Provide real selectivity estimators for inet/cidr operators.

This patch fills in the formerly-stub networksel() and networkjoinsel()
estimation functions.  Those are used for << <<= >> >>= and && operators
on inet/cidr types.  The estimation is not perfect, certainly, because
we rely on the existing statistics collected for the inet btree operators.
But it's a long way better than nothing, and it's not clear that asking
ANALYZE to collect separate stats for these operators would be a win.

Emre Hasegeli, with reviews from Dilip Kumar and Heikki Linnakangas,
and some further hacking by me

src/backend/utils/adt/network_selfuncs.c

@@ -3,7 +3,9 @@
  * network_selfuncs.c
  *	  Functions for selectivity estimation of inet/cidr operators
  *
- * Currently these are just stubs, but we hope to do better soon.
+ * This module provides estimators for the subnet inclusion and overlap
+ * operators.  Estimates are based on null fraction, most common values,
+ * and histogram of inet/cidr columns.
  *
  * Portions Copyright (c) 1996-2015, PostgreSQL Global Development Group
  * Portions Copyright (c) 1994, Regents of the University of California
@@ -16,17 +18,989 @@
  */
 #include "postgres.h"
 
+#include <math.h>
+
+#include "access/htup_details.h"
+#include "catalog/pg_operator.h"
+#include "catalog/pg_statistic.h"
 #include "utils/inet.h"
+#include "utils/lsyscache.h"
+#include "utils/selfuncs.h"
+
+
+/* Default selectivity for the inet overlap operator */
+#define DEFAULT_OVERLAP_SEL 0.01
+
+/* Default selectivity for the various inclusion operators */
+#define DEFAULT_INCLUSION_SEL 0.005
+
+/* Default selectivity for specified operator */
+#define DEFAULT_SEL(operator) \
+	((operator) == OID_INET_OVERLAP_OP ? \
+	 DEFAULT_OVERLAP_SEL : DEFAULT_INCLUSION_SEL)
 
+/* Maximum number of items to consider in join selectivity calculations */
+#define MAX_CONSIDERED_ELEMS 1024
 
+static Selectivity networkjoinsel_inner(Oid operator,
+					 VariableStatData *vardata1, VariableStatData *vardata2);
+static Selectivity networkjoinsel_semi(Oid operator,
+					VariableStatData *vardata1, VariableStatData *vardata2);
+static Selectivity mcv_population(float4 *mcv_numbers, int mcv_nvalues);
+static Selectivity inet_hist_value_sel(Datum *values, int nvalues,
+					Datum constvalue, int opr_codenum);
+static Selectivity inet_mcv_join_sel(Datum *mcv1_values,
+				  float4 *mcv1_numbers, int mcv1_nvalues, Datum *mcv2_values,
+				  float4 *mcv2_numbers, int mcv2_nvalues, Oid operator);
+static Selectivity inet_mcv_hist_sel(Datum *mcv_values, float4 *mcv_numbers,
+				  int mcv_nvalues, Datum *hist_values, int hist_nvalues,
+				  int opr_codenum);
+static Selectivity inet_hist_inclusion_join_sel(Datum *hist1_values,
+							 int hist1_nvalues,
+							 Datum *hist2_values, int hist2_nvalues,
+							 int opr_codenum);
+static Selectivity inet_semi_join_sel(Datum lhs_value,
+				   bool mcv_exists, Datum *mcv_values, int mcv_nvalues,
+				   bool hist_exists, Datum *hist_values, int hist_nvalues,
+				   double hist_weight,
+				   FmgrInfo *proc, int opr_codenum);
+static int	inet_opr_codenum(Oid operator);
+static int	inet_inclusion_cmp(inet *left, inet *right, int opr_codenum);
+static int inet_masklen_inclusion_cmp(inet *left, inet *right,
+						   int opr_codenum);
+static int inet_hist_match_divider(inet *boundary, inet *query,
+						int opr_codenum);
+
+/*
+ * Selectivity estimation for the subnet inclusion/overlap operators
+ */
 Datum
 networksel(PG_FUNCTION_ARGS)
 {
-	PG_RETURN_FLOAT8(0.001);
+	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,
+				mcv_selec,
+				non_mcv_selec;
+	Datum		constvalue,
+			   *hist_values;
+	int			hist_nvalues;
+	Form_pg_statistic stats;
+	double		sumcommon,
+				nullfrac;
+	FmgrInfo	proc;
+
+	/*
+	 * 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));
+	}
+
+	/* All of the operators handled here are strict. */
+	if (((Const *) other)->constisnull)
+	{
+		ReleaseVariableStats(vardata);
+		PG_RETURN_FLOAT8(0.0);
+	}
+	constvalue = ((Const *) other)->constvalue;
+
+	/* Otherwise, we need stats in order to produce a non-default estimate. */
+	if (!HeapTupleIsValid(vardata.statsTuple))
+	{
+		ReleaseVariableStats(vardata);
+		PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
+	}
+
+	stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
+	nullfrac = stats->stanullfrac;
+
+	/*
+	 * If we have most-common-values info, add up the fractions of the MCV
+	 * entries that satisfy MCV OP CONST.  These fractions contribute directly
+	 * to the result selectivity.  Also add up the total fraction represented
+	 * by MCV entries.
