greenplumn rangetypes_selfuncs 源码
greenplumn rangetypes_selfuncs 代码
文件路径：/src/backend/utils/adt/rangetypes_selfuncs.c
/*-------------------------------------------------------------------------
*
* rangetypes_selfuncs.c
* Functions for selectivity estimation of range operators
*
* Estimates are based on histograms of lower and upper bounds, and the
* fraction of empty ranges.
*
* Portions Copyright (c) 1996-2019, PostgreSQL Global Development Group
* Portions Copyright (c) 1994, Regents of the University of California
*
*
* IDENTIFICATION
* src/backend/utils/adt/rangetypes_selfuncs.c
*
*-------------------------------------------------------------------------
*/
#include "postgres.h"
#include <math.h>
#include "access/htup_details.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_statistic.h"
#include "catalog/pg_type.h"
#include "utils/float.h"
#include "utils/fmgrprotos.h"
#include "utils/lsyscache.h"
#include "utils/rangetypes.h"
#include "utils/selfuncs.h"
#include "utils/typcache.h"
static double calc_rangesel(TypeCacheEntry *typcache, VariableStatData *vardata,
RangeType *constval, Oid operator);
static double default_range_selectivity(Oid operator);
static double calc_hist_selectivity(TypeCacheEntry *typcache,
VariableStatData *vardata, RangeType *constval,
Oid operator);
static double calc_hist_selectivity_scalar(TypeCacheEntry *typcache,
RangeBound *constbound,
RangeBound *hist, int hist_nvalues,
bool equal);
static int rbound_bsearch(TypeCacheEntry *typcache, RangeBound *value,
RangeBound *hist, int hist_length, bool equal);
static float8 get_position(TypeCacheEntry *typcache, RangeBound *value,
RangeBound *hist1, RangeBound *hist2);
static float8 get_len_position(double value, double hist1, double hist2);
static float8 get_distance(TypeCacheEntry *typcache, RangeBound *bound1,
RangeBound *bound2);
static int length_hist_bsearch(Datum *length_hist_values,
int length_hist_nvalues, double value, bool equal);
static double calc_length_hist_frac(Datum *length_hist_values,
int length_hist_nvalues, double length1, double length2, bool equal);
static double calc_hist_selectivity_contained(TypeCacheEntry *typcache,
RangeBound *lower, RangeBound *upper,
RangeBound *hist_lower, int hist_nvalues,
Datum *length_hist_values, int length_hist_nvalues);
static double calc_hist_selectivity_contains(TypeCacheEntry *typcache,
RangeBound *lower, RangeBound *upper,
RangeBound *hist_lower, int hist_nvalues,
Datum *length_hist_values, int length_hist_nvalues);
/*
* Returns a default selectivity estimate for given operator, when we don't
* have statistics or cannot use them for some reason.
*/
static double
default_range_selectivity(Oid operator)
{
switch (operator)
{
case OID_RANGE_OVERLAP_OP:
return 0.01;
case OID_RANGE_CONTAINS_OP:
case OID_RANGE_CONTAINED_OP:
return 0.005;
case OID_RANGE_CONTAINS_ELEM_OP:
case OID_RANGE_ELEM_CONTAINED_OP:
/*
* "range @> elem" is more or less identical to a scalar
* inequality "A >= b AND A <= c".
*/
return DEFAULT_RANGE_INEQ_SEL;
case OID_RANGE_LESS_OP:
case OID_RANGE_LESS_EQUAL_OP:
case OID_RANGE_GREATER_OP:
case OID_RANGE_GREATER_EQUAL_OP:
case OID_RANGE_LEFT_OP:
case OID_RANGE_RIGHT_OP:
case OID_RANGE_OVERLAPS_LEFT_OP:
case OID_RANGE_OVERLAPS_RIGHT_OP:
/* these are similar to regular scalar inequalities */
return DEFAULT_INEQ_SEL;
default:
/* all range operators should be handled above, but just in case */
return 0.01;
}
}
/*
* rangesel -- restriction selectivity for range operators
*/
Datum
rangesel(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;
TypeCacheEntry *typcache = NULL;
RangeType *constrange = NULL;
/*
* 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_range_selectivity(operator));
/*
* Can't do anything useful if the something is not a constant, either.
