rangetypes_spgist.c

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/*-------------------------------------------------------------------------
 *
 * rangetypes_spgist.c
 *    implementation of quad tree over ranges mapped to 2d-points for SP-GiST.
 *
 * Quad tree is a data structure similar to a binary tree, but is adapted to
 * 2d data. Each inner node of a quad tree contains a point (centroid) which
 * divides the 2d-space into 4 quadrants. Each quadrant is associated with a
 * child node.
 *
 * Ranges are mapped to 2d-points so that the lower bound is one dimension,
 * and the upper bound is another. By convention, we visualize the lower bound
 * to be the horizontal axis, and upper bound the vertical axis.
 *
 * One quirk with this mapping is the handling of empty ranges. An empty range
 * doesn't have lower and upper bounds, so it cannot be mapped to 2d space in
 * a straightforward way. To cope with that, the root node can have a 5th
 * quadrant, which is reserved for empty ranges. Furthermore, there can be
 * inner nodes in the tree with no centroid. They contain only two child nodes,
 * one for empty ranges and another for non-empty ones. Such a node can appear
 * as the root node, or in the tree under the 5th child of the root node (in
 * which case it will only contain empty nodes).
 *
 * The SP-GiST picksplit function uses medians along both axes as the centroid.
 * This implementation only uses the comparison function of the range element
 * datatype, therefore it works for any range type.
 *
 * 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_spgist.c
 *
 *-------------------------------------------------------------------------
 */

#include "postgres.h"

#include "access/spgist.h"
#include "access/stratnum.h"
#include "catalog/pg_type.h"
#include "utils/builtins.h"
#include "utils/datum.h"
#include "utils/rangetypes.h"

static int16 getQuadrant(TypeCacheEntry *typcache, const RangeType *centroid,
                         const RangeType *tst);
static int  bound_cmp(const void *a, const void *b, void *arg);

static int adjacent_inner_consistent(TypeCacheEntry *typcache,
                                     const RangeBound *arg, const RangeBound *centroid,
                                     const RangeBound *prev);
static int adjacent_cmp_bounds(TypeCacheEntry *typcache, const RangeBound *arg,
                               const RangeBound *centroid);

/*
 * SP-GiST 'config' interface function.
 */
Datum
spg_range_quad_config(PG_FUNCTION_ARGS)
{
    /* spgConfigIn *cfgin = (spgConfigIn *) PG_GETARG_POINTER(0); */
    spgConfigOut *cfg = (spgConfigOut *) PG_GETARG_POINTER(1);

    cfg->prefixType = ANYRANGEOID;
    cfg->labelType = VOIDOID;   /* we don't need node labels */
    cfg->canReturnData = true;
    cfg->longValuesOK = false;
    PG_RETURN_VOID();
}

/*----------
 * Determine which quadrant a 2d-mapped range falls into, relative to the
 * centroid.
 *
 * Quadrants are numbered like this:
 *
 *   4  |  1
 *  ----+----
 *   3  |  2
 *
 * Where the lower bound of range is the horizontal axis and upper bound the
 * vertical axis.
 *
 * Ranges on one of the axes are taken to lie in the quadrant with higher value
 * along perpendicular axis. That is, a value on the horizontal axis is taken
 * to belong to quadrant 1 or 4, and a value on the vertical axis is taken to
 * belong to quadrant 1 or 2. A range equal to centroid is taken to lie in
 * quadrant 1.
 *
 * Empty ranges are taken to lie in the special quadrant 5.
 *----------
 */
static int16
getQuadrant(TypeCacheEntry *typcache, const RangeType *centroid, const RangeType *tst)
{
    RangeBound  centroidLower,
                centroidUpper;
    bool        centroidEmpty;
    RangeBound  lower,
                upper;
    bool        empty;

    range_deserialize(typcache, centroid, &centroidLower, &centroidUpper,
                      &centroidEmpty);
    range_deserialize(typcache, tst, &lower, &upper, &empty);

    if (empty)
        return 5;

    if (range_cmp_bounds(typcache, &lower, &centroidLower) >= 0)
    {
        if (range_cmp_bounds(typcache, &upper, &centroidUpper) >= 0)
            return 1;
        else
            return 2;
    }
    else
    {
        if (range_cmp_bounds(typcache, &upper, &centroidUpper) >= 0)
            return 4;
        else
            return 3;
    }
}