+	 */
+	fmgr_info(get_opcode(operator), &proc);
+	mcv_selec = mcv_selectivity(&vardata, &proc, constvalue, varonleft,
+								&sumcommon);
+
+	/*
+	 * If we have a histogram, use it to estimate the proportion of the
+	 * non-MCV population that satisfies the clause.  If we don't, apply the
+	 * default selectivity to that population.
+	 */
+	if (get_attstatsslot(vardata.statsTuple,
+						 vardata.atttype, vardata.atttypmod,
+						 STATISTIC_KIND_HISTOGRAM, InvalidOid,
+						 NULL,
+						 &hist_values, &hist_nvalues,
+						 NULL, NULL))
+	{
+		int			opr_codenum = inet_opr_codenum(operator);
+
+		/* Commute if needed, so we can consider histogram to be on the left */
+		if (!varonleft)
+			opr_codenum = -opr_codenum;
+		non_mcv_selec = inet_hist_value_sel(hist_values, hist_nvalues,
+											constvalue, opr_codenum);
+
+		free_attstatsslot(vardata.atttype, hist_values, hist_nvalues, NULL, 0);
+	}
+	else
+		non_mcv_selec = DEFAULT_SEL(operator);
+
+	/* Combine selectivities for MCV and non-MCV populations */
+	selec = mcv_selec + (1.0 - nullfrac - sumcommon) * non_mcv_selec;
+
+	/* Result should be in range, but make sure... */
+	CLAMP_PROBABILITY(selec);
+
+	ReleaseVariableStats(vardata);
+
+	PG_RETURN_FLOAT8(selec);
 }
 
+/*
+ * Join selectivity estimation for the subnet inclusion/overlap operators
+ *
+ * This function has the same structure as eqjoinsel() in selfuncs.c.
+ *
+ * Throughout networkjoinsel and its subroutines, we have a performance issue
+ * in that the amount of work to be done is O(N^2) in the length of the MCV
+ * and histogram arrays.  To keep the runtime from getting out of hand when
+ * large statistics targets have been set, we arbitrarily limit the number of
+ * values considered to 1024 (MAX_CONSIDERED_ELEMS).  For the MCV arrays, this
+ * is easy: just consider at most the first N elements.  (Since the MCVs are
+ * sorted by decreasing frequency, this correctly gets us the first N MCVs.)
+ * For the histogram arrays, we decimate; that is consider only every k'th
+ * element, where k is chosen so that no more than MAX_CONSIDERED_ELEMS
+ * elements are considered.  This should still give us a good random sample of
+ * the non-MCV population.  Decimation is done on-the-fly in the loops that
+ * iterate over the histogram arrays.
+ */
 Datum
 networkjoinsel(PG_FUNCTION_ARGS)
 {
-	PG_RETURN_FLOAT8(0.001);
+	PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
+	Oid			operator = PG_GETARG_OID(1);
+	List	   *args = (List *) PG_GETARG_POINTER(2);
+#ifdef NOT_USED
+	JoinType	jointype = (JoinType) PG_GETARG_INT16(3);
+#endif
+	SpecialJoinInfo *sjinfo = (SpecialJoinInfo *) PG_GETARG_POINTER(4);
+	double		selec;
+	VariableStatData vardata1;
+	VariableStatData vardata2;
+	bool		join_is_reversed;
+
+	get_join_variables(root, args, sjinfo,
+					   &vardata1, &vardata2, &join_is_reversed);
+
+	switch (sjinfo->jointype)
+	{
+		case JOIN_INNER:
+		case JOIN_LEFT:
+		case JOIN_FULL:
+
+			/*
+			 * Selectivity for left/full join is not exactly the same as inner
+			 * join, but we neglect the difference, as eqjoinsel does.
+			 */
+			selec = networkjoinsel_inner(operator, &vardata1, &vardata2);
+			break;
+		case JOIN_SEMI:
+		case JOIN_ANTI:
+			/* Here, it's important that we pass the outer var on the left. */
+			if (!join_is_reversed)
+				selec = networkjoinsel_semi(operator, &vardata1, &vardata2);
+			else
+				selec = networkjoinsel_semi(get_commutator(operator),
+											&vardata2, &vardata1);
+			break;
+		default:
+			/* other values not expected here */
+			elog(ERROR, "unrecognized join type: %d",
+				 (int) sjinfo->jointype);
+			selec = 0;			/* keep compiler quiet */
+			break;
+	}
+
+	ReleaseVariableStats(vardata1);
+	ReleaseVariableStats(vardata2);
+
+	CLAMP_PROBABILITY(selec);
+
+	PG_RETURN_FLOAT8((float8) selec);
+}
+
+/*
+ * Inner join selectivity estimation for subnet inclusion/overlap operators
+ *
+ * Calculates MCV vs MCV, MCV vs histogram and histogram vs histogram
+ * selectivity for join using the subnet inclusion operators.  Unlike the
+ * join selectivity function for the equality operator, eqjoinsel_inner(),
+ * one to one matching of the values is not enough.  Network inclusion
+ * operators are likely to match many to many, so we must check all pairs.