*/
if (!IsA(other, Const))
{
ReleaseVariableStats(vardata);
PG_RETURN_FLOAT8(default_range_selectivity(operator));
}
/*
* All the range 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)
{
/* we have other Op var, commute to make var Op other */
operator = get_commutator(operator);
if (!operator)
{
/* Use default selectivity (should we raise an error instead?) */
ReleaseVariableStats(vardata);
PG_RETURN_FLOAT8(default_range_selectivity(operator));
}
}
/*
* OK, there's a Var and a Const we're dealing with here. We need the
* Const to be of same range type as the 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.)
*
* If the operator is "range @> element", the constant should be of the
* element type of the range column. Convert it to a range that includes
* only that single point, so that we don't need special handling for that
* in what follows.
*/
if (operator == OID_RANGE_CONTAINS_ELEM_OP)
{
typcache = range_get_typcache(fcinfo, vardata.vartype);
if (((Const *) other)->consttype == typcache->rngelemtype->type_id)
{
RangeBound lower,
upper;
lower.inclusive = true;
lower.val = ((Const *) other)->constvalue;
lower.infinite = false;
lower.lower = true;
upper.inclusive = true;
upper.val = ((Const *) other)->constvalue;
upper.infinite = false;
upper.lower = false;
constrange = range_serialize(typcache, &lower, &upper, false);
}
}
else if (operator == OID_RANGE_ELEM_CONTAINED_OP)
{
/*
* Here, the Var is the elem, not the range. For now we just punt and
* return the default estimate. In future we could disassemble the
* range constant and apply scalarineqsel ...
*/
}
else if (((Const *) other)->consttype == vardata.vartype)
{
/* Both sides are the same range type */
typcache = range_get_typcache(fcinfo, vardata.vartype);
constrange = DatumGetRangeTypeP(((Const *) other)->constvalue);
}
/*
* If we got a valid constant on one side of the operator, proceed to
* estimate using statistics. Otherwise punt and return a default constant
* estimate. Note that calc_rangesel need not handle
* OID_RANGE_ELEM_CONTAINED_OP.
*/
if (constrange)
selec = calc_rangesel(typcache, &vardata, constrange, operator);
else
selec = default_range_selectivity(operator);
ReleaseVariableStats(vardata);
CLAMP_PROBABILITY(selec);
PG_RETURN_FLOAT8((float8) selec);
}
static double
calc_rangesel(TypeCacheEntry *typcache, VariableStatData *vardata,
RangeType *constval, Oid operator)
{
double hist_selec;
double selec;
float4 empty_frac,
null_frac;
/*
* First look up the fraction of NULLs and empty ranges from pg_statistic.
*/
if (HeapTupleIsValid(vardata->statsTuple))
{
Form_pg_statistic stats;
AttStatsSlot sslot;
stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
null_frac = stats->stanullfrac;
/* Try to get fraction of empty ranges */
if (get_attstatsslot(&sslot, vardata->statsTuple,
STATISTIC_KIND_RANGE_LENGTH_HISTOGRAM,
InvalidOid,
ATTSTATSSLOT_NUMBERS))
{
if (sslot.nnumbers != 1)
elog(ERROR, "invalid empty fraction statistic"); /* shouldn't happen */
empty_frac = sslot.numbers[0];
free_attstatsslot(&sslot);
}
else
{
/* No empty fraction statistic. Assume no empty ranges. */
empty_frac = 0.0;
}
}
else
{
/*
* No stats are available. Follow through the calculations below
* anyway, assuming no NULLs and no empty ranges. This still allows us
* to give a better-than-nothing estimate based on whether the
* constant is an empty range or not.
