentry)
702 : {
703 : int mcelem_index,
704 : i,
705 72 : unique_nitems = 0;
706 : float selec,
707 : minfreq,
708 : nullelem_freq;
709 : float *dist,
710 : *mcelem_dist,
711 : *hist_part;
712 : float avg_count,
713 : mult,
714 : rest;
715 : float *elem_selec;
716 :
717 : /*
718 : * There should be three more Numbers than Values in the MCELEM slot,
719 : * because the last three cells should hold minimal and maximal frequency
720 : * among the non-null elements, and then the frequency of null elements.
721 : * Punt if not right, because we can't do much without the element freqs.
722 : */
723 72 : if (numbers == NULL || nnumbers != nmcelem + 3)
724 0 : return DEFAULT_CONTAIN_SEL;
725 :
726 : /* Can't do much without a count histogram, either */
727 72 : if (hist == NULL || nhist < 3)
728 0 : return DEFAULT_CONTAIN_SEL;
729 :
730 : /*
731 : * Grab some of the summary statistics that compute_array_stats() stores:
732 : * lowest frequency, frequency of null elements, and average distinct
733 : * element count.
734 : */
735 72 : minfreq = numbers[nmcelem];
736 72 : nullelem_freq = numbers[nmcelem + 2];
737 72 : avg_count = hist[nhist - 1];
738 :
739 : /*
740 : * "rest" will be the sum of the frequencies of all elements not
741 : * represented in MCELEM. The average distinct element count is the sum
742 : * of the frequencies of *all* elements. Begin with that; we will proceed
743 : * to subtract the MCELEM frequencies.
744 : */
745 72 : rest = avg_count;
746 :
747 : /*
748 : * mult is a multiplier representing estimate of probability that each
749 : * mcelem that is not present in constant doesn't occur.
750 : */
751 72 : mult = 1.0f;
752 :
753 : /*
754 : * elem_selec is array of estimated frequencies for elements in the
755 : * constant.
756 : */
757 72 : elem_selec = (float *) palloc(sizeof(float) * nitems);
758 :
759 : /* Scan mcelem and array in parallel. */
760 72 : mcelem_index = 0;
761 228 : for (i = 0; i < nitems; i++)
762 : {
763 156 : bool match = false;
764 :
765 : /* Ignore any duplicates in the array data. */
766 276 : if (i > 0 &&
767 120 : element_compare(&array_data[i - 1], &array_data[i], typentry) == 0)
768 0 : continue;
769 :
770 : /*
771 : * Iterate over MCELEM until we find an entry greater than or equal to
772 : * this element of the constant. Update "rest" and "mult" for mcelem
773 : * entries skipped over.
774 : */
775 4128 : while (mcelem_index < nmcelem)
776 : {
777 4128 : int cmp = element_compare(&mcelem[mcelem_index],
778 4128 : &array_data[i],
779 : typentry);
780 :
781 4128 : if (cmp < 0)
782 : {
783 3972 : mult *= (1.0f - numbers[mcelem_index]);
784 3972 : rest -= numbers[mcelem_index];
785 3972 : mcelem_index++;
786 : }
787 : else
788 : {
789 156 : if (cmp == 0)
790 156 : match = true; /* mcelem is found */
791 156 : break;
792 : }
793 : }
794 :
795 156 : if (match)
796 : {
797 : /* MCELEM matches the array item. */
798 156 : elem_selec[unique_nitems] = numbers[mcelem_index];
799 : /* "rest" is decremented for all mcelems, matched or not */
800 156 : rest -= numbers[mcelem_index];
801 156 : mcelem_index++;
802 : }
803 : else
804 : {
805 : /*
806 : * The element is not in MCELEM. Punt, but assume that the
807 : * selectivity cannot be more than minfreq / 2.
808 : */
809 0 : elem_selec[unique_nitems] = Min(DEFAULT_CONTAIN_SEL,
810 : minfreq / 2);
811 : }
812 :
813 156 : unique_nitems++;
814 : }
815 :
816 : /*
817 : * If we handled all constant elements without exhausting the MCELEM
818 : * array, finish walking it to complete calculation of "rest" and "mult".
819 : */
820 7668 : while (mcelem_index < nmcelem)
821 : {
822 7596 : mult *= (1.0f - numbers[mcelem_index]);
823 7596 : rest -= numbers[mcelem_index];
824 7596 : mcelem_index++;
825 : }
826 :
827 : /*
828 : * The presence of many distinct rare elements materially decreases
829 : * selectivity. Use the Poisson distribution to estimate the probability
830 : * of a column value having zero occurrences of such elements. See above
831 : * for the definition of "rest".