+ * (Note: it might be possible to exploit understanding of the histogram's
+ * btree ordering to reduce the work needed, but we don't currently try.)
+ * Also, MCV vs histogram selectivity is not neglected as in eqjoinsel_inner().
+ */
+static Selectivity
+networkjoinsel_inner(Oid operator,
+					 VariableStatData *vardata1, VariableStatData *vardata2)
+{
+	Form_pg_statistic stats;
+	double		nullfrac1 = 0.0,
+				nullfrac2 = 0.0;
+	Selectivity selec = 0.0,
+				sumcommon1 = 0.0,
+				sumcommon2 = 0.0;
+	bool		mcv1_exists = false,
+				mcv2_exists = false,
+				hist1_exists = false,
+				hist2_exists = false;
+	int			opr_codenum;
+	int			mcv1_nvalues,
+				mcv2_nvalues,
+				mcv1_nnumbers,
+				mcv2_nnumbers,
+				hist1_nvalues,
+				hist2_nvalues,
+				mcv1_length = 0,
+				mcv2_length = 0;
+	Datum	   *mcv1_values,
+			   *mcv2_values,
+			   *hist1_values,
+			   *hist2_values;
+	float4	   *mcv1_numbers,
+			   *mcv2_numbers;
+
+	if (HeapTupleIsValid(vardata1->statsTuple))
+	{
+		stats = (Form_pg_statistic) GETSTRUCT(vardata1->statsTuple);
+		nullfrac1 = stats->stanullfrac;
+
+		mcv1_exists = get_attstatsslot(vardata1->statsTuple,
+									   vardata1->atttype, vardata1->atttypmod,
+									   STATISTIC_KIND_MCV, InvalidOid,
+									   NULL,
+									   &mcv1_values, &mcv1_nvalues,
+									   &mcv1_numbers, &mcv1_nnumbers);
+		hist1_exists = get_attstatsslot(vardata1->statsTuple,
+									  vardata1->atttype, vardata1->atttypmod,
+										STATISTIC_KIND_HISTOGRAM, InvalidOid,
+										NULL,
+										&hist1_values, &hist1_nvalues,
+										NULL, NULL);
+		/* Arbitrarily limit number of MCVs considered */
+		mcv1_length = Min(mcv1_nvalues, MAX_CONSIDERED_ELEMS);
+		if (mcv1_exists)
+			sumcommon1 = mcv_population(mcv1_numbers, mcv1_length);
+	}
+
+	if (HeapTupleIsValid(vardata2->statsTuple))
+	{
+		stats = (Form_pg_statistic) GETSTRUCT(vardata2->statsTuple);
+		nullfrac2 = stats->stanullfrac;
+
+		mcv2_exists = get_attstatsslot(vardata2->statsTuple,
+									   vardata2->atttype, vardata2->atttypmod,
+									   STATISTIC_KIND_MCV, InvalidOid,
+									   NULL,
+									   &mcv2_values, &mcv2_nvalues,
+									   &mcv2_numbers, &mcv2_nnumbers);
+		hist2_exists = get_attstatsslot(vardata2->statsTuple,
+									  vardata2->atttype, vardata2->atttypmod,
+										STATISTIC_KIND_HISTOGRAM, InvalidOid,
+										NULL,
+										&hist2_values, &hist2_nvalues,
+										NULL, NULL);
+		/* Arbitrarily limit number of MCVs considered */
+		mcv2_length = Min(mcv2_nvalues, MAX_CONSIDERED_ELEMS);
+		if (mcv2_exists)
+			sumcommon2 = mcv_population(mcv2_numbers, mcv2_length);
+	}
+
+	opr_codenum = inet_opr_codenum(operator);
+
+	/*
+	 * Calculate selectivity for MCV vs MCV matches.
+	 */
+	if (mcv1_exists && mcv2_exists)
+		selec += inet_mcv_join_sel(mcv1_values, mcv1_numbers, mcv1_length,
+								   mcv2_values, mcv2_numbers, mcv2_length,
+								   operator);
+
+	/*
+	 * Add in selectivities for MCV vs histogram matches, scaling according to
+	 * the fractions of the populations represented by the histograms. Note
+	 * that the second case needs to commute the operator.
