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/*-------------------------------------------------------------------------
*
* array_selfuncs.c
* Functions for selectivity estimation of array operators
*
* Portions Copyright (c) 1996-2020, PostgreSQL Global Development Group
* Portions Copyright (c) 1994, Regents of the University of California
*
*
* IDENTIFICATION
* src/backend/utils/adt/array_selfuncs.c
*
*-------------------------------------------------------------------------
*/
#include "postgres.h"
#include <math.h>
#include "access/htup_details.h"
#include "catalog/pg_collation.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_statistic.h"
#include "utils/array.h"
#include "utils/builtins.h"
#include "utils/lsyscache.h"
#include "utils/selfuncs.h"
#include "utils/typcache.h"
/* Default selectivity constant for "@>" and "<@" operators */
#define DEFAULT_CONTAIN_SEL 0.005
/* Default selectivity constant for "&&" operator */
#define DEFAULT_OVERLAP_SEL 0.01
/* Default selectivity for given operator */
#define DEFAULT_SEL(operator) \
((operator) == OID_ARRAY_OVERLAP_OP ? \
DEFAULT_OVERLAP_SEL : DEFAULT_CONTAIN_SEL)
static Selectivity calc_arraycontsel(VariableStatData *vardata, Datum constval,
Oid elemtype, Oid operator);
static Selectivity mcelem_array_selec(ArrayType *array,
TypeCacheEntry *typentry,
Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
float4 *hist, int nhist,
Oid operator);
static Selectivity mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
Datum *array_data, int nitems,
Oid operator, TypeCacheEntry *typentry);
static Selectivity mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
Datum *array_data, int nitems,
float4 *hist, int nhist,
Oid operator, TypeCacheEntry *typentry);
static float *calc_hist(const float4 *hist, int nhist, int n);
static float *calc_distr(const float *p, int n, int m, float rest);
static int floor_log2(uint32 n);
static bool find_next_mcelem(Datum *mcelem, int nmcelem, Datum value,
int *index, TypeCacheEntry *typentry);
static int element_compare(const void *key1, const void *key2, void *arg);
static int float_compare_desc(const void *key1, const void *key2);
/*
* scalararraysel_containment
* Estimate selectivity of ScalarArrayOpExpr via array containment.
*
* If we have const =/<> ANY/ALL (array_var) then we can estimate the
* selectivity as though this were an array containment operator,
* array_var op ARRAY[const].
*
* scalararraysel() has already verified that the ScalarArrayOpExpr's operator
* is the array element type's default equality or inequality operator, and
* has aggressively simplified both inputs to constants.
*
* Returns selectivity (0..1), or -1 if we fail to estimate selectivity.
*/
Selectivity
scalararraysel_containment(PlannerInfo *root,
Node *leftop, Node *rightop,
Oid elemtype, bool isEquality, bool useOr,
int varRelid)
{
Selectivity selec;
VariableStatData vardata;
Datum constval;
TypeCacheEntry *typentry;
FmgrInfo *cmpfunc;
/*
* rightop must be a variable, else punt.
*/
examine_variable(root, rightop, varRelid, &vardata);
if (!vardata.rel)
{
ReleaseVariableStats(vardata);
return -1.0;
}
/*
* leftop must be a constant, else punt.
*/
if (!IsA(leftop, Const))
{
ReleaseVariableStats(vardata);
return -1.0;
}
if (((Const *) leftop)->constisnull)
{
/* qual can't succeed if null on left */
ReleaseVariableStats(vardata);
return (Selectivity) 0.0;
}
constval = ((Const *) leftop)->constvalue;
/* Get element type's default comparison function */
typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
{
ReleaseVariableStats(vardata);
return -1.0;
}
cmpfunc = &typentry->cmp_proc_finfo;
/*
* If the operator is <>, swap ANY/ALL, then invert the result later.