/*
 * Choose SP-GiST function: choose path for addition of new range.
 */
Datum
spg_range_quad_choose(PG_FUNCTION_ARGS)
{
    spgChooseIn *in = (spgChooseIn *) PG_GETARG_POINTER(0);
    spgChooseOut *out = (spgChooseOut *) PG_GETARG_POINTER(1);
    RangeType  *inRange = DatumGetRangeTypeP(in->datum),
               *centroid;
    int16       quadrant;
    TypeCacheEntry *typcache;

    if (in->allTheSame)
    {
        out->resultType = spgMatchNode;
        /* nodeN will be set by core */
        out->result.matchNode.levelAdd = 0;
        out->result.matchNode.restDatum = RangeTypePGetDatum(inRange);
        PG_RETURN_VOID();
    }

    typcache = range_get_typcache(fcinfo, RangeTypeGetOid(inRange));

    /*
     * A node with no centroid divides ranges purely on whether they're empty
     * or not. All empty ranges go to child node 0, all non-empty ranges go to
     * node 1.
     */
    if (!in->hasPrefix)
    {
        out->resultType = spgMatchNode;
        if (RangeIsEmpty(inRange))
            out->result.matchNode.nodeN = 0;
        else
            out->result.matchNode.nodeN = 1;
        out->result.matchNode.levelAdd = 1;
        out->result.matchNode.restDatum = RangeTypePGetDatum(inRange);
        PG_RETURN_VOID();
    }

    centroid = DatumGetRangeTypeP(in->prefixDatum);
    quadrant = getQuadrant(typcache, centroid, inRange);

    Assert(quadrant <= in->nNodes);

    /* Select node matching to quadrant number */
    out->resultType = spgMatchNode;
    out->result.matchNode.nodeN = quadrant - 1;
    out->result.matchNode.levelAdd = 1;
    out->result.matchNode.restDatum = RangeTypePGetDatum(inRange);

    PG_RETURN_VOID();
}

/*
 * Bound comparison for sorting.
 */
static int
bound_cmp(const void *a, const void *b, void *arg)
{
    RangeBound *ba = (RangeBound *) a;
    RangeBound *bb = (RangeBound *) b;
    TypeCacheEntry *typcache = (TypeCacheEntry *) arg;

    return range_cmp_bounds(typcache, ba, bb);
}

/*
 * Picksplit SP-GiST function: split ranges into nodes. Select "centroid"
 * range and distribute ranges according to quadrants.
 */
Datum
spg_range_quad_picksplit(PG_FUNCTION_ARGS)
{
    spgPickSplitIn *in = (spgPickSplitIn *) PG_GETARG_POINTER(0);
    spgPickSplitOut *out = (spgPickSplitOut *) PG_GETARG_POINTER(1);
    int         i;
    int         j;
    int         nonEmptyCount;
    RangeType  *centroid;
    bool        empty;
    TypeCacheEntry *typcache;

    /* Use the median values of lower and upper bounds as the centroid range */
    RangeBound *lowerBounds,
               *upperBounds;

    typcache = range_get_typcache(fcinfo,
                                  RangeTypeGetOid(DatumGetRangeTypeP(in->datums[0])));

    /* Allocate memory for bounds */
    lowerBounds = palloc(sizeof(RangeBound) * in->nTuples);
    upperBounds = palloc(sizeof(RangeBound) * in->nTuples);
    j = 0;

    /* Deserialize bounds of ranges, count non-empty ranges */
    for (i = 0; i < in->nTuples; i++)
    {
        range_deserialize(typcache, DatumGetRangeTypeP(in->datums[i]),
                          &lowerBounds[j], &upperBounds[j], &empty);
        if (!empty)
            j++;
    }
    nonEmptyCount = j;