+	 */
+	if (mcv1_exists && hist2_exists)
+		selec += (1.0 - nullfrac2 - sumcommon2) *
+			inet_mcv_hist_sel(mcv1_values, mcv1_numbers, mcv1_length,
+							  hist2_values, hist2_nvalues,
+							  opr_codenum);
+	if (mcv2_exists && hist1_exists)
+		selec += (1.0 - nullfrac1 - sumcommon1) *
+			inet_mcv_hist_sel(mcv2_values, mcv2_numbers, mcv2_length,
+							  hist1_values, hist1_nvalues,
+							  -opr_codenum);
+
+	/*
+	 * Add in selectivity for histogram vs histogram matches, again scaling
+	 * appropriately.
+	 */
+	if (hist1_exists && hist2_exists)
+		selec += (1.0 - nullfrac1 - sumcommon1) *
+			(1.0 - nullfrac2 - sumcommon2) *
+			inet_hist_inclusion_join_sel(hist1_values, hist1_nvalues,
+										 hist2_values, hist2_nvalues,
+										 opr_codenum);
+
+	/*
+	 * If useful statistics are not available then use the default estimate.
+	 * We can apply null fractions if known, though.
+	 */
+	if ((!mcv1_exists && !hist1_exists) || (!mcv2_exists && !hist2_exists))
+		selec = (1.0 - nullfrac1) * (1.0 - nullfrac2) * DEFAULT_SEL(operator);
+
+	/* Release stats. */
+	if (mcv1_exists)
+		free_attstatsslot(vardata1->atttype, mcv1_values, mcv1_nvalues,
+						  mcv1_numbers, mcv1_nnumbers);
+	if (mcv2_exists)
+		free_attstatsslot(vardata2->atttype, mcv2_values, mcv2_nvalues,
+						  mcv2_numbers, mcv2_nnumbers);
+	if (hist1_exists)
+		free_attstatsslot(vardata1->atttype, hist1_values, hist1_nvalues,
+						  NULL, 0);
+	if (hist2_exists)
+		free_attstatsslot(vardata2->atttype, hist2_values, hist2_nvalues,
+						  NULL, 0);
+
+	return selec;
+}
+
+/*
+ * Semi join selectivity estimation for subnet inclusion/overlap operators
+ *
+ * Calculates MCV vs MCV, MCV vs histogram, histogram vs MCV, and histogram vs
+ * histogram selectivity for semi/anti join cases.
+ */
+static Selectivity
+networkjoinsel_semi(Oid operator,
+					VariableStatData *vardata1, VariableStatData *vardata2)
+{
+	Form_pg_statistic stats;
+	Selectivity selec = 0.0,
+				sumcommon1 = 0.0,
+				sumcommon2 = 0.0;
+	double		nullfrac1 = 0.0,
+				nullfrac2 = 0.0,
+				hist2_weight = 0.0;
+	bool		mcv1_exists = false,
+				mcv2_exists = false,
+				hist1_exists = false,
+				hist2_exists = false;
+	int			opr_codenum;
+	FmgrInfo	proc;
+	int			i,
+				mcv1_nvalues,
+				mcv2_nvalues,
+				mcv1_nnumbers,
+				mcv2_nnumbers,
+				hist1_nvalues,
+				hist2_nvalues,
+				mcv1_length = 0,
+				mcv2_length = 0;
+	Datum	   *mcv1_values,
+			   *mcv2_values,
+			   *hist1_values,
+			   *hist2_values;
+	float4	   *mcv1_numbers,
+			   *mcv2_numbers;
+
+	if (HeapTupleIsValid(vardata1->statsTuple))
+	{
+		stats = (Form_pg_statistic) GETSTRUCT(vardata1->statsTuple);
+		nullfrac1 = stats->stanullfrac;
+
+		mcv1_exists = get_attstatsslot(vardata1->statsTuple,
+									   vardata1->atttype, vardata1->atttypmod,
+									   STATISTIC_KIND_MCV, InvalidOid,
+									   NULL,
+									   &mcv1_values, &mcv1_nvalues,
+									   &mcv1_numbers, &mcv1_nnumbers);
+		hist1_exists = get_attstatsslot(vardata1->statsTuple,
+									  vardata1->atttype, vardata1->atttypmod,
+										STATISTIC_KIND_HISTOGRAM, InvalidOid,
+										NULL,
+										&hist1_values, &hist1_nvalues,
+										NULL, NULL);
+		/* Arbitrarily limit number of MCVs considered */
+		mcv1_length = Min(mcv1_nvalues, MAX_CONSIDERED_ELEMS);
+		if (mcv1_exists)
+			sumcommon1 = mcv_population(mcv1_numbers, mcv1_length);
+	}
+
+	if (HeapTupleIsValid(vardata2->statsTuple))
+	{
+		stats = (Form_pg_statistic) GETSTRUCT(vardata2->statsTuple);
+		nullfrac2 = stats->stanullfrac;
+
+		mcv2_exists = get_attstatsslot(vardata2->statsTuple,
+									   vardata2->atttype, vardata2->atttypmod,
+									   STATISTIC_KIND_MCV, InvalidOid,
+									   NULL,
+									   &mcv2_values, &mcv2_nvalues,
+									   &mcv2_numbers, &mcv2_nnumbers);
+		hist2_exists = get_attstatsslot(vardata2->statsTuple,
+									  vardata2->atttype, vardata2->atttypmod,
+										STATISTIC_KIND_HISTOGRAM, InvalidOid,
+										NULL,
+										&hist2_values, &hist2_nvalues,
+										NULL, NULL);
+		/* Arbitrarily limit number of MCVs considered */
+		mcv2_length = Min(mcv2_nvalues, MAX_CONSIDERED_ELEMS);
+		if (mcv2_exists)
+			sumcommon2 = mcv_population(mcv2_numbers, mcv2_length);
+	}
+
+	opr_codenum = inet_opr_codenum(operator);
+	fmgr_info(get_opcode(operator), &proc);
+
+	/* Estimate number of input rows represented by RHS histogram. */
+	if (hist2_exists && vardata2->rel)
+		hist2_weight = (1.0 - nullfrac2 - sumcommon2) * vardata2->rel->rows;
+
+	/*
+	 * Consider each element of the LHS MCV list, matching it to whatever RHS
+	 * stats we have.  Scale according to the known frequency of the MCV.