*/
if (!isEquality)
useOr = !useOr;
/* Get array element stats for var, if available */
if (HeapTupleIsValid(vardata.statsTuple) &&
statistic_proc_security_check(&vardata, cmpfunc->fn_oid))
{
Form_pg_statistic stats;
AttStatsSlot sslot;
AttStatsSlot hslot;
stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
/* MCELEM will be an array of same type as element */
if (get_attstatsslot(&sslot, vardata.statsTuple,
STATISTIC_KIND_MCELEM, InvalidOid,
ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
{
/* For ALL case, also get histogram of distinct-element counts */
if (useOr ||
!get_attstatsslot(&hslot, vardata.statsTuple,
STATISTIC_KIND_DECHIST, InvalidOid,
ATTSTATSSLOT_NUMBERS))
memset(&hslot, 0, sizeof(hslot));
/*
* For = ANY, estimate as var @> ARRAY[const].
*
* For = ALL, estimate as var <@ ARRAY[const].
*/
if (useOr)
selec = mcelem_array_contain_overlap_selec(sslot.values,
sslot.nvalues,
sslot.numbers,
sslot.nnumbers,
&constval, 1,
OID_ARRAY_CONTAINS_OP,
typentry);
else
selec = mcelem_array_contained_selec(sslot.values,
sslot.nvalues,
sslot.numbers,
sslot.nnumbers,
&constval, 1,
hslot.numbers,
hslot.nnumbers,
OID_ARRAY_CONTAINED_OP,
typentry);
free_attstatsslot(&hslot);
free_attstatsslot(&sslot);
}
else
{
/* No most-common-elements info, so do without */
if (useOr)
selec = mcelem_array_contain_overlap_selec(NULL, 0,
NULL, 0,
&constval, 1,
OID_ARRAY_CONTAINS_OP,
typentry);
else
selec = mcelem_array_contained_selec(NULL, 0,
NULL, 0,
&constval, 1,
NULL, 0,
OID_ARRAY_CONTAINED_OP,
typentry);
}
/*
* MCE stats count only non-null rows, so adjust for null rows.
*/
selec *= (1.0 - stats->stanullfrac);
}
else
{
/* No stats at all, so do without */
if (useOr)
selec = mcelem_array_contain_overlap_selec(NULL, 0,
NULL, 0,
&constval, 1,
OID_ARRAY_CONTAINS_OP,
typentry);
else
selec = mcelem_array_contained_selec(NULL, 0,
NULL, 0,
&constval, 1,
NULL, 0,
OID_ARRAY_CONTAINED_OP,
typentry);
/* we assume no nulls here, so no stanullfrac correction */
}
ReleaseVariableStats(vardata);
/*
* If the operator is <>, invert the results.
*/
if (!isEquality)
selec = 1.0 - selec;
CLAMP_PROBABILITY(selec);
return selec;
}
/*
* arraycontsel -- restriction selectivity for array @>, &&, <@ operators
*/
Datum
arraycontsel(PG_FUNCTION_ARGS)
{
PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
Oid operator = PG_GETARG_OID(1);
List *args = (List *) PG_GETARG_POINTER(2);
int varRelid = PG_GETARG_INT32(3);
VariableStatData vardata;
Node *other;
bool varonleft;
Selectivity selec;
Oid element_typeid;
/*
* If expression is not (variable op something) or (something op
* variable), then punt and return a default estimate.
*/
if (!get_restriction_variable(root, args, varRelid,
&vardata, &other, &varonleft))
PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
/*
* Can't do anything useful if the something is not a constant, either.
*/
if (!IsA(other, Const))
{
ReleaseVariableStats(vardata);
PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
}
/*
* The "&&", "@>" and "<@" operators are strict, so we can cope with a
* NULL constant right away.
*/
if (((Const *) other)->constisnull)
{
ReleaseVariableStats(vardata);
PG_RETURN_FLOAT8(0.0);
}
/*
* If var is on the right, commute the operator, so that we can assume the
* var is on the left in what follows.
*/
if (!varonleft)
{
if (operator == OID_ARRAY_CONTAINS_OP)
operator = OID_ARRAY_CONTAINED_OP;
else if (operator == OID_ARRAY_CONTAINED_OP)
operator = OID_ARRAY_CONTAINS_OP;
}
/*
* OK, there's a Var and a Const we're dealing with here. We need the
* Const to be an array with same element type as column, else we can't do
* anything useful. (Such cases will likely fail at runtime, but here
* we'd rather just return a default estimate.)
