freepage.c

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/*-------------------------------------------------------------------------
 *
 * freepage.c
 *    Management of free memory pages.
 *
 * The intention of this code is to provide infrastructure for memory
 * allocators written specifically for PostgreSQL.  At least in the case
 * of dynamic shared memory, we can't simply use malloc() or even
 * relatively thin wrappers like palloc() which sit on top of it, because
 * no allocator built into the operating system will deal with relative
 * pointers.  In the future, we may find other cases in which greater
 * control over our own memory management seems desirable.
 *
 * A FreePageManager keeps track of which 4kB pages of memory are currently
 * unused from the point of view of some higher-level memory allocator.
 * Unlike a user-facing allocator such as palloc(), a FreePageManager can
 * only allocate and free in units of whole pages, and freeing an
 * allocation can only be done given knowledge of its length in pages.
 *
 * Since a free page manager has only a fixed amount of dedicated memory,
 * and since there is no underlying allocator, it uses the free pages
 * it is given to manage to store its bookkeeping data.  It keeps multiple
 * freelists of runs of pages, sorted by the size of the run; the head of
 * each freelist is stored in the FreePageManager itself, and the first
 * page of each run contains a relative pointer to the next run. See
 * FreePageManagerGetInternal for more details on how the freelists are
 * managed.
 *
 * To avoid memory fragmentation, it's important to consolidate adjacent
 * spans of pages whenever possible; otherwise, large allocation requests
 * might not be satisfied even when sufficient contiguous space is
 * available.  Therefore, in addition to the freelists, we maintain an
 * in-memory btree of free page ranges ordered by page number.  If a
 * range being freed precedes or follows a range that is already free,
 * the existing range is extended; if it exactly bridges the gap between
 * free ranges, then the two existing ranges are consolidated with the
 * newly-freed range to form one great big range of free pages.
 *
 * When there is only one range of free pages, the btree is trivial and
 * is stored within the FreePageManager proper; otherwise, pages are
 * allocated from the area under management as needed.  Even in cases
 * where memory fragmentation is very severe, only a tiny fraction of
 * the pages under management are consumed by this btree.
 *
 * Portions Copyright (c) 1996-2017, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *    src/backend/utils/mmgr/freepage.c
 *
 *-------------------------------------------------------------------------
 */

#include "postgres.h"
#include "lib/stringinfo.h"
#include "miscadmin.h"

#include "utils/freepage.h"
#include "utils/relptr.h"


/* Magic numbers to identify various page types */
#define FREE_PAGE_SPAN_LEADER_MAGIC     0xea4020f0
#define FREE_PAGE_LEAF_MAGIC            0x98eae728
#define FREE_PAGE_INTERNAL_MAGIC        0x19aa32c9

/* Doubly linked list of spans of free pages; stored in first page of span. */
struct FreePageSpanLeader
{
    int         magic;          /* always FREE_PAGE_SPAN_LEADER_MAGIC */
    Size        npages;         /* number of pages in span */
    RelptrFreePageSpanLeader prev;
    RelptrFreePageSpanLeader next;
};

/* Common header for btree leaf and internal pages. */
typedef struct FreePageBtreeHeader
{
    int         magic;          /* FREE_PAGE_LEAF_MAGIC or
                                 * FREE_PAGE_INTERNAL_MAGIC */
    Size        nused;          /* number of items used */
    RelptrFreePageBtree parent; /* uplink */
} FreePageBtreeHeader;

/* Internal key; points to next level of btree. */
typedef struct FreePageBtreeInternalKey
{
    Size        first_page;     /* low bound for keys on child page */
    RelptrFreePageBtree child;  /* downlink */
} FreePageBtreeInternalKey;

/* Leaf key; no payload data. */
typedef struct FreePageBtreeLeafKey
{
    Size        first_page;     /* first page in span */
    Size        npages;         /* number of pages in span */
} FreePageBtreeLeafKey;

/* Work out how many keys will fit on a page. */
#define FPM_ITEMS_PER_INTERNAL_PAGE \
    ((FPM_PAGE_SIZE - sizeof(FreePageBtreeHeader)) / \
        sizeof(FreePageBtreeInternalKey))
#define FPM_ITEMS_PER_LEAF_PAGE \
    ((FPM_PAGE_SIZE - sizeof(FreePageBtreeHeader)) / \
        sizeof(FreePageBtreeLeafKey))