    /*
     * All the ranges are empty. The best we can do is to construct an inner
     * node with no centroid, and put all ranges into node 0. If non-empty
     * ranges are added later, they will be routed to node 1.
     */
    if (nonEmptyCount == 0)
    {
        out->nNodes = 2;
        out->hasPrefix = false;
        /* Prefix is empty */
        out->prefixDatum = PointerGetDatum(NULL);
        out->nodeLabels = NULL;

        out->mapTuplesToNodes = palloc(sizeof(int) * in->nTuples);
        out->leafTupleDatums = palloc(sizeof(Datum) * in->nTuples);

        /* Place all ranges into node 0 */
        for (i = 0; i < in->nTuples; i++)
        {
            RangeType  *range = DatumGetRangeTypeP(in->datums[i]);

            out->leafTupleDatums[i] = RangeTypePGetDatum(range);
            out->mapTuplesToNodes[i] = 0;
        }
        PG_RETURN_VOID();
    }

    /* Sort range bounds in order to find medians */
    qsort_arg(lowerBounds, nonEmptyCount, sizeof(RangeBound),
              bound_cmp, typcache);
    qsort_arg(upperBounds, nonEmptyCount, sizeof(RangeBound),
              bound_cmp, typcache);

    /* Construct "centroid" range from medians of lower and upper bounds */
    centroid = range_serialize(typcache, &lowerBounds[nonEmptyCount / 2],
                               &upperBounds[nonEmptyCount / 2], false);
    out->hasPrefix = true;
    out->prefixDatum = RangeTypePGetDatum(centroid);

    /* Create node for empty ranges only if it is a root node */
    out->nNodes = (in->level == 0) ? 5 : 4;
    out->nodeLabels = NULL;     /* we don't need node labels */

    out->mapTuplesToNodes = palloc(sizeof(int) * in->nTuples);
    out->leafTupleDatums = palloc(sizeof(Datum) * in->nTuples);

    /*
     * Assign ranges to corresponding nodes according to quadrants relative to
     * "centroid" range.
     */
    for (i = 0; i < in->nTuples; i++)
    {
        RangeType  *range = DatumGetRangeTypeP(in->datums[i]);
        int16       quadrant = getQuadrant(typcache, centroid, range);

        out->leafTupleDatums[i] = RangeTypePGetDatum(range);
        out->mapTuplesToNodes[i] = quadrant - 1;
    }

    PG_RETURN_VOID();
}

/*
 * SP-GiST consistent function for inner nodes: check which nodes are
 * consistent with given set of queries.
 */
Datum
spg_range_quad_inner_consistent(PG_FUNCTION_ARGS)
{
    spgInnerConsistentIn *in = (spgInnerConsistentIn *) PG_GETARG_POINTER(0);
    spgInnerConsistentOut *out = (spgInnerConsistentOut *) PG_GETARG_POINTER(1);
    int         which;
    int         i;
    MemoryContext oldCtx;

    /*
     * For adjacent search we need also previous centroid (if any) to improve
     * the precision of the consistent check. In this case needPrevious flag
     * is set and centroid is passed into traversalValue.
     */
    bool        needPrevious = false;

    if (in->allTheSame)
    {
        /* Report that all nodes should be visited */
        out->nNodes = in->nNodes;
        out->nodeNumbers = (int *) palloc(sizeof(int) * in->nNodes);
        for (i = 0; i < in->nNodes; i++)
            out->nodeNumbers[i] = i;
        PG_RETURN_VOID();
    }

    if (!in->hasPrefix)
    {
        /*
         * No centroid on this inner node. Such a node has two child nodes,
         * the first for empty ranges, and the second for non-empty ones.
         */
        Assert(in->nNodes == 2);

        /*
         * Nth bit of which variable means that (N - 1)th node should be
         * visited. Initially all bits are set. Bits of nodes which should be
         * skipped will be unset.
         */
        which = (1 << 1) | (1 << 2);
        for (i = 0; i < in->nkeys; i++)
        {
            StrategyNumber strategy = in->scankeys[i].sk_strategy;
            bool        empty;