*/
element_typeid = get_base_element_type(((Const *) other)->consttype);
if (element_typeid != InvalidOid &&
element_typeid == get_base_element_type(vardata.vartype))
{
selec = calc_arraycontsel(&vardata, ((Const *) other)->constvalue,
element_typeid, operator);
}
else
{
selec = DEFAULT_SEL(operator);
}
ReleaseVariableStats(vardata);
CLAMP_PROBABILITY(selec);
PG_RETURN_FLOAT8((float8) selec);
}
/*
* arraycontjoinsel -- join selectivity for array @>, &&, <@ operators
*/
Datum
arraycontjoinsel(PG_FUNCTION_ARGS)
{
/* For the moment this is just a stub */
Oid operator = PG_GETARG_OID(1);
PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
}
/*
* Calculate selectivity for "arraycolumn @> const", "arraycolumn && const"
* or "arraycolumn <@ const" based on the statistics
*
* This function is mainly responsible for extracting the pg_statistic data
* to be used; we then pass the problem on to mcelem_array_selec().
*/
static Selectivity
calc_arraycontsel(VariableStatData *vardata, Datum constval,
Oid elemtype, Oid operator)
{
Selectivity selec;
TypeCacheEntry *typentry;
FmgrInfo *cmpfunc;
ArrayType *array;
/* Get element type's default comparison function */
typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
return DEFAULT_SEL(operator);
cmpfunc = &typentry->cmp_proc_finfo;
/*
* The caller made sure the const is an array with same element type, so
* get it now
*/
array = DatumGetArrayTypeP(constval);
if (HeapTupleIsValid(vardata->statsTuple) &&
statistic_proc_security_check(vardata, cmpfunc->fn_oid))
{
Form_pg_statistic stats;
AttStatsSlot sslot;
AttStatsSlot hslot;
stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
/* MCELEM will be an array of same type as column */
if (get_attstatsslot(&sslot, vardata->statsTuple,
STATISTIC_KIND_MCELEM, InvalidOid,
ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
{
/*
* For "array <@ const" case we also need histogram of distinct
* element counts.
*/
if (operator != OID_ARRAY_CONTAINED_OP ||
!get_attstatsslot(&hslot, vardata->statsTuple,
STATISTIC_KIND_DECHIST, InvalidOid,
ATTSTATSSLOT_NUMBERS))
memset(&hslot, 0, sizeof(hslot));
/* Use the most-common-elements slot for the array Var. */
selec = mcelem_array_selec(array, typentry,
sslot.values, sslot.nvalues,
sslot.numbers, sslot.nnumbers,
hslot.numbers, hslot.nnumbers,
operator);
free_attstatsslot(&hslot);
free_attstatsslot(&sslot);
}
else
{
/* No most-common-elements info, so do without */
selec = mcelem_array_selec(array, typentry,
NULL, 0, NULL, 0, NULL, 0,
operator);
}
/*
* MCE stats count only non-null rows, so adjust for null rows.
*/
selec *= (1.0 - stats->stanullfrac);
}
else
{
/* No stats at all, so do without */
selec = mcelem_array_selec(array, typentry,
NULL, 0, NULL, 0, NULL, 0,
operator);
/* we assume no nulls here, so no stanullfrac correction */
}
/* If constant was toasted, release the copy we made */
if (PointerGetDatum(array) != constval)
pfree(array);
return selec;
}
/*
* Array selectivity estimation based on most common elements statistics
*
* This function just deconstructs and sorts the array constant's contents,
* and then passes the problem on to mcelem_array_contain_overlap_selec or
* mcelem_array_contained_selec depending on the operator.
*/
static Selectivity
mcelem_array_selec(ArrayType *array, TypeCacheEntry *typentry,
Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
float4 *hist, int nhist,
Oid operator)
{
Selectivity selec;
int num_elems;
Datum *elem_values;
bool *elem_nulls;
bool null_present;
int nonnull_nitems;
int i;
/*
* Prepare constant array data for sorting. Sorting lets us find unique
* elements and efficiently merge with the MCELEM array.