*/
null_frac = 0.0;
empty_frac = 0.0;
}
if (RangeIsEmpty(constval))
{
/*
* An empty range matches all ranges, all empty ranges, or nothing,
* depending on the operator
*/
switch (operator)
{
/* these return false if either argument is empty */
case OID_RANGE_OVERLAP_OP:
case OID_RANGE_OVERLAPS_LEFT_OP:
case OID_RANGE_OVERLAPS_RIGHT_OP:
case OID_RANGE_LEFT_OP:
case OID_RANGE_RIGHT_OP:
/* nothing is less than an empty range */
case OID_RANGE_LESS_OP:
selec = 0.0;
break;
/* only empty ranges can be contained by an empty range */
case OID_RANGE_CONTAINED_OP:
/* only empty ranges are <= an empty range */
case OID_RANGE_LESS_EQUAL_OP:
selec = empty_frac;
break;
/* everything contains an empty range */
case OID_RANGE_CONTAINS_OP:
/* everything is >= an empty range */
case OID_RANGE_GREATER_EQUAL_OP:
selec = 1.0;
break;
/* all non-empty ranges are > an empty range */
case OID_RANGE_GREATER_OP:
selec = 1.0 - empty_frac;
break;
/* an element cannot be empty */
case OID_RANGE_CONTAINS_ELEM_OP:
default:
elog(ERROR, "unexpected operator %u", operator);
selec = 0.0; /* keep compiler quiet */
break;
}
}
else
{
/*
* Calculate selectivity using bound histograms. If that fails for
* some reason, e.g no histogram in pg_statistic, use the default
* constant estimate for the fraction of non-empty values. This is
* still somewhat better than just returning the default estimate,
* because this still takes into account the fraction of empty and
* NULL tuples, if we had statistics for them.
*/
hist_selec = calc_hist_selectivity(typcache, vardata, constval,
operator);
if (hist_selec < 0.0)
hist_selec = default_range_selectivity(operator);
/*
* Now merge the results for the empty ranges and histogram
* calculations, realizing that the histogram covers only the
* non-null, non-empty values.
*/
if (operator == OID_RANGE_CONTAINED_OP)
{
/* empty is contained by anything non-empty */
selec = (1.0 - empty_frac) * hist_selec + empty_frac;
}
else
{
/* with any other operator, empty Op non-empty matches nothing */
selec = (1.0 - empty_frac) * hist_selec;
}
}
/* all range operators are strict */
selec *= (1.0 - null_frac);
/* result should be in range, but make sure... */
CLAMP_PROBABILITY(selec);
return selec;
}
/*
* Calculate range operator selectivity using histograms of range bounds.
*
* This estimate is for the portion of values that are not empty and not
* NULL.
*/
static double
calc_hist_selectivity(TypeCacheEntry *typcache, VariableStatData *vardata,
RangeType *constval, Oid operator)
{
AttStatsSlot hslot;
AttStatsSlot lslot;
int nhist;
RangeBound *hist_lower;
RangeBound *hist_upper;
int i;
RangeBound const_lower;
RangeBound const_upper;
bool empty;
double hist_selec;
/* Can't use the histogram with insecure range support functions */
if (!statistic_proc_security_check(vardata,
typcache->rng_cmp_proc_finfo.fn_oid))
return -1;
if (OidIsValid(typcache->rng_subdiff_finfo.fn_oid) &&
!statistic_proc_security_check(vardata,
typcache->rng_subdiff_finfo.fn_oid))
return -1;
/* Try to get histogram of ranges */
if (!(HeapTupleIsValid(vardata->statsTuple) &&
get_attstatsslot(&hslot, vardata->statsTuple,
STATISTIC_KIND_BOUNDS_HISTOGRAM, InvalidOid,
ATTSTATSSLOT_VALUES)))
return -1.0;
/*
* Convert histogram of ranges into histograms of its lower and upper
* bounds.
*/
nhist = hslot.nvalues;
hist_lower = (RangeBound *) palloc(sizeof(RangeBound) * nhist);
hist_upper = (RangeBound *) palloc(sizeof(RangeBound) * nhist);
for (i = 0; i < nhist; i++)
{
range_deserialize(typcache, DatumGetRangeTypeP(hslot.values[i]),
&hist_lower[i], &hist_upper[i], &empty);
/* The histogram should not contain any empty ranges */
if (empty)
elog(ERROR, "bounds histogram contains an empty range");
}
/* @> and @< also need a histogram of range lengths */
if (operator == OID_RANGE_CONTAINS_OP ||
operator == OID_RANGE_CONTAINED_OP)
{
if (!(HeapTupleIsValid(vardata->statsTuple) &&
get_attstatsslot(&lslot, vardata->statsTuple,
STATISTIC_KIND_RANGE_LENGTH_HISTOGRAM,
InvalidOid,
ATTSTATSSLOT_VALUES)))
{
free_attstatsslot(&hslot);
return -1.0;
}
/* check that it's a histogram, not just a dummy entry */
if (lslot.nvalues < 2)
{
free_attstatsslot(&lslot);
free_attstatsslot(&hslot);
return -1.0;
}
}
else
memset(&lslot, 0, sizeof(lslot));
/* Extract the bounds of the constant value. */
range_deserialize(typcache, constval, &const_lower, &const_upper, &empty);
Assert(!empty);
/*
* Calculate selectivity comparing the lower or upper bound of the
* constant with the histogram of lower or upper bounds.