832 : */
833 72 : mult *= exp(-rest);
834 :
835 : /*----------
836 : * Using the distinct element count histogram requires
837 : * O(unique_nitems * (nmcelem + unique_nitems))
838 : * operations. Beyond a certain computational cost threshold, it's
839 : * reasonable to sacrifice accuracy for decreased planning time. We limit
840 : * the number of operations to EFFORT * nmcelem; since nmcelem is limited
841 : * by the column's statistics target, the work done is user-controllable.
842 : *
843 : * If the number of operations would be too large, we can reduce it
844 : * without losing all accuracy by reducing unique_nitems and considering
845 : * only the most-common elements of the constant array. To make the
846 : * results exactly match what we would have gotten with only those
847 : * elements to start with, we'd have to remove any discarded elements'
848 : * frequencies from "mult", but since this is only an approximation
849 : * anyway, we don't bother with that. Therefore it's sufficient to qsort
850 : * elem_selec[] and take the largest elements. (They will no longer match
851 : * up with the elements of array_data[], but we don't care.)
852 : *----------
853 : */
854 : #define EFFORT 100
855 :
856 72 : if ((nmcelem + unique_nitems) > 0 &&
857 72 : unique_nitems > EFFORT * nmcelem / (nmcelem + unique_nitems))
858 : {
859 : /*
860 : * Use the quadratic formula to solve for largest allowable N. We
861 : * have A = 1, B = nmcelem, C = - EFFORT * nmcelem.
862 : */
863 0 : double b = (double) nmcelem;
864 : int n;
865 :
866 0 : n = (int) ((sqrt(b * b + 4 * EFFORT * b) - b) / 2);
867 :
868 : /* Sort, then take just the first n elements */
869 0 : qsort(elem_selec, unique_nitems, sizeof(float),
870 : float_compare_desc);
871 0 : unique_nitems = n;
872 : }
873 :
874 : /*
875 : * Calculate probabilities of each distinct element count for both mcelems
876 : * and constant elements. At this point, assume independent element
877 : * occurrence.
878 : */
879 72 : dist = calc_distr(elem_selec, unique_nitems, unique_nitems, 0.0f);
880 72 : mcelem_dist = calc_distr(numbers, nmcelem, unique_nitems, rest);
881 :
882 : /* ignore hist[nhist-1], which is the average not a histogram member */
883 72 : hist_part = calc_hist(hist, nhist - 1, unique_nitems);
884 :
885 72 : selec = 0.0f;
886 300 : for (i = 0; i <= unique_nitems; i++)
887 : {
888 : /*
889 : * mult * dist[i] / mcelem_dist[i] gives us probability of qual
890 : * matching from assumption of independent element occurrence with the
891 : * condition that distinct element count = i.
892 : */
893 228 : if (mcelem_dist[i] > 0)
894 228 : selec += hist_part[i] * mult * dist[i] / mcelem_dist[i];
895 : }
896 :
897 72 : pfree(dist);
898 72 : pfree(mcelem_dist);
899 72 : pfree(hist_part);
900 72 : pfree(elem_selec);
901 :
902 : /* Take into account occurrence of NULL element. */
903 72 : selec *= (1.0f - nullelem_freq);
904 :
905 72 : CLAMP_PROBABILITY(selec);
906 :
907 72 : return selec;
908 : }
909 :
910 : /*
911 : * Calculate the first n distinct element count probabilities from a
912 : * histogram of distinct element counts.
913 : *
914 : * Returns a palloc'd array of n+1 entries, with array[k] being the
915 : * probability of element count k, k in [0..n].
916 : *
917 : * We assume that a histogram box with bounds a and b gives 1 / ((b - a + 1) *
918 : * (nhist - 1)) probability to each value in (a,b) and an additional half of
919 : * that to a and b themselves.
920 : */
921 : static float *
922 72 : calc_hist(const float4 *hist, int nhist, int n)
923 : {
924 : float *hist_part;
925 : int k,
926 72 : i = 0;
927 72 : float prev_interval = 0,
928 : next_interval;
929 : float frac;
930 :
931 72 : hist_part = (float *) palloc((n + 1) * sizeof(float));
932 :
933 : /*
934 : * frac is a probability contribution for each interval between histogram
935 : * values. We have nhist - 1 intervals, so contribution of each one will
936 : * be 1 / (nhist - 1).