*/
deconstruct_array(array,
typentry->type_id,
typentry->typlen,
typentry->typbyval,
typentry->typalign,
&elem_values, &elem_nulls, &num_elems);
/* Collapse out any null elements */
nonnull_nitems = 0;
null_present = false;
for (i = 0; i < num_elems; i++)
{
if (elem_nulls[i])
null_present = true;
else
elem_values[nonnull_nitems++] = elem_values[i];
}
/*
* Query "column @> '{anything, null}'" matches nothing. For the other
* two operators, presence of a null in the constant can be ignored.
*/
if (null_present && operator == OID_ARRAY_CONTAINS_OP)
{
pfree(elem_values);
pfree(elem_nulls);
return (Selectivity) 0.0;
}
/* Sort extracted elements using their default comparison function. */
qsort_arg(elem_values, nonnull_nitems, sizeof(Datum),
element_compare, typentry);
/* Separate cases according to operator */
if (operator == OID_ARRAY_CONTAINS_OP || operator == OID_ARRAY_OVERLAP_OP)
selec = mcelem_array_contain_overlap_selec(mcelem, nmcelem,
numbers, nnumbers,
elem_values, nonnull_nitems,
operator, typentry);
else if (operator == OID_ARRAY_CONTAINED_OP)
selec = mcelem_array_contained_selec(mcelem, nmcelem,
numbers, nnumbers,
elem_values, nonnull_nitems,
hist, nhist,
operator, typentry);
else
{
elog(ERROR, "arraycontsel called for unrecognized operator %u",
operator);
selec = 0.0; /* keep compiler quiet */
}
pfree(elem_values);
pfree(elem_nulls);
return selec;
}
/*
* Estimate selectivity of "column @> const" and "column && const" based on
* most common element statistics. This estimation assumes element
* occurrences are independent.
*
* mcelem (of length nmcelem) and numbers (of length nnumbers) are from
* the array column's MCELEM statistics slot, or are NULL/0 if stats are
* not available. array_data (of length nitems) is the constant's elements.
*
* Both the mcelem and array_data arrays are assumed presorted according
* to the element type's cmpfunc. Null elements are not present.
*
* TODO: this estimate probably could be improved by using the distinct
* elements count histogram. For example, excepting the special case of
* "column @> '{}'", we can multiply the calculated selectivity by the
* fraction of nonempty arrays in the column.
*/
static Selectivity
mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
Datum *array_data, int nitems,
Oid operator, TypeCacheEntry *typentry)
{
Selectivity selec,
elem_selec;
int mcelem_index,
i;
bool use_bsearch;
float4 minfreq;
/*
* There should be three more Numbers than Values, because the last three
* cells should hold minimal and maximal frequency among the non-null
* elements, and then the frequency of null elements. Ignore the Numbers
* if not right.
*/
if (nnumbers != nmcelem + 3)
{
numbers = NULL;
nnumbers = 0;
}
if (numbers)
{
/* Grab the lowest observed frequency */
minfreq = numbers[nmcelem];
}
else
{
/* Without statistics make some default assumptions */
minfreq = 2 * (float4) DEFAULT_CONTAIN_SEL;
}
/* Decide whether it is faster to use binary search or not. */
if (nitems * floor_log2((uint32) nmcelem) < nmcelem + nitems)
use_bsearch = true;
else
use_bsearch = false;
if (operator == OID_ARRAY_CONTAINS_OP)
{
/*
* Initial selectivity for "column @> const" query is 1.0, and it will
* be decreased with each element of constant array.
*/
selec = 1.0;
}
else
{
/*
* Initial selectivity for "column && const" query is 0.0, and it will
* be increased with each element of constant array.