/* A btree page of either sort */
struct FreePageBtree
{
    FreePageBtreeHeader hdr;
    union
    {
        FreePageBtreeInternalKey internal_key[FPM_ITEMS_PER_INTERNAL_PAGE];
        FreePageBtreeLeafKey leaf_key[FPM_ITEMS_PER_LEAF_PAGE];
    }           u;
};

/* Results of a btree search */
typedef struct FreePageBtreeSearchResult
{
    FreePageBtree *page;
    Size        index;
    bool        found;
    unsigned    split_pages;
} FreePageBtreeSearchResult;

/* Helper functions */
static void FreePageBtreeAdjustAncestorKeys(FreePageManager *fpm,
                                FreePageBtree *btp);
static Size FreePageBtreeCleanup(FreePageManager *fpm);
static FreePageBtree *FreePageBtreeFindLeftSibling(char *base,
                             FreePageBtree *btp);
static FreePageBtree *FreePageBtreeFindRightSibling(char *base,
                              FreePageBtree *btp);
static Size FreePageBtreeFirstKey(FreePageBtree *btp);
static FreePageBtree *FreePageBtreeGetRecycled(FreePageManager *fpm);
static void FreePageBtreeInsertInternal(char *base, FreePageBtree *btp,
                            Size index, Size first_page, FreePageBtree *child);
static void FreePageBtreeInsertLeaf(FreePageBtree *btp, Size index,
                        Size first_page, Size npages);
static void FreePageBtreeRecycle(FreePageManager *fpm, Size pageno);
static void FreePageBtreeRemove(FreePageManager *fpm, FreePageBtree *btp,
                    Size index);
static void FreePageBtreeRemovePage(FreePageManager *fpm, FreePageBtree *btp);
static void FreePageBtreeSearch(FreePageManager *fpm, Size first_page,
                    FreePageBtreeSearchResult *result);
static Size FreePageBtreeSearchInternal(FreePageBtree *btp, Size first_page);
static Size FreePageBtreeSearchLeaf(FreePageBtree *btp, Size first_page);
static FreePageBtree *FreePageBtreeSplitPage(FreePageManager *fpm,
                       FreePageBtree *btp);
static void FreePageBtreeUpdateParentPointers(char *base, FreePageBtree *btp);
static void FreePageManagerDumpBtree(FreePageManager *fpm, FreePageBtree *btp,
                         FreePageBtree *parent, int level, StringInfo buf);
static void FreePageManagerDumpSpans(FreePageManager *fpm,
                         FreePageSpanLeader *span, Size expected_pages,
                         StringInfo buf);
static bool FreePageManagerGetInternal(FreePageManager *fpm, Size npages,
                           Size *first_page);
static Size FreePageManagerPutInternal(FreePageManager *fpm, Size first_page,
                           Size npages, bool soft);
static void FreePagePopSpanLeader(FreePageManager *fpm, Size pageno);
static void FreePagePushSpanLeader(FreePageManager *fpm, Size first_page,
                       Size npages);
static Size FreePageManagerLargestContiguous(FreePageManager *fpm);
static void FreePageManagerUpdateLargest(FreePageManager *fpm);

#if FPM_EXTRA_ASSERTS
static Size sum_free_pages(FreePageManager *fpm);
#endif

/*
 * Initialize a new, empty free page manager.
 *
 * 'fpm' should reference caller-provided memory large enough to contain a
 * FreePageManager.  We'll initialize it here.
 *
 * 'base' is the address to which all pointers are relative.  When managing
 * a dynamic shared memory segment, it should normally be the base of the
 * segment.  When managing backend-private memory, it can be either NULL or,
 * if managing a single contiguous extent of memory, the start of that extent.
 */
void
FreePageManagerInitialize(FreePageManager *fpm, char *base)
{
    Size        f;

    relptr_store(base, fpm->self, fpm);
    relptr_store(base, fpm->btree_root, (FreePageBtree *) NULL);
    relptr_store(base, fpm->btree_recycle, (FreePageSpanLeader *) NULL);
    fpm->btree_depth = 0;
    fpm->btree_recycle_count = 0;
    fpm->singleton_first_page = 0;
    fpm->singleton_npages = 0;
    fpm->contiguous_pages = 0;
    fpm->contiguous_pages_dirty = true;
#ifdef FPM_EXTRA_ASSERTS
    fpm->free_pages = 0;
#endif

    for (f = 0; f < FPM_NUM_FREELISTS; f++)
        relptr_store(base, fpm->freelist[f], (FreePageSpanLeader *) NULL);
}