937 : */
938 72 : frac = 1.0f / ((float) (nhist - 1));
939 :
940 300 : for (k = 0; k <= n; k++)
941 : {
942 228 : int count = 0;
943 :
944 : /*
945 : * Count the histogram boundaries equal to k. (Although the histogram
946 : * should theoretically contain only exact integers, entries are
947 : * floats so there could be roundoff error in large values. Treat any
948 : * fractional value as equal to the next larger k.)
949 : */
950 2016 : while (i < nhist && hist[i] <= k)
951 : {
952 1788 : count++;
953 1788 : i++;
954 : }
955 :
956 228 : if (count > 0)
957 : {
958 : /* k is an exact bound for at least one histogram box. */
959 : float val;
960 :
961 : /* Find length between current histogram value and the next one */
962 216 : if (i < nhist)
963 216 : next_interval = hist[i] - hist[i - 1];
964 : else
965 0 : next_interval = 0;
966 :
967 : /*
968 : * count - 1 histogram boxes contain k exclusively. They
969 : * contribute a total of (count - 1) * frac probability. Also
970 : * factor in the partial histogram boxes on either side.
971 : */
972 216 : val = (float) (count - 1);
973 216 : if (next_interval > 0)
974 216 : val += 0.5f / next_interval;
975 216 : if (prev_interval > 0)
976 144 : val += 0.5f / prev_interval;
977 216 : hist_part[k] = frac * val;
978 :
979 216 : prev_interval = next_interval;
980 : }
981 : else
982 : {
983 : /* k does not appear as an exact histogram bound. */
984 12 : if (prev_interval > 0)
985 12 : hist_part[k] = frac / prev_interval;
986 : else
987 0 : hist_part[k] = 0.0f;
988 : }
989 : }
990 :
991 72 : return hist_part;
992 : }
993 :
994 : /*
995 : * Consider n independent events with probabilities p[]. This function
996 : * calculates probabilities of exact k of events occurrence for k in [0..m].
997 : * Returns a palloc'd array of size m+1.
998 : *
999 : * "rest" is the sum of the probabilities of all low-probability events not
1000 : * included in p.
1001 : *
1002 : * Imagine matrix M of size (n + 1) x (m + 1). Element M[i,j] denotes the
1003 : * probability that exactly j of first i events occur. Obviously M[0,0] = 1.
1004 : * For any constant j, each increment of i increases the probability iff the
1005 : * event occurs. So, by the law of total probability:
1006 : * M[i,j] = M[i - 1, j] * (1 - p[i]) + M[i - 1, j - 1] * p[i]
1007 : * for i > 0, j > 0.
1008 : * M[i,0] = M[i - 1, 0] * (1 - p[i]) for i > 0.
1009 : */
1010 : static float *
1011 144 : calc_distr(const float *p, int n, int m, float rest)
1012 : {
1013 : float *row,
1014 : *prev_row,
1015 : *tmp;
1016 : int i,
1017 : j;
1018 :
1019 : /*
1020 : * Since we return only the last row of the matrix and need only the
1021 : * current and previous row for calculations, allocate two rows.
1022 : */
1023 144 : row = (float *) palloc((m + 1) * sizeof(float));
1024 144 : prev_row = (float *) palloc((m + 1) * sizeof(float));
1025 :
1026 : /* M[0,0] = 1 */
1027 144 : row[0] = 1.0f;
1028 12024 : for (i = 1; i <= n; i++)
1029 : {
1030 11880 : float t = p[i - 1];
1031 :
1032 : /* Swap rows */
1033 11880 : tmp = row;
1034 11880 : row = prev_row;
1035 11880 : prev_row = tmp;
1036 :
1037 : /* Calculate next row */
1038 53148 : for (j = 0; j <= i && j <= m; j++)
1039 : {
1040 41268 : float val = 0.0f;
1041 :
1042 41268 : if (j < i)
1043 40956 : val += prev_row[j] * (1.0f - t);
1044 41268 : if (j > 0)
1045 29388 : val += prev_row[j - 1] * t;
1046 41268 : row[j] = val;
1047 : }
1048 : }
1049 :
1050 : /*
1051 : * The presence of many distinct rare (not in "p") elements materially
1052 : * decreases selectivity. Model their collective occurrence with the
1053 : * Poisson distribution.