*/
switch (operator)
{
case OID_RANGE_LESS_OP:
/*
* The regular b-tree comparison operators (<, <=, >, >=) compare
* the lower bounds first, and the upper bounds for values with
* equal lower bounds. Estimate that by comparing the lower bounds
* only. This gives a fairly accurate estimate assuming there
* aren't many rows with a lower bound equal to the constant's
* lower bound.
*/
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, false);
break;
case OID_RANGE_LESS_EQUAL_OP:
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, true);
break;
case OID_RANGE_GREATER_OP:
hist_selec =
1 - calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, false);
break;
case OID_RANGE_GREATER_EQUAL_OP:
hist_selec =
1 - calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, true);
break;
case OID_RANGE_LEFT_OP:
/* var << const when upper(var) < lower(const) */
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_lower,
hist_upper, nhist, false);
break;
case OID_RANGE_RIGHT_OP:
/* var >> const when lower(var) > upper(const) */
hist_selec =
1 - calc_hist_selectivity_scalar(typcache, &const_upper,
hist_lower, nhist, true);
break;
case OID_RANGE_OVERLAPS_RIGHT_OP:
/* compare lower bounds */
hist_selec =
1 - calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, false);
break;
case OID_RANGE_OVERLAPS_LEFT_OP:
/* compare upper bounds */
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_upper,
hist_upper, nhist, true);
break;
case OID_RANGE_OVERLAP_OP:
case OID_RANGE_CONTAINS_ELEM_OP:
/*
* A && B <=> NOT (A << B OR A >> B).
*
* Since A << B and A >> B are mutually exclusive events we can
* sum their probabilities to find probability of (A << B OR A >>
* B).
*
* "range @> elem" is equivalent to "range && [elem,elem]". The
* caller already constructed the singular range from the element
* constant, so just treat it the same as &&.
*/
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_lower, hist_upper,
nhist, false);
hist_selec +=
(1.0 - calc_hist_selectivity_scalar(typcache, &const_upper, hist_lower,
nhist, true));
hist_selec = 1.0 - hist_selec;
break;
case OID_RANGE_CONTAINS_OP:
hist_selec =
calc_hist_selectivity_contains(typcache, &const_lower,
&const_upper, hist_lower, nhist,
lslot.values, lslot.nvalues);
break;
case OID_RANGE_CONTAINED_OP:
if (const_lower.infinite)
{
/*
* Lower bound no longer matters. Just estimate the fraction
* with an upper bound <= const upper bound
*/
hist_selec =
calc_hist_selectivity_scalar(typcache, &const_upper,
hist_upper, nhist, true);
}
else if (const_upper.infinite)
{
hist_selec =
1.0 - calc_hist_selectivity_scalar(typcache, &const_lower,
hist_lower, nhist, false);
}
else
{
hist_selec =
calc_hist_selectivity_contained(typcache, &const_lower,
&const_upper, hist_lower, nhist,
lslot.values, lslot.nvalues);
}
break;
default:
elog(ERROR, "unknown range operator %u", operator);
hist_selec = -1.0; /* keep compiler quiet */
break;
}
free_attstatsslot(&lslot);
free_attstatsslot(&hslot);
return hist_selec;
}
/*
* Look up the fraction of values less than (or equal, if 'equal' argument
* is true) a given const in a histogram of range bounds.
*/
static double
calc_hist_selectivity_scalar(TypeCacheEntry *typcache, RangeBound *constbound,
RangeBound *hist, int hist_nvalues, bool equal)
{
Selectivity selec;
int index;
/*
* Find the histogram bin the given constant falls into. Estimate
* selectivity as the number of preceding whole bins.