*/
selec = 0.0;
}
/* Scan mcelem and array in parallel. */
mcelem_index = 0;
for (i = 0; i < nitems; i++)
{
bool match = false;
/* Ignore any duplicates in the array data. */
if (i > 0 &&
element_compare(&array_data[i - 1], &array_data[i], typentry) == 0)
continue;
/* Find the smallest MCELEM >= this array item. */
if (use_bsearch)
{
match = find_next_mcelem(mcelem, nmcelem, array_data[i],
&mcelem_index, typentry);
}
else
{
while (mcelem_index < nmcelem)
{
int cmp = element_compare(&mcelem[mcelem_index],
&array_data[i],
typentry);
if (cmp < 0)
mcelem_index++;
else
{
if (cmp == 0)
match = true; /* mcelem is found */
break;
}
}
}
if (match && numbers)
{
/* MCELEM matches the array item; use its frequency. */
elem_selec = numbers[mcelem_index];
mcelem_index++;
}
else
{
/*
* The element is not in MCELEM. Punt, but assume that the
* selectivity cannot be more than minfreq / 2.
*/
elem_selec = Min(DEFAULT_CONTAIN_SEL, minfreq / 2);
}
/*
* Update overall selectivity using the current element's selectivity
* and an assumption of element occurrence independence.
*/
if (operator == OID_ARRAY_CONTAINS_OP)
selec *= elem_selec;
else
selec = selec + elem_selec - selec * elem_selec;
/* Clamp intermediate results to stay sane despite roundoff error */
CLAMP_PROBABILITY(selec);
}
return selec;
}
/*
* Estimate selectivity of "column <@ const" based on most common element
* statistics.
*
* mcelem (of length nmcelem) and numbers (of length nnumbers) are from
* the array column's MCELEM statistics slot, or are NULL/0 if stats are
* not available. array_data (of length nitems) is the constant's elements.
* hist (of length nhist) is from the array column's DECHIST statistics slot,
* or is NULL/0 if those stats are not available.
*
* Both the mcelem and array_data arrays are assumed presorted according
* to the element type's cmpfunc. Null elements are not present.
*
* Independent element occurrence would imply a particular distribution of
* distinct element counts among matching rows. Real data usually falsifies
* that assumption. For example, in a set of 11-element integer arrays having
* elements in the range [0..10], element occurrences are typically not
* independent. If they were, a sufficiently-large set would include all
* distinct element counts 0 through 11. We correct for this using the
* histogram of distinct element counts.
*
* In the "column @> const" and "column && const" cases, we usually have a
* "const" with low number of elements (otherwise we have selectivity close
* to 0 or 1 respectively). That's why the effect of dependence related
* to distinct element count distribution is negligible there. In the
* "column <@ const" case, number of elements is usually high (otherwise we
* have selectivity close to 0). That's why we should do a correction with
* the array distinct element count distribution here.
*
* Using the histogram of distinct element counts produces a different
* distribution law than independent occurrences of elements. This
* distribution law can be described as follows:
*
* P(o1, o2, ..., on) = f1^o1 * (1 - f1)^(1 - o1) * f2^o2 *
* (1 - f2)^(1 - o2) * ... * fn^on * (1 - fn)^(1 - on) * hist[m] / ind[m]
*
* where:
* o1, o2, ..., on - occurrences of elements 1, 2, ..., n
* (1 - occurrence, 0 - no occurrence) in row
* f1, f2, ..., fn - frequencies of elements 1, 2, ..., n
* (scalar values in [0..1]) according to collected statistics
* m = o1 + o2 + ... + on = total number of distinct elements in row
* hist[m] - histogram data for occurrence of m elements.
* ind[m] - probability of m occurrences from n events assuming their
* probabilities to be equal to frequencies of array elements.
*
* ind[m] = sum(f1^o1 * (1 - f1)^(1 - o1) * f2^o2 * (1 - f2)^(1 - o2) *
* ... * fn^on * (1 - fn)^(1 - on), o1, o2, ..., on) | o1 + o2 + .. on = m
*/
static Selectivity
mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
float4 *numbers, int nnumbers,
Datum *array_data, int nitems,
float4 *hist, int nhist,
Oid operator, TypeCacheEntry *typentry)
{
int mcelem_index,
i,
unique_nitems = 0;
float selec,
minfreq,
nullelem_freq;
float *dist,
*mcelem_dist,
*hist_part;
float avg_count,
mult,
rest;
float *elem_selec;
/*
* There should be three more Numbers than Values in the MCELEM slot,
* because the last three cells should hold minimal and maximal frequency
* among the non-null elements, and then the frequency of null elements.