+	 */
+	if (mcv1_exists && (mcv2_exists || hist2_exists))
+	{
+		for (i = 0; i < mcv1_length; i++)
+		{
+			selec += mcv1_numbers[i] *
+				inet_semi_join_sel(mcv1_values[i],
+								   mcv2_exists, mcv2_values, mcv2_length,
+								   hist2_exists, hist2_values, hist2_nvalues,
+								   hist2_weight,
+								   &proc, opr_codenum);
+		}
+	}
+
+	/*
+	 * Consider each element of the LHS histogram, except for the first and
+	 * last elements, which we exclude on the grounds that they're outliers
+	 * and thus not very representative.  Scale on the assumption that each
+	 * such histogram element represents an equal share of the LHS histogram
+	 * population (which is a bit bogus, because the members of its bucket may
+	 * not all act the same with respect to the join clause, but it's hard to
+	 * do better).
+	 *
+	 * If there are too many histogram elements, decimate to limit runtime.
+	 */
+	if (hist1_exists && hist1_nvalues > 2 && (mcv2_exists || hist2_exists))
+	{
+		double		hist_selec_sum = 0.0;
+		int			k,
+					n;
+
+		k = (hist1_nvalues - 3) / MAX_CONSIDERED_ELEMS + 1;
+
+		n = 0;
+		for (i = 1; i < hist1_nvalues - 1; i += k)
+		{
+			hist_selec_sum +=
+				inet_semi_join_sel(hist1_values[i],
+								   mcv2_exists, mcv2_values, mcv2_length,
+								   hist2_exists, hist2_values, hist2_nvalues,
+								   hist2_weight,
+								   &proc, opr_codenum);
+			n++;
+		}
+
+		selec += (1.0 - nullfrac1 - sumcommon1) * hist_selec_sum / n;
+	}
+
+	/*
+	 * If useful statistics are not available then use the default estimate.
+	 * We can apply null fractions if known, though.
+	 */
+	if ((!mcv1_exists && !hist1_exists) || (!mcv2_exists && !hist2_exists))
+		selec = (1.0 - nullfrac1) * (1.0 - nullfrac2) * DEFAULT_SEL(operator);
+
+	/* Release stats. */
+	if (mcv1_exists)
+		free_attstatsslot(vardata1->atttype, mcv1_values, mcv1_nvalues,
+						  mcv1_numbers, mcv1_nnumbers);
+	if (mcv2_exists)
+		free_attstatsslot(vardata2->atttype, mcv2_values, mcv2_nvalues,
+						  mcv2_numbers, mcv2_nnumbers);
+	if (hist1_exists)
+		free_attstatsslot(vardata1->atttype, hist1_values, hist1_nvalues,
+						  NULL, 0);
+	if (hist2_exists)
+		free_attstatsslot(vardata2->atttype, hist2_values, hist2_nvalues,
+						  NULL, 0);
+
+	return selec;
+}
+
+/*
+ * Compute the fraction of a relation's population that is represented
+ * by the MCV list.
+ */
+static Selectivity
+mcv_population(float4 *mcv_numbers, int mcv_nvalues)
+{
+	Selectivity sumcommon = 0.0;
+	int			i;
+
+	for (i = 0; i < mcv_nvalues; i++)
+	{
+		sumcommon += mcv_numbers[i];
+	}
+
+	return sumcommon;
+}
+
+/*
+ * Inet histogram vs single value selectivity estimation
+ *
+ * Estimate the fraction of the histogram population that satisfies
+ * "value OPR CONST".  (The result needs to be scaled to reflect the
+ * proportion of the total population represented by the histogram.)
+ *
+ * The histogram is originally for the inet btree comparison operators.