/*
 * Allocate a run of pages of the given length from the free page manager.
 * The return value indicates whether we were able to satisfy the request;
 * if true, the first page of the allocation is stored in *first_page.
 */
bool
FreePageManagerGet(FreePageManager *fpm, Size npages, Size *first_page)
{
    bool        result;
    Size        contiguous_pages;

    result = FreePageManagerGetInternal(fpm, npages, first_page);

    /*
     * It's a bit counterintuitive, but allocating pages can actually create
     * opportunities for cleanup that create larger ranges.  We might pull a
     * key out of the btree that enables the item at the head of the btree
     * recycle list to be inserted; and then if there are more items behind it
     * one of those might cause two currently-separated ranges to merge,
     * creating a single range of contiguous pages larger than any that
     * existed previously.  It might be worth trying to improve the cleanup
     * algorithm to avoid such corner cases, but for now we just notice the
     * condition and do the appropriate reporting.
     */
    contiguous_pages = FreePageBtreeCleanup(fpm);
    if (fpm->contiguous_pages < contiguous_pages)
        fpm->contiguous_pages = contiguous_pages;

    /*
     * FreePageManagerGetInternal may have set contiguous_pages_dirty.
     * Recompute contigous_pages if so.
     */
    FreePageManagerUpdateLargest(fpm);

#ifdef FPM_EXTRA_ASSERTS
    if (result)
    {
        Assert(fpm->free_pages >= npages);
        fpm->free_pages -= npages;
    }
    Assert(fpm->free_pages == sum_free_pages(fpm));
    Assert(fpm->contiguous_pages == FreePageManagerLargestContiguous(fpm));
#endif
    return result;
}

#ifdef FPM_EXTRA_ASSERTS
static void
sum_free_pages_recurse(FreePageManager *fpm, FreePageBtree *btp, Size *sum)
{
    char       *base = fpm_segment_base(fpm);

    Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC ||
           btp->hdr.magic == FREE_PAGE_LEAF_MAGIC);
    ++*sum;
    if (btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC)
    {
        Size        index;


        for (index = 0; index < btp->hdr.nused; ++index)
        {
            FreePageBtree *child;

            child = relptr_access(base, btp->u.internal_key[index].child);
            sum_free_pages_recurse(fpm, child, sum);
        }
    }
}
static Size
sum_free_pages(FreePageManager *fpm)
{
    FreePageSpanLeader *recycle;
    char       *base = fpm_segment_base(fpm);
    Size        sum = 0;
    int         list;

    /* Count the spans by scanning the freelists. */
    for (list = 0; list < FPM_NUM_FREELISTS; ++list)
    {

        if (!relptr_is_null(fpm->freelist[list]))
        {
            FreePageSpanLeader *candidate =
            relptr_access(base, fpm->freelist[list]);

            do
            {
                sum += candidate->npages;
                candidate = relptr_access(base, candidate->next);
            } while (candidate != NULL);
        }
    }

    /* Count btree internal pages. */
    if (fpm->btree_depth > 0)
    {
        FreePageBtree *root = relptr_access(base, fpm->btree_root);

        sum_free_pages_recurse(fpm, root, &sum);
    }

    /* Count the recycle list. */
    for (recycle = relptr_access(base, fpm->btree_recycle);
         recycle != NULL;
         recycle = relptr_access(base, recycle->next))
    {
        Assert(recycle->npages == 1);
        ++sum;
    }

    return sum;
}
#endif

/*
 * Compute the size of the largest run of pages that the user could
 * successfully get.
 */
static Size
FreePageManagerLargestContiguous(FreePageManager *fpm)
{
    char       *base;
    Size        largest;

    base = fpm_segment_base(fpm);
    largest = 0;
    if (!relptr_is_null(fpm->freelist[FPM_NUM_FREELISTS - 1]))
    {
        FreePageSpanLeader *candidate;

        candidate = relptr_access(base, fpm->freelist[FPM_NUM_FREELISTS - 1]);
        do
        {
            if (candidate->npages > largest)
                largest = candidate->npages;
            candidate = relptr_access(base, candidate->next);
        } while (candidate != NULL);
    }
    else
    {
        Size        f = FPM_NUM_FREELISTS - 1;

        do
        {
            --f;
            if (!relptr_is_null(fpm->freelist[f]))
            {
                largest = f + 1;
                break;
            }
        } while (f > 0);
    }

    return largest;
}

/*
 * Recompute the size of the largest run of pages that the user could
 * successfully get, if it has been marked dirty.
 */
static void
FreePageManagerUpdateLargest(FreePageManager *fpm)
{
    if (!fpm->contiguous_pages_dirty)
        return;

    fpm->contiguous_pages = FreePageManagerLargestContiguous(fpm);
    fpm->contiguous_pages_dirty = false;
}