* Punt if not right, because we can't do much without the element freqs.
*/
if (numbers == NULL || nnumbers != nmcelem + 3)
return DEFAULT_CONTAIN_SEL;
/* Can't do much without a count histogram, either */
if (hist == NULL || nhist < 3)
return DEFAULT_CONTAIN_SEL;
/*
* Grab some of the summary statistics that compute_array_stats() stores:
* lowest frequency, frequency of null elements, and average distinct
* element count.
*/
minfreq = numbers[nmcelem];
nullelem_freq = numbers[nmcelem + 2];
avg_count = hist[nhist - 1];
/*
* "rest" will be the sum of the frequencies of all elements not
* represented in MCELEM. The average distinct element count is the sum
* of the frequencies of *all* elements. Begin with that; we will proceed
* to subtract the MCELEM frequencies.
*/
rest = avg_count;
/*
* mult is a multiplier representing estimate of probability that each
* mcelem that is not present in constant doesn't occur.
*/
mult = 1.0f;
/*
* elem_selec is array of estimated frequencies for elements in the
* constant.
*/
elem_selec = (float *) palloc(sizeof(float) * nitems);
/* Scan mcelem and array in parallel. */
mcelem_index = 0;
for (i = 0; i < nitems; i++)
{
bool match = false;
/* Ignore any duplicates in the array data. */
if (i > 0 &&
element_compare(&array_data[i - 1], &array_data[i], typentry) == 0)
continue;
/*
* Iterate over MCELEM until we find an entry greater than or equal to
* this element of the constant. Update "rest" and "mult" for mcelem
* entries skipped over.
*/
while (mcelem_index < nmcelem)
{
int cmp = element_compare(&mcelem[mcelem_index],
&array_data[i],
typentry);
if (cmp < 0)
{
mult *= (1.0f - numbers[mcelem_index]);
rest -= numbers[mcelem_index];
mcelem_index++;
}
else
{
if (cmp == 0)
match = true; /* mcelem is found */
break;
}
}
if (match)
{
/* MCELEM matches the array item. */
elem_selec[unique_nitems] = numbers[mcelem_index];
/* "rest" is decremented for all mcelems, matched or not */
rest -= numbers[mcelem_index];
mcelem_index++;
}
else
{
/*
* The element is not in MCELEM. Punt, but assume that the
* selectivity cannot be more than minfreq / 2.
*/
elem_selec[unique_nitems] = Min(DEFAULT_CONTAIN_SEL,
minfreq / 2);
}
unique_nitems++;
}
/*
* If we handled all constant elements without exhausting the MCELEM
* array, finish walking it to complete calculation of "rest" and "mult".
*/
while (mcelem_index < nmcelem)
{
mult *= (1.0f - numbers[mcelem_index]);
rest -= numbers[mcelem_index];
mcelem_index++;
}
/*
* The presence of many distinct rare elements materially decreases
* selectivity. Use the Poisson distribution to estimate the probability
* of a column value having zero occurrences of such elements. See above
* for the definition of "rest".
*/
mult *= exp(-rest);
/*----------
* Using the distinct element count histogram requires
* O(unique_nitems * (nmcelem + unique_nitems))
* operations. Beyond a certain computational cost threshold, it's
* reasonable to sacrifice accuracy for decreased planning time. We limit
* the number of operations to EFFORT * nmcelem; since nmcelem is limited
* by the column's statistics target, the work done is user-controllable.
*
* If the number of operations would be too large, we can reduce it
* without losing all accuracy by reducing unique_nitems and considering
* only the most-common elements of the constant array. To make the
* results exactly match what we would have gotten with only those
* elements to start with, we'd have to remove any discarded elements'
* frequencies from "mult", but since this is only an approximation
* anyway, we don't bother with that. Therefore it's sufficient to qsort
* elem_selec[] and take the largest elements. (They will no longer match
* up with the elements of array_data[], but we don't care.)
*----------
*/
#define EFFORT 100
if ((nmcelem + unique_nitems) > 0 &&
unique_nitems > EFFORT * nmcelem / (nmcelem + unique_nitems))
{
/*
* Use the quadratic formula to solve for largest allowable N. We
* have A = 1, B = nmcelem, C = - EFFORT * nmcelem.