*/
index = rbound_bsearch(typcache, constbound, hist, hist_nvalues, equal);
selec = (Selectivity) (Max(index, 0)) / (Selectivity) (hist_nvalues - 1);
/* Adjust using linear interpolation within the bin */
if (index >= 0 && index < hist_nvalues - 1)
selec += get_position(typcache, constbound, &hist[index],
&hist[index + 1]) / (Selectivity) (hist_nvalues - 1);
return selec;
}
/*
* Binary search on an array of range bounds. Returns greatest index of range
* bound in array which is less(less or equal) than given range bound. If all
* range bounds in array are greater or equal(greater) than given range bound,
* return -1. When "equal" flag is set conditions in brackets are used.
*
* This function is used in scalar operator selectivity estimation. Another
* goal of this function is to find a histogram bin where to stop
* interpolation of portion of bounds which are less or equal to given bound.
*/
static int
rbound_bsearch(TypeCacheEntry *typcache, RangeBound *value, RangeBound *hist,
int hist_length, bool equal)
{
int lower = -1,
upper = hist_length - 1,
cmp,
middle;
while (lower < upper)
{
middle = (lower + upper + 1) / 2;
cmp = range_cmp_bounds(typcache, &hist[middle], value);
if (cmp < 0 || (equal && cmp == 0))
lower = middle;
else
upper = middle - 1;
}
return lower;
}
/*
* Binary search on length histogram. Returns greatest index of range length in
* histogram which is less than (less than or equal) the given length value. If
* all lengths in the histogram are greater than (greater than or equal) the
* given length, returns -1.
*/
static int
length_hist_bsearch(Datum *length_hist_values, int length_hist_nvalues,
double value, bool equal)
{
int lower = -1,
upper = length_hist_nvalues - 1,
middle;
while (lower < upper)
{
double middleval;
middle = (lower + upper + 1) / 2;
middleval = DatumGetFloat8(length_hist_values[middle]);
if (middleval < value || (equal && middleval <= value))
lower = middle;
else
upper = middle - 1;
}
return lower;
}
/*
* Get relative position of value in histogram bin in [0,1] range.
*/
static float8
get_position(TypeCacheEntry *typcache, RangeBound *value, RangeBound *hist1,
RangeBound *hist2)
{
bool has_subdiff = OidIsValid(typcache->rng_subdiff_finfo.fn_oid);
float8 position;
if (!hist1->infinite && !hist2->infinite)
{
float8 bin_width;
/*
* Both bounds are finite. Assuming the subtype's comparison function
* works sanely, the value must be finite, too, because it lies
* somewhere between the bounds. If it doesn't, just return something.
*/
if (value->infinite)
return 0.5;
/* Can't interpolate without subdiff function */
if (!has_subdiff)
return 0.5;
/* Calculate relative position using subdiff function. */
bin_width = DatumGetFloat8(FunctionCall2Coll(
&typcache->rng_subdiff_finfo,
typcache->rng_collation,
hist2->val,
hist1->val));
if (bin_width <= 0.0)
return 0.5; /* zero width bin */
position = DatumGetFloat8(FunctionCall2Coll(
&typcache->rng_subdiff_finfo,
typcache->rng_collation,
value->val,
hist1->val))
/ bin_width;
/* Relative position must be in [0,1] range */
position = Max(position, 0.0);
position = Min(position, 1.0);
return position;
}
else if (hist1->infinite && !hist2->infinite)
{
/*
* Lower bin boundary is -infinite, upper is finite. If the value is
* -infinite, return 0.0 to indicate it's equal to the lower bound.
* Otherwise return 1.0 to indicate it's infinitely far from the lower
* bound.
*/
return ((value->infinite && value->lower) ? 0.0 : 1.0);
}
else if (!hist1->infinite && hist2->infinite)
{
/* same as above, but in reverse */
return ((value->infinite && !value->lower) ? 1.0 : 0.0);
}
else
{
/*
* If both bin boundaries are infinite, they should be equal to each
* other, and the value should also be infinite and equal to both
* bounds. (But don't Assert that, to avoid crashing if a user creates
* a datatype with a broken comparison function).