/*
 * Transfer a run of pages to the free page manager.
 */
void
FreePageManagerPut(FreePageManager *fpm, Size first_page, Size npages)
{
    Size        contiguous_pages;

    Assert(npages > 0);

    /* Record the new pages. */
    contiguous_pages =
        FreePageManagerPutInternal(fpm, first_page, npages, false);

    /*
     * If the new range we inserted into the page manager was contiguous with
     * an existing range, it may have opened up cleanup opportunities.
     */
    if (contiguous_pages > npages)
    {
        Size        cleanup_contiguous_pages;

        cleanup_contiguous_pages = FreePageBtreeCleanup(fpm);
        if (cleanup_contiguous_pages > contiguous_pages)
            contiguous_pages = cleanup_contiguous_pages;
    }

    /* See if we now have a new largest chunk. */
    if (fpm->contiguous_pages < contiguous_pages)
        fpm->contiguous_pages = contiguous_pages;

    /*
     * The earlier call to FreePageManagerPutInternal may have set
     * contiguous_pages_dirty if it needed to allocate internal pages, so
     * recompute contiguous_pages if necessary.
     */
    FreePageManagerUpdateLargest(fpm);

#ifdef FPM_EXTRA_ASSERTS
    fpm->free_pages += npages;
    Assert(fpm->free_pages == sum_free_pages(fpm));
    Assert(fpm->contiguous_pages == FreePageManagerLargestContiguous(fpm));
#endif
}

/*
 * Produce a debugging dump of the state of a free page manager.
 */
char *
FreePageManagerDump(FreePageManager *fpm)
{
    char       *base = fpm_segment_base(fpm);
    StringInfoData buf;
    FreePageSpanLeader *recycle;
    bool        dumped_any_freelist = false;
    Size        f;

    /* Initialize output buffer. */
    initStringInfo(&buf);

    /* Dump general stuff. */
    appendStringInfo(&buf, "metadata: self %zu max contiguous pages = %zu\n",
                     fpm->self.relptr_off, fpm->contiguous_pages);

    /* Dump btree. */
    if (fpm->btree_depth > 0)
    {
        FreePageBtree *root;

        appendStringInfo(&buf, "btree depth %u:\n", fpm->btree_depth);
        root = relptr_access(base, fpm->btree_root);
        FreePageManagerDumpBtree(fpm, root, NULL, 0, &buf);
    }
    else if (fpm->singleton_npages > 0)
    {
        appendStringInfo(&buf, "singleton: %zu(%zu)\n",
                         fpm->singleton_first_page, fpm->singleton_npages);
    }

    /* Dump btree recycle list. */
    recycle = relptr_access(base, fpm->btree_recycle);
    if (recycle != NULL)
    {
        appendStringInfo(&buf, "btree recycle:");
        FreePageManagerDumpSpans(fpm, recycle, 1, &buf);
    }

    /* Dump free lists. */
    for (f = 0; f < FPM_NUM_FREELISTS; ++f)
    {
        FreePageSpanLeader *span;

        if (relptr_is_null(fpm->freelist[f]))
            continue;
        if (!dumped_any_freelist)
        {
            appendStringInfo(&buf, "freelists:\n");
            dumped_any_freelist = true;
        }
        appendStringInfo(&buf, "  %zu:", f + 1);
        span = relptr_access(base, fpm->freelist[f]);
        FreePageManagerDumpSpans(fpm, span, f + 1, &buf);
    }

    /* And return result to caller. */
    return buf.data;
}


/*
 * The first_page value stored at index zero in any non-root page must match
 * the first_page value stored in its parent at the index which points to that
 * page.  So when the value stored at index zero in a btree page changes, we've
 * got to walk up the tree adjusting ancestor keys until we reach an ancestor
 * where that key isn't index zero.  This function should be called after
 * updating the first key on the target page; it will propagate the change
 * upward as far as needed.
 *
 * We assume here that the first key on the page has not changed enough to
 * require changes in the ordering of keys on its ancestor pages.  Thus,
 * if we search the parent page for the first key greater than or equal to
 * the first key on the current page, the downlink to this page will be either
 * the exact index returned by the search (if the first key decreased)
 * or one less (if the first key increased).
 */
static void
FreePageBtreeAdjustAncestorKeys(FreePageManager *fpm, FreePageBtree *btp)
{
    char       *base = fpm_segment_base(fpm);
    Size        first_page;
    FreePageBtree *parent;
    FreePageBtree *child;