*/
double b = (double) nmcelem;
int n;
n = (int) ((sqrt(b * b + 4 * EFFORT * b) - b) / 2);
/* Sort, then take just the first n elements */
qsort(elem_selec, unique_nitems, sizeof(float),
float_compare_desc);
unique_nitems = n;
}
/*
* Calculate probabilities of each distinct element count for both mcelems
* and constant elements. At this point, assume independent element
* occurrence.
*/
dist = calc_distr(elem_selec, unique_nitems, unique_nitems, 0.0f);
mcelem_dist = calc_distr(numbers, nmcelem, unique_nitems, rest);
/* ignore hist[nhist-1], which is the average not a histogram member */
hist_part = calc_hist(hist, nhist - 1, unique_nitems);
selec = 0.0f;
for (i = 0; i <= unique_nitems; i++)
{
/*
* mult * dist[i] / mcelem_dist[i] gives us probability of qual
* matching from assumption of independent element occurrence with the
* condition that distinct element count = i.
*/
if (mcelem_dist[i] > 0)
selec += hist_part[i] * mult * dist[i] / mcelem_dist[i];
}
pfree(dist);
pfree(mcelem_dist);
pfree(hist_part);
pfree(elem_selec);
/* Take into account occurrence of NULL element. */
selec *= (1.0f - nullelem_freq);
CLAMP_PROBABILITY(selec);
return selec;
}
/*
* Calculate the first n distinct element count probabilities from a
* histogram of distinct element counts.
*
* Returns a palloc'd array of n+1 entries, with array[k] being the
* probability of element count k, k in [0..n].
*
* We assume that a histogram box with bounds a and b gives 1 / ((b - a + 1) *
* (nhist - 1)) probability to each value in (a,b) and an additional half of
* that to a and b themselves.
*/
static float *
calc_hist(const float4 *hist, int nhist, int n)
{
float *hist_part;
int k,
i = 0;
float prev_interval = 0,
next_interval;
float frac;
hist_part = (float *) palloc((n + 1) * sizeof(float));
/*
* frac is a probability contribution for each interval between histogram
* values. We have nhist - 1 intervals, so contribution of each one will
* be 1 / (nhist - 1).
*/
frac = 1.0f / ((float) (nhist - 1));
for (k = 0; k <= n; k++)
{
int count = 0;
/*
* Count the histogram boundaries equal to k. (Although the histogram
* should theoretically contain only exact integers, entries are
* floats so there could be roundoff error in large values. Treat any
* fractional value as equal to the next larger k.)
*/
while (i < nhist && hist[i] <= k)
{
count++;
i++;
}
if (count > 0)
{
/* k is an exact bound for at least one histogram box. */
float val;
/* Find length between current histogram value and the next one */
if (i < nhist)
next_interval = hist[i] - hist[i - 1];
else
next_interval = 0;
/*
* count - 1 histogram boxes contain k exclusively. They
* contribute a total of (count - 1) * frac probability. Also
* factor in the partial histogram boxes on either side.
*/
val = (float) (count - 1);
if (next_interval > 0)
val += 0.5f / next_interval;
if (prev_interval > 0)
val += 0.5f / prev_interval;
hist_part[k] = frac * val;
prev_interval = next_interval;
}
else
{
/* k does not appear as an exact histogram bound. */
if (prev_interval > 0)
hist_part[k] = frac / prev_interval;
else
hist_part[k] = 0.0f;
}
}
return hist_part;
}
/*
* Consider n independent events with probabilities p[]. This function
* calculates probabilities of exact k of events occurrence for k in [0..m].
* Returns a palloc'd array of size m+1.
*
* "rest" is the sum of the probabilities of all low-probability events not
* included in p.
*
* Imagine matrix M of size (n + 1) x (m + 1). Element M[i,j] denotes the
* probability that exactly j of first i events occur. Obviously M[0,0] = 1.
* For any constant j, each increment of i increases the probability iff the
* event occurs. So, by the law of total probability:
* M[i,j] = M[i - 1, j] * (1 - p[i]) + M[i - 1, j - 1] * p[i]
* for i > 0, j > 0.