            /*
             * The only strategy when second argument of operator is not range
             * is RANGESTRAT_CONTAINS_ELEM.
             */
            if (strategy != RANGESTRAT_CONTAINS_ELEM)
                empty = RangeIsEmpty(
                                     DatumGetRangeTypeP(in->scankeys[i].sk_argument));
            else
                empty = false;

            switch (strategy)
            {
                case RANGESTRAT_BEFORE:
                case RANGESTRAT_OVERLEFT:
                case RANGESTRAT_OVERLAPS:
                case RANGESTRAT_OVERRIGHT:
                case RANGESTRAT_AFTER:
                case RANGESTRAT_ADJACENT:
                    /* These strategies return false if any argument is empty */
                    if (empty)
                        which = 0;
                    else
                        which &= (1 << 2);
                    break;

                case RANGESTRAT_CONTAINS:

                    /*
                     * All ranges contain an empty range. Only non-empty
                     * ranges can contain a non-empty range.
                     */
                    if (!empty)
                        which &= (1 << 2);
                    break;

                case RANGESTRAT_CONTAINED_BY:

                    /*
                     * Only an empty range is contained by an empty range.
                     * Both empty and non-empty ranges can be contained by a
                     * non-empty range.
                     */
                    if (empty)
                        which &= (1 << 1);
                    break;

                case RANGESTRAT_CONTAINS_ELEM:
                    which &= (1 << 2);
                    break;

                case RANGESTRAT_EQ:
                    if (empty)
                        which &= (1 << 1);
                    else
                        which &= (1 << 2);
                    break;

                default:
                    elog(ERROR, "unrecognized range strategy: %d", strategy);
                    break;
            }
            if (which == 0)
                break;          /* no need to consider remaining conditions */
        }
    }
    else
    {
        RangeBound  centroidLower,
                    centroidUpper;
        bool        centroidEmpty;
        TypeCacheEntry *typcache;
        RangeType  *centroid;

        /* This node has a centroid. Fetch it. */
        centroid = DatumGetRangeTypeP(in->prefixDatum);
        typcache = range_get_typcache(fcinfo,
                                      RangeTypeGetOid(DatumGetRangeTypeP(centroid)));
        range_deserialize(typcache, centroid, &centroidLower, &centroidUpper,
                          &centroidEmpty);

        Assert(in->nNodes == 4 || in->nNodes == 5);

        /*
         * Nth bit of which variable means that (N - 1)th node (Nth quadrant)
         * should be visited. Initially all bits are set. Bits of nodes which
         * can be skipped will be unset.
         */
        which = (1 << 1) | (1 << 2) | (1 << 3) | (1 << 4) | (1 << 5);

        for (i = 0; i < in->nkeys; i++)
        {
            StrategyNumber strategy;
            RangeBound  lower,
                        upper;
            bool        empty;
            RangeType  *range = NULL;

            RangeType  *prevCentroid = NULL;
            RangeBound  prevLower,
                        prevUpper;
            bool        prevEmpty;

            /* Restrictions on range bounds according to scan strategy */
            RangeBound *minLower = NULL,
                       *maxLower = NULL,
                       *minUpper = NULL,
                       *maxUpper = NULL;

            /* Are the restrictions on range bounds inclusive? */
            bool        inclusive = true;
            bool        strictEmpty = true;
            int         cmp,
                        which1,
                        which2;

            strategy = in->scankeys[i].sk_strategy;

            /*
             * RANGESTRAT_CONTAINS_ELEM is just like RANGESTRAT_CONTAINS, but
             * the argument is a single element. Expand the single element to
             * a range containing only the element, and treat it like
             * RANGESTRAT_CONTAINS.
             */
            if (strategy == RANGESTRAT_CONTAINS_ELEM)
            {
                lower.inclusive = true;
                lower.infinite = false;
                lower.lower = true;
                lower.val = in->scankeys[i].sk_argument;

                upper.inclusive = true;
                upper.infinite = false;
                upper.lower = false;
                upper.val = in->scankeys[i].sk_argument;

                empty = false;

                strategy = RANGESTRAT_CONTAINS;
            }
            else
            {
                range = DatumGetRangeTypeP(in->scankeys[i].sk_argument);
                range_deserialize(typcache, range, &lower, &upper, &empty);
            }