    /* This might be either a leaf or an internal page. */
    Assert(btp->hdr.nused > 0);
    if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
    {
        Assert(btp->hdr.nused <= FPM_ITEMS_PER_LEAF_PAGE);
        first_page = btp->u.leaf_key[0].first_page;
    }
    else
    {
        Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        Assert(btp->hdr.nused <= FPM_ITEMS_PER_INTERNAL_PAGE);
        first_page = btp->u.internal_key[0].first_page;
    }
    child = btp;

    /* Loop until we find an ancestor that does not require adjustment. */
    for (;;)
    {
        Size        s;

        parent = relptr_access(base, child->hdr.parent);
        if (parent == NULL)
            break;
        s = FreePageBtreeSearchInternal(parent, first_page);

        /* Key is either at index s or index s-1; figure out which. */
        if (s >= parent->hdr.nused)
        {
            Assert(s == parent->hdr.nused);
            --s;
        }
        else
        {
            FreePageBtree *check;

            check = relptr_access(base, parent->u.internal_key[s].child);
            if (check != child)
            {
                Assert(s > 0);
                --s;
            }
        }

#ifdef USE_ASSERT_CHECKING
        /* Debugging double-check. */
        {
            FreePageBtree *check;

            check = relptr_access(base, parent->u.internal_key[s].child);
            Assert(s < parent->hdr.nused);
            Assert(child == check);
        }
#endif

        /* Update the parent key. */
        parent->u.internal_key[s].first_page = first_page;

        /*
         * If this is the first key in the parent, go up another level; else
         * done.
         */
        if (s > 0)
            break;
        child = parent;
    }
}

/*
 * Attempt to reclaim space from the free-page btree.  The return value is
 * the largest range of contiguous pages created by the cleanup operation.
 */
static Size
FreePageBtreeCleanup(FreePageManager *fpm)
{
    char       *base = fpm_segment_base(fpm);
    Size        max_contiguous_pages = 0;

    /* Attempt to shrink the depth of the btree. */
    while (!relptr_is_null(fpm->btree_root))
    {
        FreePageBtree *root = relptr_access(base, fpm->btree_root);

        /* If the root contains only one key, reduce depth by one. */
        if (root->hdr.nused == 1)
        {
            /* Shrink depth of tree by one. */
            Assert(fpm->btree_depth > 0);
            --fpm->btree_depth;
            if (root->hdr.magic == FREE_PAGE_LEAF_MAGIC)
            {
                /* If root is a leaf, convert only entry to singleton range. */
                relptr_store(base, fpm->btree_root, (FreePageBtree *) NULL);
                fpm->singleton_first_page = root->u.leaf_key[0].first_page;
                fpm->singleton_npages = root->u.leaf_key[0].npages;
            }
            else
            {
                FreePageBtree *newroot;

                /* If root is an internal page, make only child the root. */
                Assert(root->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
                relptr_copy(fpm->btree_root, root->u.internal_key[0].child);
                newroot = relptr_access(base, fpm->btree_root);
                relptr_store(base, newroot->hdr.parent, (FreePageBtree *) NULL);
            }
            FreePageBtreeRecycle(fpm, fpm_pointer_to_page(base, root));
        }
        else if (root->hdr.nused == 2 &&
                 root->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        {
            Size        end_of_first;
            Size        start_of_second;

            end_of_first = root->u.leaf_key[0].first_page +
                root->u.leaf_key[0].npages;
            start_of_second = root->u.leaf_key[1].first_page;

            if (end_of_first + 1 == start_of_second)
            {
                Size        root_page = fpm_pointer_to_page(base, root);

                if (end_of_first == root_page)
                {
                    FreePagePopSpanLeader(fpm, root->u.leaf_key[0].first_page);
                    FreePagePopSpanLeader(fpm, root->u.leaf_key[1].first_page);
                    fpm->singleton_first_page = root->u.leaf_key[0].first_page;
                    fpm->singleton_npages = root->u.leaf_key[0].npages +
                        root->u.leaf_key[1].npages + 1;
                    fpm->btree_depth = 0;
                    relptr_store(base, fpm->btree_root,
                                 (FreePageBtree *) NULL);
                    FreePagePushSpanLeader(fpm, fpm->singleton_first_page,
                                           fpm->singleton_npages);
                    Assert(max_contiguous_pages == 0);
                    max_contiguous_pages = fpm->singleton_npages;
                }
            }