*
* Assume the value to lie in the middle of the infinite bounds.
*/
return 0.5;
}
}
/*
* Get relative position of value in a length histogram bin in [0,1] range.
*/
static double
get_len_position(double value, double hist1, double hist2)
{
if (!isinf(hist1) && !isinf(hist2))
{
/*
* Both bounds are finite. The value should be finite too, because it
* lies somewhere between the bounds. If it doesn't, just return
* something.
*/
if (isinf(value))
return 0.5;
return 1.0 - (hist2 - value) / (hist2 - hist1);
}
else if (isinf(hist1) && !isinf(hist2))
{
/*
* Lower bin boundary is -infinite, upper is finite. Return 1.0 to
* indicate the value is infinitely far from the lower bound.
*/
return 1.0;
}
else if (isinf(hist1) && isinf(hist2))
{
/* same as above, but in reverse */
return 0.0;
}
else
{
/*
* If both bin boundaries are infinite, they should be equal to each
* other, and the value should also be infinite and equal to both
* bounds. (But don't Assert that, to avoid crashing unnecessarily if
* the caller messes up)
*
* Assume the value to lie in the middle of the infinite bounds.
*/
return 0.5;
}
}
/*
* Measure distance between two range bounds.
*/
static float8
get_distance(TypeCacheEntry *typcache, RangeBound *bound1, RangeBound *bound2)
{
bool has_subdiff = OidIsValid(typcache->rng_subdiff_finfo.fn_oid);
if (!bound1->infinite && !bound2->infinite)
{
/*
* No bounds are infinite, use subdiff function or return default
* value of 1.0 if no subdiff is available.
*/
if (has_subdiff)
return
DatumGetFloat8(FunctionCall2Coll(&typcache->rng_subdiff_finfo,
typcache->rng_collation,
bound2->val,
bound1->val));
else
return 1.0;
}
else if (bound1->infinite && bound2->infinite)
{
/* Both bounds are infinite */
if (bound1->lower == bound2->lower)
return 0.0;
else
return get_float8_infinity();
}
else
{
/* One bound is infinite, another is not */
return get_float8_infinity();
}
}
/*
* Calculate the average of function P(x), in the interval [length1, length2],
* where P(x) is the fraction of tuples with length < x (or length <= x if
* 'equal' is true).
*/
static double
calc_length_hist_frac(Datum *length_hist_values, int length_hist_nvalues,
double length1, double length2, bool equal)
{
double frac;
double A,
B,
PA,
PB;
double pos;
int i;
double area;
Assert(length2 >= length1);
if (length2 < 0.0)
return 0.0; /* shouldn't happen, but doesn't hurt to check */
/* All lengths in the table are <= infinite. */
if (isinf(length2) && equal)
return 1.0;
/*----------
* The average of a function between A and B can be calculated by the
* formula:
*
* B
* 1 /
* ------- | P(x)dx
* B - A /
* A
*
* The geometrical interpretation of the integral is the area under the
* graph of P(x). P(x) is defined by the length histogram. We calculate
* the area in a piecewise fashion, iterating through the length histogram
* bins. Each bin is a trapezoid:
*
* P(x2)
* /|
* / |
* P(x1)/ |
* | |
* | |
* ---+---+--
* x1 x2
*
* where x1 and x2 are the boundaries of the current histogram, and P(x1)
* and P(x1) are the cumulative fraction of tuples at the boundaries.
*
* The area of each trapezoid is 1/2 * (P(x2) + P(x1)) * (x2 - x1)
*
* The first bin contains the lower bound passed by the caller, so we
* use linear interpolation between the previous and next histogram bin
* boundary to calculate P(x1). Likewise for the last bin: we use linear
* interpolation to calculate P(x2). For the bins in between, x1 and x2
* lie on histogram bin boundaries, so P(x1) and P(x2) are simply:
* P(x1) = (bin index) / (number of bins)
* P(x2) = (bin index + 1 / (number of bins)
*/
/* First bin, the one that contains lower bound */
i = length_hist_bsearch(length_hist_values, length_hist_nvalues, length1, equal);
if (i >= length_hist_nvalues - 1)
return 1.0;
if (i < 0)
{
i = 0;
pos = 0.0;
}
else
{
/* interpolate length1's position in the bin */
pos = get_len_position(length1,
DatumGetFloat8(length_hist_values[i]),
DatumGetFloat8(length_hist_values[i + 1]));
}
PB = (((double) i) + pos) / (double) (length_hist_nvalues - 1);
B = length1;
/*
* In the degenerate case that length1 == length2, simply return
* P(length1). This is not merely an optimization: if length1 == length2,
* we'd divide by zero later on.