            /*
             * Most strategies are handled by forming a bounding box from the
             * search key, defined by a minLower, maxLower, minUpper,
             * maxUpper. Some modify 'which' directly, to specify exactly
             * which quadrants need to be visited.
             *
             * For most strategies, nothing matches an empty search key, and
             * an empty range never matches a non-empty key. If a strategy
             * does not behave like that wrt. empty ranges, set strictEmpty to
             * false.
             */
            switch (strategy)
            {
                case RANGESTRAT_BEFORE:

                    /*
                     * Range A is before range B if upper bound of A is lower
                     * than lower bound of B.
                     */
                    maxUpper = &lower;
                    inclusive = false;
                    break;

                case RANGESTRAT_OVERLEFT:

                    /*
                     * Range A is overleft to range B if upper bound of A is
                     * less or equal to upper bound of B.
                     */
                    maxUpper = &upper;
                    break;

                case RANGESTRAT_OVERLAPS:

                    /*
                     * Non-empty ranges overlap, if lower bound of each range
                     * is lower or equal to upper bound of the other range.
                     */
                    maxLower = &upper;
                    minUpper = &lower;
                    break;

                case RANGESTRAT_OVERRIGHT:

                    /*
                     * Range A is overright to range B if lower bound of A is
                     * greater or equal to lower bound of B.
                     */
                    minLower = &lower;
                    break;

                case RANGESTRAT_AFTER:

                    /*
                     * Range A is after range B if lower bound of A is greater
                     * than upper bound of B.
                     */
                    minLower = &upper;
                    inclusive = false;
                    break;

                case RANGESTRAT_ADJACENT:
                    if (empty)
                        break;  /* Skip to strictEmpty check. */

                    /*
                     * Previously selected quadrant could exclude possibility
                     * for lower or upper bounds to be adjacent. Deserialize
                     * previous centroid range if present for checking this.
                     */
                    if (in->traversalValue)
                    {
                        prevCentroid = DatumGetRangeTypeP(in->traversalValue);
                        range_deserialize(typcache, prevCentroid,
                                          &prevLower, &prevUpper, &prevEmpty);
                    }

                    /*
                     * For a range's upper bound to be adjacent to the
                     * argument's lower bound, it will be found along the line
                     * adjacent to (and just below) Y=lower. Therefore, if the
                     * argument's lower bound is less than the centroid's
                     * upper bound, the line falls in quadrants 2 and 3; if
                     * greater, the line falls in quadrants 1 and 4. (see
                     * adjacent_cmp_bounds for description of edge cases).
                     */
                    cmp = adjacent_inner_consistent(typcache, &lower,
                                                    &centroidUpper,
                                                    prevCentroid ? &prevUpper : NULL);
                    if (cmp > 0)
                        which1 = (1 << 1) | (1 << 4);
                    else if (cmp < 0)
                        which1 = (1 << 2) | (1 << 3);
                    else
                        which1 = 0;

                    /*
                     * Also search for ranges's adjacent to argument's upper
                     * bound. They will be found along the line adjacent to
                     * (and just right of) X=upper, which falls in quadrants 3
                     * and 4, or 1 and 2.
                     */
                    cmp = adjacent_inner_consistent(typcache, &upper,
                                                    &centroidLower,
                                                    prevCentroid ? &prevLower : NULL);
                    if (cmp > 0)
                        which2 = (1 << 1) | (1 << 2);
                    else if (cmp < 0)
                        which2 = (1 << 3) | (1 << 4);
                    else
                        which2 = 0;

                    /* We must chase down ranges adjacent to either bound. */
                    which &= which1 | which2;

                    needPrevious = true;
                    break;

                case RANGESTRAT_CONTAINS:

                    /*
                     * Non-empty range A contains non-empty range B if lower
                     * bound of A is lower or equal to lower bound of range B
                     * and upper bound of range A is greater or equal to upper
                     * bound of range A.
                     *
                     * All non-empty ranges contain an empty range.
                     */
                    strictEmpty = false;
                    if (!empty)
                    {
                        which &= (1 << 1) | (1 << 2) | (1 << 3) | (1 << 4);
                        maxLower = &lower;
                        minUpper = &upper;
                    }
                    break;

                case RANGESTRAT_CONTAINED_BY:
                    /* The opposite of contains. */
                    strictEmpty = false;
                    if (empty)
                    {
                        /* An empty range is only contained by an empty range */
                        which &= (1 << 5);
                    }
                    else
                    {
                        minLower = &lower;
                        maxUpper = &upper;
                    }
                    break;

                case RANGESTRAT_EQ:

                    /*
                     * Equal range can be only in the same quadrant where
                     * argument would be placed to.
                     */
                    strictEmpty = false;
                    which &= (1 << getQuadrant(typcache, centroid, range));
                    break;

                default:
                    elog(ERROR, "unrecognized range strategy: %d", strategy);
                    break;
            }

            if (strictEmpty)
            {
                if (empty)
                {
                    /* Scan key is empty, no branches are satisfying */
                    which = 0;
                    break;
                }
                else
                {
                    /* Shouldn't visit tree branch with empty ranges */
                    which &= (1 << 1) | (1 << 2) | (1 << 3) | (1 << 4);
                }
            }

            /*
             * Using the bounding box, see which quadrants we have to descend
             * into.
             */
            if (minLower)
            {
                /*
                 * If the centroid's lower bound is less than or equal to the
                 * minimum lower bound, anything in the 3rd and 4th quadrants
                 * will have an even smaller lower bound, and thus can't
                 * match.
                 */
                if (range_cmp_bounds(typcache, &centroidLower, minLower) <= 0)
                    which &= (1 << 1) | (1 << 2) | (1 << 5);
            }
            if (maxLower)
            {
                /*
                 * If the centroid's lower bound is greater than the maximum
                 * lower bound, anything in the 1st and 2nd quadrants will
                 * also have a greater than or equal lower bound, and thus
                 * can't match. If the centroid's lower bound is equal to the
                 * maximum lower bound, we can still exclude the 1st and 2nd
                 * quadrants if we're looking for a value strictly greater
                 * than the maximum.
                 */
                int         cmp;

                cmp = range_cmp_bounds(typcache, &centroidLower, maxLower);
                if (cmp > 0 || (!inclusive && cmp == 0))
                    which &= (1 << 3) | (1 << 4) | (1 << 5);
            }
            if (minUpper)
            {
                /*
                 * If the centroid's upper bound is less than or equal to the
                 * minimum upper bound, anything in the 2nd and 3rd quadrants
                 * will have an even smaller upper bound, and thus can't
                 * match.
                 */
                if (range_cmp_bounds(typcache, &centroidUpper, minUpper) <= 0)
                    which &= (1 << 1) | (1 << 4) | (1 << 5);
            }
            if (maxUpper)
            {
                /*
                 * If the centroid's upper bound is greater than the maximum
                 * upper bound, anything in the 1st and 4th quadrants will
                 * also have a greater than or equal upper bound, and thus
                 * can't match. If the centroid's upper bound is equal to the
                 * maximum upper bound, we can still exclude the 1st and 4th
                 * quadrants if we're looking for a value strictly greater
                 * than the maximum.
                 */
                int         cmp;

                cmp = range_cmp_bounds(typcache, &centroidUpper, maxUpper);
                if (cmp > 0 || (!inclusive && cmp == 0))
                    which &= (1 << 2) | (1 << 3) | (1 << 5);
            }

            if (which == 0)
                break;          /* no need to consider remaining conditions */
        }
    }

    /* We must descend into the quadrant(s) identified by 'which' */
    out->nodeNumbers = (int *) palloc(sizeof(int) * in->nNodes);
    if (needPrevious)
        out->traversalValues = (void **) palloc(sizeof(void *) * in->nNodes);
    out->nNodes = 0;

    /*
     * Elements of traversalValues should be allocated in
     * traversalMemoryContext
     */
    oldCtx = MemoryContextSwitchTo(in->traversalMemoryContext);

    for (i = 1; i <= in->nNodes; i++)
    {
        if (which & (1 << i))
        {
            /* Save previous prefix if needed */
            if (needPrevious)
            {
                Datum       previousCentroid;