* M[i,0] = M[i - 1, 0] * (1 - p[i]) for i > 0.
*/
static float *
calc_distr(const float *p, int n, int m, float rest)
{
float *row,
*prev_row,
*tmp;
int i,
j;
/*
* Since we return only the last row of the matrix and need only the
* current and previous row for calculations, allocate two rows.
*/
row = (float *) palloc((m + 1) * sizeof(float));
prev_row = (float *) palloc((m + 1) * sizeof(float));
/* M[0,0] = 1 */
row[0] = 1.0f;
for (i = 1; i <= n; i++)
{
float t = p[i - 1];
/* Swap rows */
tmp = row;
row = prev_row;
prev_row = tmp;
/* Calculate next row */
for (j = 0; j <= i && j <= m; j++)
{
float val = 0.0f;
if (j < i)
val += prev_row[j] * (1.0f - t);
if (j > 0)
val += prev_row[j - 1] * t;
row[j] = val;
}
}
/*
* The presence of many distinct rare (not in "p") elements materially
* decreases selectivity. Model their collective occurrence with the
* Poisson distribution.
*/
if (rest > DEFAULT_CONTAIN_SEL)
{
float t;
/* Swap rows */
tmp = row;
row = prev_row;
prev_row = tmp;
for (i = 0; i <= m; i++)
row[i] = 0.0f;
/* Value of Poisson distribution for 0 occurrences */
t = exp(-rest);
/*
* Calculate convolution of previously computed distribution and the
* Poisson distribution.
*/
for (i = 0; i <= m; i++)
{
for (j = 0; j <= m - i; j++)
row[j + i] += prev_row[j] * t;
/* Get Poisson distribution value for (i + 1) occurrences */
t *= rest / (float) (i + 1);
}
}
pfree(prev_row);
return row;
}
/* Fast function for floor value of 2 based logarithm calculation. */
static int
floor_log2(uint32 n)
{
int logval = 0;
if (n == 0)
return -1;
if (n >= (1 << 16))
{
n >>= 16;
logval += 16;
}
if (n >= (1 << 8))
{
n >>= 8;
logval += 8;
}
if (n >= (1 << 4))
{
n >>= 4;
logval += 4;
}
if (n >= (1 << 2))
{
n >>= 2;
logval += 2;
}
if (n >= (1 << 1))
{
logval += 1;
}
return logval;
}
/*
* find_next_mcelem binary-searches a most common elements array, starting
* from *index, for the first member >= value. It saves the position of the
* match into *index and returns true if it's an exact match. (Note: we
* assume the mcelem elements are distinct so there can't be more than one
* exact match.)
*/
static bool
find_next_mcelem(Datum *mcelem, int nmcelem, Datum value, int *index,
TypeCacheEntry *typentry)
{
int l = *index,
r = nmcelem - 1,
i,
res;
while (l <= r)
{
i = (l + r) / 2;
res = element_compare(&mcelem[i], &value, typentry);
if (res == 0)
{
*index = i;
return true;
}
else if (res < 0)
l = i + 1;
else
r = i - 1;
}
*index = l;
return false;
}
/*
* Comparison function for elements.
*
* We use the element type's default btree opclass, and its default collation
* if the type is collation-sensitive.
*
* XXX consider using SortSupport infrastructure
*/
static int
element_compare(const void *key1, const void *key2, void *arg)
{
Datum d1 = *((const Datum *) key1);
Datum d2 = *((const Datum *) key2);
TypeCacheEntry *typentry = (TypeCacheEntry *) arg;
FmgrInfo *cmpfunc = &typentry->cmp_proc_finfo;
Datum c;
c = FunctionCall2Coll(cmpfunc, typentry->typcollation, d1, d2);
return DatumGetInt32(c);
}
/*
* Comparison function for sorting floats into descending order.
*/
static int
float_compare_desc(const void *key1, const void *key2)
{
float d1 = *((const float *) key1);
float d2 = *((const float *) key2);
if (d1 > d2)
return -1;
else if (d1 < d2)
return 1;
else
return 0;
}