1054 : */
1055 144 : if (rest > DEFAULT_CONTAIN_SEL)
1056 : {
1057 : float t;
1058 :
1059 : /* Swap rows */
1060 0 : tmp = row;
1061 0 : row = prev_row;
1062 0 : prev_row = tmp;
1063 :
1064 0 : for (i = 0; i <= m; i++)
1065 0 : row[i] = 0.0f;
1066 :
1067 : /* Value of Poisson distribution for 0 occurrences */
1068 0 : t = exp(-rest);
1069 :
1070 : /*
1071 : * Calculate convolution of previously computed distribution and the
1072 : * Poisson distribution.
1073 : */
1074 0 : for (i = 0; i <= m; i++)
1075 : {
1076 0 : for (j = 0; j <= m - i; j++)
1077 0 : row[j + i] += prev_row[j] * t;
1078 :
1079 : /* Get Poisson distribution value for (i + 1) occurrences */
1080 0 : t *= rest / (float) (i + 1);
1081 : }
1082 : }
1083 :
1084 144 : pfree(prev_row);
1085 144 : return row;
1086 : }
1087 :
1088 : /* Fast function for floor value of 2 based logarithm calculation. */
1089 : static int
1090 996 : floor_log2(uint32 n)
1091 : {
1092 996 : int logval = 0;
1093 :
1094 996 : if (n == 0)
1095 600 : return -1;
1096 396 : if (n >= (1 << 16))
1097 : {
1098 0 : n >>= 16;
1099 0 : logval += 16;
1100 : }
1101 396 : if (n >= (1 << 8))
1102 : {
1103 120 : n >>= 8;
1104 120 : logval += 8;
1105 : }
1106 396 : if (n >= (1 << 4))
1107 : {
1108 276 : n >>= 4;
1109 276 : logval += 4;
1110 : }
1111 396 : if (n >= (1 << 2))
1112 : {
1113 264 : n >>= 2;
1114 264 : logval += 2;
1115 : }
1116 396 : if (n >= (1 << 1))
1117 : {
1118 186 : logval += 1;
1119 : }
1120 396 : return logval;
1121 : }
1122 :
1123 : /*
1124 : * find_next_mcelem binary-searches a most common elements array, starting
1125 : * from *index, for the first member >= value. It saves the position of the
1126 : * match into *index and returns true if it's an exact match. (Note: we
1127 : * assume the mcelem elements are distinct so there can't be more than one
1128 : * exact match.)
1129 : */
1130 : static bool
1131 840 : find_next_mcelem(Datum *mcelem, int nmcelem, Datum value, int *index,
1132 : TypeCacheEntry *typentry)
1133 : {
1134 840 : int l = *index,
1135 840 : r = nmcelem - 1,
1136 : i,
1137 : res;
1138 :
1139 3324 : while (l <= r)
1140 : {
1141 2892 : i = (l + r) / 2;
1142 2892 : res = element_compare(&mcelem[i], &value, typentry);
1143 2892 : if (res == 0)
1144 : {
1145 408 : *index = i;
1146 408 : return true;
1147 : }
1148 2484 : else if (res < 0)
1149 882 : l = i + 1;
1150 : else
1151 1602 : r = i - 1;
1152 : }
1153 432 : *index = l;
1154 432 : return false;
1155 : }
1156 :
1157 : /*
1158 : * Comparison function for elements.
1159 : *
1160 : * We use the element type's default btree opclass, and its default collation
1161 : * if the type is collation-sensitive.
1162 : *
1163 : * XXX consider using SortSupport infrastructure
1164 : */
1165 : static int
1166 7608 : element_compare(const void *key1, const void *key2, void *arg)
1167 : {
1168 7608 : Datum d1 = *((const Datum *) key1);
1169 7608 : Datum d2 = *((const Datum *) key2);
1170 7608 : TypeCacheEntry *typentry = (TypeCacheEntry *) arg;
1171 7608 : FmgrInfo *cmpfunc = &typentry->cmp_proc_finfo;
1172 : Datum c;
1173 :
1174 7608 : c = FunctionCall2Coll(cmpfunc, typentry->typcollation, d1, d2);
1175 7608 : return DatumGetInt32(c);
1176 : }
1177 :
1178 : /*
1179 : * Comparison function for sorting floats into descending order.
1180 : */
1181 : static int
1182 0 : float_compare_desc(const void *key1, const void *key2)
1183 : {
1184 0 : float d1 = *((const float *) key1);
1185 0 : float d2 = *((const float *) key2);
1186 :
1187 0 : if (d1 > d2)
1188 0 : return -1;
1189 0 : else if (d1 < d2)
1190 0 : return 1;
1191 : else
1192 0 : return 0;
1193 : }
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