                /*
                 * We know, that in->prefixDatum in this place is varlena,
                 * because it's range
                 */
                previousCentroid = datumCopy(in->prefixDatum, false, -1);
                out->traversalValues[out->nNodes] = (void *) previousCentroid;
            }
            out->nodeNumbers[out->nNodes] = i - 1;
            out->nNodes++;
        }
    }

    MemoryContextSwitchTo(oldCtx);

    PG_RETURN_VOID();
}

/*
 * adjacent_cmp_bounds
 *
 * Given an argument and centroid bound, this function determines if any
 * bounds that are adjacent to the argument are smaller than, or greater than
 * or equal to centroid. For brevity, we call the arg < centroid "left", and
 * arg >= centroid case "right". This corresponds to how the quadrants are
 * arranged, if you imagine that "left" is equivalent to "down" and "right"
 * is equivalent to "up".
 *
 * For the "left" case, returns -1, and for the "right" case, returns 1.
 */
static int
adjacent_cmp_bounds(TypeCacheEntry *typcache, const RangeBound *arg,
                    const RangeBound *centroid)
{
    int         cmp;

    Assert(arg->lower != centroid->lower);

    cmp = range_cmp_bounds(typcache, arg, centroid);

    if (centroid->lower)
    {
        /*------
         * The argument is an upper bound, we are searching for adjacent lower
         * bounds. A matching adjacent lower bound must be *larger* than the
         * argument, but only just.
         *
         * The following table illustrates the desired result with a fixed
         * argument bound, and different centroids. The CMP column shows
         * the value of 'cmp' variable, and ADJ shows whether the argument
         * and centroid are adjacent, per bounds_adjacent(). (N) means we
         * don't need to check for that case, because it's implied by CMP.
         * With the argument range [..., 500), the adjacent range we're
         * searching for is [500, ...):
         *
         *  ARGUMENT   CENTROID     CMP   ADJ
         *  [..., 500) [498, ...)    >    (N)   [500, ...) is to the right
         *  [..., 500) [499, ...)    =    (N)   [500, ...) is to the right
         *  [..., 500) [500, ...)    <     Y    [500, ...) is to the right
         *  [..., 500) [501, ...)    <     N    [500, ...) is to the left
         *
         * So, we must search left when the argument is smaller than, and not
         * adjacent, to the centroid. Otherwise search right.
         *------
         */
        if (cmp < 0 && !bounds_adjacent(typcache, *arg, *centroid))
            return -1;
        else
            return 1;
    }
    else
    {
        /*------
         * The argument is a lower bound, we are searching for adjacent upper
         * bounds. A matching adjacent upper bound must be *smaller* than the
         * argument, but only just.
         *
         *  ARGUMENT   CENTROID     CMP   ADJ
         *  [500, ...) [..., 499)    >    (N)   [..., 500) is to the right
         *  [500, ...) [..., 500)    >    (Y)   [..., 500) is to the right
         *  [500, ...) [..., 501)    =    (N)   [..., 500) is to the left
         *  [500, ...) [..., 502)    <    (N)   [..., 500) is to the left
         *
         * We must search left when the argument is smaller than or equal to
         * the centroid. Otherwise search right. We don't need to check
         * whether the argument is adjacent with the centroid, because it
         * doesn't matter.
         *------
         */
        if (cmp <= 0)
            return -1;
        else
            return 1;
    }
}