+ * Only the common bits of the network part and the length of the network part
+ * (masklen) are interesting for the subnet inclusion operators.  Fortunately,
+ * btree comparison treats the network part as the major sort key.  Even so,
+ * the length of the network part would not really be significant in the
+ * histogram.  This would lead to big mistakes for data sets with uneven
+ * masklen distribution.  To reduce this problem, comparisons with the left
+ * and the right sides of the buckets are used together.
+ *
+ * Histogram bucket matches are calculated in two forms.  If the constant
+ * matches both bucket endpoints the bucket is considered as fully matched.
+ * The second form is to match the bucket partially; we recognize this when
+ * the constant matches just one endpoint, or the two endpoints fall on
+ * opposite sides of the constant.  (Note that when the constant matches an
+ * interior histogram element, it gets credit for partial matches to the
+ * buckets on both sides, while a match to a histogram endpoint gets credit
+ * for only one partial match.  This is desirable.)
+ *
+ * The divider in the partial bucket match is imagined as the distance
+ * between the decisive bits and the common bits of the addresses.  It will
+ * be used as a power of two as it is the natural scale for the IP network
+ * inclusion.  This partial bucket match divider calculation is an empirical
+ * formula and subject to change with more experiment.
+ *
+ * For a partial match, we try to calculate dividers for both of the
+ * boundaries.  If the address family of a boundary value does not match the
+ * constant or comparison of the length of the network parts is not correct
+ * for the operator, the divider for that boundary will not be taken into
+ * account.  If both of the dividers are valid, the greater one will be used
+ * to minimize the mistake in buckets that have disparate masklens.  This
+ * calculation is unfair when dividers can be calculated for both of the
+ * boundaries but they are far from each other; but it is not a common
+ * situation as the boundaries are expected to share most of their significant
+ * bits of their masklens.  The mistake would be greater, if we would use the
+ * minimum instead of the maximum, and we don't know a sensible way to combine
+ * them.
+ *
+ * For partial match in buckets that have different address families on the
+ * left and right sides, only the boundary with the same address family is
+ * taken into consideration.  This can cause more mistakes for these buckets
+ * if the masklens of their boundaries are also disparate.  But this can only
+ * happen in one bucket, since only two address families exist.  It seems a
+ * better option than not considering these buckets at all.
+ */
+static Selectivity
+inet_hist_value_sel(Datum *values, int nvalues, Datum constvalue,
+					int opr_codenum)
+{
+	Selectivity match = 0.0;
+	inet	   *query,
+			   *left,
+			   *right;
+	int			i,
+				k,
+				n;
+	int			left_order,
+				right_order,
+				left_divider,
+				right_divider;
+
+	/* guard against zero-divide below */
+	if (nvalues <= 1)
+		return 0.0;
+
+	/* if there are too many histogram elements, decimate to limit runtime */
+	k = (nvalues - 2) / MAX_CONSIDERED_ELEMS + 1;
+
+	query = DatumGetInetPP(constvalue);
+
+	/* "left" is the left boundary value of the current bucket ... */
+	left = DatumGetInetPP(values[0]);
+	left_order = inet_inclusion_cmp(left, query, opr_codenum);
+
+	n = 0;
+	for (i = k; i < nvalues; i += k)
+	{
+		/* ... and "right" is the right boundary value */
+		right = DatumGetInetPP(values[i]);
+		right_order = inet_inclusion_cmp(right, query, opr_codenum);
+
+		if (left_order == 0 && right_order == 0)
+		{
+			/* The whole bucket matches, since both endpoints do. */
+			match += 1.0;
+		}
+		else if ((left_order <= 0 && right_order >= 0) ||
+				 (left_order >= 0 && right_order <= 0))
+		{
+			/* Partial bucket match. */
+			left_divider = inet_hist_match_divider(left, query, opr_codenum);
+			right_divider = inet_hist_match_divider(right, query, opr_codenum);
+
+			if (left_divider >= 0 || right_divider >= 0)
+				match += 1.0 / pow(2.0, Max(left_divider, right_divider));
+		}
+
+		/* Shift the variables. */
+		left = right;
+		left_order = right_order;
+
+		/* Count the number of buckets considered. */
+		n++;
+	}
+
+	return match / n;
+}
+
+/*
+ * Inet MCV vs MCV join selectivity estimation
+ *
+ * We simply add up the fractions of the populations that satisfy the clause.
+ * The result is exact and does not need to be scaled further.
+ */
+static Selectivity
+inet_mcv_join_sel(Datum *mcv1_values, float4 *mcv1_numbers, int mcv1_nvalues,
+				  Datum *mcv2_values, float4 *mcv2_numbers, int mcv2_nvalues,
+				  Oid operator)
+{
+	Selectivity selec = 0.0;
+	FmgrInfo	proc;
+	int			i,
+				j;
+
+	fmgr_info(get_opcode(operator), &proc);
+
+	for (i = 0; i < mcv1_nvalues; i++)
+	{
+		for (j = 0; j < mcv2_nvalues; j++)
+			if (DatumGetBool(FunctionCall2(&proc,
+										   mcv1_values[i],
+										   mcv2_values[j])))
+				selec += mcv1_numbers[i] * mcv2_numbers[j];
+	}
+	return selec;
+}
+
+/*
+ * Inet MCV vs histogram join selectivity estimation
+ *
+ * For each MCV on the lefthand side, estimate the fraction of the righthand's
+ * histogram population that satisfies the join clause, and add those up,
+ * scaling by the MCV's frequency.  The result still needs to be scaled
+ * according to the fraction of the righthand's population represented by
+ * the histogram.