*/
if (length2 == length1)
return PB;
/*
* Loop through all the bins, until we hit the last bin, the one that
* contains the upper bound. (if lower and upper bounds are in the same
* bin, this falls out immediately)
*/
area = 0.0;
for (; i < length_hist_nvalues - 1; i++)
{
double bin_upper = DatumGetFloat8(length_hist_values[i + 1]);
/* check if we've reached the last bin */
if (!(bin_upper < length2 || (equal && bin_upper <= length2)))
break;
/* the upper bound of previous bin is the lower bound of this bin */
A = B;
PA = PB;
B = bin_upper;
PB = (double) i / (double) (length_hist_nvalues - 1);
/*
* Add the area of this trapezoid to the total. The point of the
* if-check is to avoid NaN, in the corner case that PA == PB == 0,
* and B - A == Inf. The area of a zero-height trapezoid (PA == PB ==
* 0) is zero, regardless of the width (B - A).
*/
if (PA > 0 || PB > 0)
area += 0.5 * (PB + PA) * (B - A);
}
/* Last bin */
A = B;
PA = PB;
B = length2; /* last bin ends at the query upper bound */
if (i >= length_hist_nvalues - 1)
pos = 0.0;
else
{
if (DatumGetFloat8(length_hist_values[i]) == DatumGetFloat8(length_hist_values[i + 1]))
pos = 0.0;
else
pos = get_len_position(length2, DatumGetFloat8(length_hist_values[i]), DatumGetFloat8(length_hist_values[i + 1]));
}
PB = (((double) i) + pos) / (double) (length_hist_nvalues - 1);
if (PA > 0 || PB > 0)
area += 0.5 * (PB + PA) * (B - A);
/*
* Ok, we have calculated the area, ie. the integral. Divide by width to
* get the requested average.
*
* Avoid NaN arising from infinite / infinite. This happens at least if
* length2 is infinite. It's not clear what the correct value would be in
* that case, so 0.5 seems as good as any value.
*/
if (isinf(area) && isinf(length2))
frac = 0.5;
else
frac = area / (length2 - length1);
return frac;
}
/*
* Calculate selectivity of "var <@ const" operator, ie. estimate the fraction
* of ranges that fall within the constant lower and upper bounds. This uses
* the histograms of range lower bounds and range lengths, on the assumption
* that the range lengths are independent of the lower bounds.
*
* The caller has already checked that constant lower and upper bounds are
* finite.
*/
static double
calc_hist_selectivity_contained(TypeCacheEntry *typcache,
RangeBound *lower, RangeBound *upper,
RangeBound *hist_lower, int hist_nvalues,
Datum *length_hist_values, int length_hist_nvalues)
{
int i,
upper_index;
float8 prev_dist;
double bin_width;
double upper_bin_width;
double sum_frac;
/*
* Begin by finding the bin containing the upper bound, in the lower bound
* histogram. Any range with a lower bound > constant upper bound can't
* match, ie. there are no matches in bins greater than upper_index.
*/
upper->inclusive = !upper->inclusive;
upper->lower = true;
upper_index = rbound_bsearch(typcache, upper, hist_lower, hist_nvalues,
false);
/*
* Calculate upper_bin_width, ie. the fraction of the (upper_index,
* upper_index + 1) bin which is greater than upper bound of query range
* using linear interpolation of subdiff function.
*/
if (upper_index >= 0 && upper_index < hist_nvalues - 1)
upper_bin_width = get_position(typcache, upper,
&hist_lower[upper_index],
&hist_lower[upper_index + 1]);
else
upper_bin_width = 0.0;
/*
* In the loop, dist and prev_dist are the distance of the "current" bin's
* lower and upper bounds from the constant upper bound.