/*----------
 * adjacent_inner_consistent
 *
 * Like adjacent_cmp_bounds, but also takes into account the previous
 * level's centroid. We might've traversed left (or right) at the previous
 * node, in search for ranges adjacent to the other bound, even though we
 * already ruled out the possibility for any matches in that direction for
 * this bound. By comparing the argument with the previous centroid, and
 * the previous centroid with the current centroid, we can determine which
 * direction we should've moved in at previous level, and which direction we
 * actually moved.
 *
 * If there can be any matches to the left, returns -1. If to the right,
 * returns 1. If there can be no matches below this centroid, because we
 * already ruled them out at the previous level, returns 0.
 *
 * XXX: Comparing just the previous and current level isn't foolproof; we
 * might still search some branches unnecessarily. For example, imagine that
 * we are searching for value 15, and we traverse the following centroids
 * (only considering one bound for the moment):
 *
 * Level 1: 20
 * Level 2: 50
 * Level 3: 25
 *
 * At this point, previous centroid is 50, current centroid is 25, and the
 * target value is to the left. But because we already moved right from
 * centroid 20 to 50 in the first level, there cannot be any values < 20 in
 * the current branch. But we don't know that just by looking at the previous
 * and current centroid, so we traverse left, unnecessarily. The reason we are
 * down this branch is that we're searching for matches with the *other*
 * bound. If we kept track of which bound we are searching for explicitly,
 * instead of deducing that from the previous and current centroid, we could
 * avoid some unnecessary work.
 *----------
 */
static int
adjacent_inner_consistent(TypeCacheEntry *typcache, const RangeBound *arg,
                          const RangeBound *centroid, const RangeBound *prev)
{
    if (prev)
    {
        int         prevcmp;
        int         cmp;

        /*
         * Which direction were we supposed to traverse at previous level,
         * left or right?
         */
        prevcmp = adjacent_cmp_bounds(typcache, arg, prev);

        /* and which direction did we actually go? */
        cmp = range_cmp_bounds(typcache, centroid, prev);

        /* if the two don't agree, there's nothing to see here */
        if ((prevcmp < 0 && cmp >= 0) || (prevcmp > 0 && cmp < 0))
            return 0;
    }

    return adjacent_cmp_bounds(typcache, arg, centroid);
}

/*
 * SP-GiST consistent function for leaf nodes: check leaf value against query
 * using corresponding function.
 */
Datum
spg_range_quad_leaf_consistent(PG_FUNCTION_ARGS)
{
    spgLeafConsistentIn *in = (spgLeafConsistentIn *) PG_GETARG_POINTER(0);
    spgLeafConsistentOut *out = (spgLeafConsistentOut *) PG_GETARG_POINTER(1);
    RangeType  *leafRange = DatumGetRangeTypeP(in->leafDatum);
    TypeCacheEntry *typcache;
    bool        res;
    int         i;

    /* all tests are exact */
    out->recheck = false;

    /* leafDatum is what it is... */
    out->leafValue = in->leafDatum;

    typcache = range_get_typcache(fcinfo, RangeTypeGetOid(leafRange));

    /* Perform the required comparison(s) */
    res = true;
    for (i = 0; i < in->nkeys; i++)
    {
        Datum       keyDatum = in->scankeys[i].sk_argument;

        /* Call the function corresponding to the scan strategy */
        switch (in->scankeys[i].sk_strategy)
        {
            case RANGESTRAT_BEFORE:
                res = range_before_internal(typcache, leafRange,
                                            DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_OVERLEFT:
                res = range_overleft_internal(typcache, leafRange,
                                              DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_OVERLAPS:
                res = range_overlaps_internal(typcache, leafRange,
                                              DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_OVERRIGHT:
                res = range_overright_internal(typcache, leafRange,
                                               DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_AFTER:
                res = range_after_internal(typcache, leafRange,
                                           DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_ADJACENT:
                res = range_adjacent_internal(typcache, leafRange,
                                              DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_CONTAINS:
                res = range_contains_internal(typcache, leafRange,
                                              DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_CONTAINED_BY:
                res = range_contained_by_internal(typcache, leafRange,
                                                  DatumGetRangeTypeP(keyDatum));
                break;
            case RANGESTRAT_CONTAINS_ELEM:
                res = range_contains_elem_internal(typcache, leafRange,
                                                   keyDatum);
                break;
            case RANGESTRAT_EQ:
                res = range_eq_internal(typcache, leafRange,
                                        DatumGetRangeTypeP(keyDatum));
                break;
            default:
                elog(ERROR, "unrecognized range strategy: %d",
                     in->scankeys[i].sk_strategy);
                break;
        }

        /*
         * If leaf datum doesn't match to a query key, no need to check
         * subsequent keys.
         */
        if (!res)
            break;
    }

    PG_RETURN_BOOL(res);
}