+ */
+static Selectivity
+inet_mcv_hist_sel(Datum *mcv_values, float4 *mcv_numbers, int mcv_nvalues,
+				  Datum *hist_values, int hist_nvalues,
+				  int opr_codenum)
+{
+	Selectivity selec = 0.0;
+	int			i;
+
+	/*
+	 * We'll call inet_hist_value_selec with the histogram on the left, so we
+	 * must commute the operator.
+	 */
+	opr_codenum = -opr_codenum;
+
+	for (i = 0; i < mcv_nvalues; i++)
+	{
+		selec += mcv_numbers[i] *
+			inet_hist_value_sel(hist_values, hist_nvalues, mcv_values[i],
+								opr_codenum);
+	}
+	return selec;
+}
+
+/*
+ * Inet histogram vs histogram join selectivity estimation
+ *
+ * Here, we take all values listed in the second histogram (except for the
+ * first and last elements, which are excluded on the grounds of possibly
+ * not being very representative) and treat them as a uniform sample of
+ * the non-MCV population for that relation.  For each one, we apply
+ * inet_hist_value_selec to see what fraction of the first histogram
+ * it matches.
+ *
+ * We could alternatively do this the other way around using the operator's
+ * commutator.  XXX would it be worthwhile to do it both ways and take the
+ * average?  That would at least avoid non-commutative estimation results.
+ */
+static Selectivity
+inet_hist_inclusion_join_sel(Datum *hist1_values, int hist1_nvalues,
+							 Datum *hist2_values, int hist2_nvalues,
+							 int opr_codenum)
+{
+	double		match = 0.0;
+	int			i,
+				k,
+				n;
+
+	if (hist2_nvalues <= 2)
+		return 0.0;				/* no interior histogram elements */
+
+	/* if there are too many histogram elements, decimate to limit runtime */
+	k = (hist2_nvalues - 3) / MAX_CONSIDERED_ELEMS + 1;
+
+	n = 0;
+	for (i = 1; i < hist2_nvalues - 1; i += k)
+	{
+		match += inet_hist_value_sel(hist1_values, hist1_nvalues,
+									 hist2_values[i], opr_codenum);
+		n++;
+	}
+
+	return match / n;
+}
+
+/*
+ * Inet semi join selectivity estimation for one value
+ *
+ * The function calculates the probability that there is at least one row
+ * in the RHS table that satisfies the "lhs_value op column" condition.
+ * It is used in semi join estimation to check a sample from the left hand
+ * side table.
+ *
+ * The MCV and histogram from the right hand side table should be provided as
+ * arguments with the lhs_value from the left hand side table for the join.
+ * hist_weight is the total number of rows represented by the histogram.
+ * For example, if the table has 1000 rows, and 10% of the rows are in the MCV
+ * list, and another 10% are NULLs, hist_weight would be 800.
+ *
+ * First, the lhs_value will be matched to the most common values.  If it
+ * matches any of them, 1.0 will be returned, because then there is surely
+ * a match.
+ *
+ * Otherwise, the histogram will be used to estimate the number of rows in
+ * the second table that match the condition.  If the estimate is greater
+ * than 1.0, 1.0 will be returned, because it means there is a greater chance
+ * that the lhs_value will match more than one row in the table.  If it is
+ * between 0.0 and 1.0, it will be returned as the probability.
+ */
+static Selectivity
+inet_semi_join_sel(Datum lhs_value,
+				   bool mcv_exists, Datum *mcv_values, int mcv_nvalues,
+				   bool hist_exists, Datum *hist_values, int hist_nvalues,
+				   double hist_weight,
+				   FmgrInfo *proc, int opr_codenum)
+{
+	if (mcv_exists)
+	{
+		int			i;
+
+		for (i = 0; i < mcv_nvalues; i++)
+		{
+			if (DatumGetBool(FunctionCall2(proc,
+										   lhs_value,
+										   mcv_values[i])))
+				return 1.0;
+		}
+	}
+
+	if (hist_exists && hist_weight > 0)
+	{
+		Selectivity hist_selec;
+
+		/* Commute operator, since we're passing lhs_value on the right */
+		hist_selec = inet_hist_value_sel(hist_values, hist_nvalues,
+										 lhs_value, -opr_codenum);
+
+		if (hist_selec > 0)
+			return Min(1.0, hist_weight * hist_selec);
+	}
+
+	return 0.0;
+}
+
+/*
+ * Assign useful code numbers for the subnet inclusion/overlap operators
+ *
+ * Only inet_masklen_inclusion_cmp() and inet_hist_match_divider() depend
+ * on the exact codes assigned here; but many other places in this file
+ * know that they can negate a code to obtain the code for the commutator
+ * operator.