*
* bin_width represents the width of the current bin. Normally it is 1.0,
* meaning a full width bin, but can be less in the corner cases: start
* and end of the loop. We start with bin_width = upper_bin_width, because
* we begin at the bin containing the upper bound.
*/
prev_dist = 0.0;
bin_width = upper_bin_width;
sum_frac = 0.0;
for (i = upper_index; i >= 0; i--)
{
double dist;
double length_hist_frac;
bool final_bin = false;
/*
* dist -- distance from upper bound of query range to lower bound of
* the current bin in the lower bound histogram. Or to the lower bound
* of the constant range, if this is the final bin, containing the
* constant lower bound.
*/
if (range_cmp_bounds(typcache, &hist_lower[i], lower) < 0)
{
dist = get_distance(typcache, lower, upper);
/*
* Subtract from bin_width the portion of this bin that we want to
* ignore.
*/
bin_width -= get_position(typcache, lower, &hist_lower[i],
&hist_lower[i + 1]);
if (bin_width < 0.0)
bin_width = 0.0;
final_bin = true;
}
else
dist = get_distance(typcache, &hist_lower[i], upper);
/*
* Estimate the fraction of tuples in this bin that are narrow enough
* to not exceed the distance to the upper bound of the query range.
*/
length_hist_frac = calc_length_hist_frac(length_hist_values,
length_hist_nvalues,
prev_dist, dist, true);
/*
* Add the fraction of tuples in this bin, with a suitable length, to
* the total.
*/
sum_frac += length_hist_frac * bin_width / (double) (hist_nvalues - 1);
if (final_bin)
break;
bin_width = 1.0;
prev_dist = dist;
}
return sum_frac;
}
/*
* Calculate selectivity of "var @> const" operator, ie. estimate the fraction
* of ranges that contain the constant lower and upper bounds. This uses
* the histograms of range lower bounds and range lengths, on the assumption
* that the range lengths are independent of the lower bounds.
*
* Note, this is "var @> const", ie. estimate the fraction of ranges that
* contain the constant lower and upper bounds.
*/
static double
calc_hist_selectivity_contains(TypeCacheEntry *typcache,
RangeBound *lower, RangeBound *upper,
RangeBound *hist_lower, int hist_nvalues,
Datum *length_hist_values, int length_hist_nvalues)
{
int i,
lower_index;
double bin_width,
lower_bin_width;
double sum_frac;
float8 prev_dist;
/* Find the bin containing the lower bound of query range. */
lower_index = rbound_bsearch(typcache, lower, hist_lower, hist_nvalues,
true);
/*
* Calculate lower_bin_width, ie. the fraction of the of (lower_index,
* lower_index + 1) bin which is greater than lower bound of query range
* using linear interpolation of subdiff function.
*/
if (lower_index >= 0 && lower_index < hist_nvalues - 1)
lower_bin_width = get_position(typcache, lower, &hist_lower[lower_index],
&hist_lower[lower_index + 1]);
else
lower_bin_width = 0.0;
/*
* Loop through all the lower bound bins, smaller than the query lower
* bound. In the loop, dist and prev_dist are the distance of the
* "current" bin's lower and upper bounds from the constant upper bound.
* We begin from query lower bound, and walk backwards, so the first bin's
* upper bound is the query lower bound, and its distance to the query
* upper bound is the length of the query range.
*
* bin_width represents the width of the current bin. Normally it is 1.0,
* meaning a full width bin, except for the first bin, which is only
* counted up to the constant lower bound.
*/
prev_dist = get_distance(typcache, lower, upper);
sum_frac = 0.0;
bin_width = lower_bin_width;
for (i = lower_index; i >= 0; i--)
{
float8 dist;
double length_hist_frac;
/*
* dist -- distance from upper bound of query range to current value
* of lower bound histogram or lower bound of query range (if we've
* reach it).
*/
dist = get_distance(typcache, &hist_lower[i], upper);
/*
* Get average fraction of length histogram which covers intervals
* longer than (or equal to) distance to upper bound of query range.
*/
length_hist_frac =
1.0 - calc_length_hist_frac(length_hist_values,
length_hist_nvalues,
prev_dist, dist, false);
sum_frac += length_hist_frac * bin_width / (double) (hist_nvalues - 1);
bin_width = 1.0;
prev_dist = dist;
}
return sum_frac;
}
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