+ */
+static int
+inet_opr_codenum(Oid operator)
+{
+	switch (operator)
+	{
+		case OID_INET_SUP_OP:
+			return -2;
+		case OID_INET_SUPEQ_OP:
+			return -1;
+		case OID_INET_OVERLAP_OP:
+			return 0;
+		case OID_INET_SUBEQ_OP:
+			return 1;
+		case OID_INET_SUB_OP:
+			return 2;
+		default:
+			elog(ERROR, "unrecognized operator %u for inet selectivity",
+				 operator);
+	}
+	return 0;					/* unreached, but keep compiler quiet */
+}
+
+/*
+ * Comparison function for the subnet inclusion/overlap operators
+ *
+ * If the comparison is okay for the specified inclusion operator, the return
+ * value will be 0.  Otherwise the return value will be less than or greater
+ * than 0 as appropriate for the operator.
+ *
+ * Comparison is compatible with the basic comparison function for the inet
+ * type.  See network_cmp_internal() in network.c for the original.  Basic
+ * comparison operators are implemented with the network_cmp_internal()
+ * function.  It is possible to implement the subnet inclusion operators with
+ * this function.
+ *
+ * Comparison is first on the common bits of the network part, then on the
+ * length of the network part (masklen) as in the network_cmp_internal()
+ * function.  Only the first part is in this function.  The second part is
+ * separated to another function for reusability.  The difference between the
+ * second part and the original network_cmp_internal() is that the inclusion
+ * operator is considered while comparing the lengths of the network parts.
+ * See the inet_masklen_inclusion_cmp() function below.
+ */
+static int
+inet_inclusion_cmp(inet *left, inet *right, int opr_codenum)
+{
+	if (ip_family(left) == ip_family(right))
+	{
+		int			order;
+
+		order = bitncmp(ip_addr(left), ip_addr(right),
+						Min(ip_bits(left), ip_bits(right)));
+		if (order != 0)
+			return order;
+
+		return inet_masklen_inclusion_cmp(left, right, opr_codenum);
+	}
+
+	return ip_family(left) - ip_family(right);
+}
+
+/*
+ * Masklen comparison function for the subnet inclusion/overlap operators
+ *
+ * Compares the lengths of the network parts of the inputs.  If the comparison
+ * is okay for the specified inclusion operator, the return value will be 0.
+ * Otherwise the return value will be less than or greater than 0 as
+ * appropriate for the operator.
+ */
+static int
+inet_masklen_inclusion_cmp(inet *left, inet *right, int opr_codenum)
+{
+	int			order;
+
+	order = (int) ip_bits(left) - (int) ip_bits(right);
+
+	/*
+	 * Return 0 if the operator would accept this combination of masklens.
+	 * Note that opr_codenum zero (overlaps) will accept all cases.
+	 */
+	if ((order > 0 && opr_codenum >= 0) ||
+		(order == 0 && opr_codenum >= -1 && opr_codenum <= 1) ||
+		(order < 0 && opr_codenum <= 0))
+		return 0;
+
+	/*
+	 * Otherwise, return a negative value for sup/supeq (notionally, the RHS
+	 * needs to have a larger masklen than it has, which would make it sort
+	 * later), or a positive value for sub/subeq (vice versa).
+	 */
+	return opr_codenum;
+}
+
+/*
+ * Inet histogram partial match divider calculation
+ *
+ * First the families and the lengths of the network parts are compared using
+ * the subnet inclusion operator.  If those are acceptable for the operator,
+ * the divider will be calculated using the masklens and the common bits of
+ * the addresses.  -1 will be returned if it cannot be calculated.
+ *
+ * See commentary for inet_hist_value_sel() for some rationale for this.
+ */
+static int
+inet_hist_match_divider(inet *boundary, inet *query, int opr_codenum)
+{
+	if (ip_family(boundary) == ip_family(query) &&
+		inet_masklen_inclusion_cmp(boundary, query, opr_codenum) == 0)
+	{
+		int			min_bits,
+					decisive_bits;
+
+		min_bits = Min(ip_bits(boundary), ip_bits(query));
+
+		/*
+		 * Set decisive_bits to the masklen of the one that should contain the
+		 * other according to the operator.
+		 */
+		if (opr_codenum < 0)
+			decisive_bits = ip_bits(boundary);
+		else if (opr_codenum > 0)
+			decisive_bits = ip_bits(query);
+		else
+			decisive_bits = min_bits;
+
+		/*
+		 * Now return the number of non-common decisive bits.  (This will be
+		 * zero if the boundary and query in fact match, else positive.)
+		 */
+		if (min_bits > 0)
+			return decisive_bits - bitncommon(ip_addr(boundary),
+											  ip_addr(query),
+											  min_bits);
+		return decisive_bits;
+	}
+
+	return -1;
 }