            /* Whether it worked or not, it's time to stop. */
            break;
        }
        else
        {
            /* Nothing more to do.  Stop. */
            break;
        }
    }

    /*
     * Attempt to free recycled btree pages.  We skip this if releasing the
     * recycled page would require a btree page split, because the page we're
     * trying to recycle would be consumed by the split, which would be
     * counterproductive.
     *
     * We also currently only ever attempt to recycle the first page on the
     * list; that could be made more aggressive, but it's not clear that the
     * complexity would be worthwhile.
     */
    while (fpm->btree_recycle_count > 0)
    {
        FreePageBtree *btp;
        Size        first_page;
        Size        contiguous_pages;

        btp = FreePageBtreeGetRecycled(fpm);
        first_page = fpm_pointer_to_page(base, btp);
        contiguous_pages = FreePageManagerPutInternal(fpm, first_page, 1, true);
        if (contiguous_pages == 0)
        {
            FreePageBtreeRecycle(fpm, first_page);
            break;
        }
        else
        {
            if (contiguous_pages > max_contiguous_pages)
                max_contiguous_pages = contiguous_pages;
        }
    }

    return max_contiguous_pages;
}

/*
 * Consider consolidating the given page with its left or right sibling,
 * if it's fairly empty.
 */
static void
FreePageBtreeConsolidate(FreePageManager *fpm, FreePageBtree *btp)
{
    char       *base = fpm_segment_base(fpm);
    FreePageBtree *np;
    Size        max;

    /*
     * We only try to consolidate pages that are less than a third full. We
     * could be more aggressive about this, but that might risk performing
     * consolidation only to end up splitting again shortly thereafter.  Since
     * the btree should be very small compared to the space under management,
     * our goal isn't so much to ensure that it always occupies the absolutely
     * smallest possible number of pages as to reclaim pages before things get
     * too egregiously out of hand.
     */
    if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        max = FPM_ITEMS_PER_LEAF_PAGE;
    else
    {
        Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        max = FPM_ITEMS_PER_INTERNAL_PAGE;
    }
    if (btp->hdr.nused >= max / 3)
        return;

    /*
     * If we can fit our right sibling's keys onto this page, consolidate.
     */
    np = FreePageBtreeFindRightSibling(base, btp);
    if (np != NULL && btp->hdr.nused + np->hdr.nused <= max)
    {
        if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        {
            memcpy(&btp->u.leaf_key[btp->hdr.nused], &np->u.leaf_key[0],
                   sizeof(FreePageBtreeLeafKey) * np->hdr.nused);
            btp->hdr.nused += np->hdr.nused;
        }
        else
        {
            memcpy(&btp->u.internal_key[btp->hdr.nused], &np->u.internal_key[0],
                   sizeof(FreePageBtreeInternalKey) * np->hdr.nused);
            btp->hdr.nused += np->hdr.nused;
            FreePageBtreeUpdateParentPointers(base, btp);
        }
        FreePageBtreeRemovePage(fpm, np);
        return;
    }

    /*
     * If we can fit our keys onto our left sibling's page, consolidate. In
     * this case, we move our keys onto the other page rather than visca
     * versa, to avoid having to adjust ancestor keys.
     */
    np = FreePageBtreeFindLeftSibling(base, btp);
    if (np != NULL && btp->hdr.nused + np->hdr.nused <= max)
    {
        if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        {
            memcpy(&np->u.leaf_key[np->hdr.nused], &btp->u.leaf_key[0],
                   sizeof(FreePageBtreeLeafKey) * btp->hdr.nused);
            np->hdr.nused += btp->hdr.nused;
        }
        else
        {
            memcpy(&np->u.internal_key[np->hdr.nused], &btp->u.internal_key[0],
                   sizeof(FreePageBtreeInternalKey) * btp->hdr.nused);
            np->hdr.nused += btp->hdr.nused;
            FreePageBtreeUpdateParentPointers(base, np);
        }
        FreePageBtreeRemovePage(fpm, btp);
        return;
    }
}

/*
 * Find the passed page's left sibling; that is, the page at the same level
 * of the tree whose keyspace immediately precedes ours.
 */
static FreePageBtree *
FreePageBtreeFindLeftSibling(char *base, FreePageBtree *btp)
{
    FreePageBtree *p = btp;
    int         levels = 0;

    /* Move up until we can move left. */
    for (;;)
    {
        Size        first_page;
        Size        index;

        first_page = FreePageBtreeFirstKey(p);
        p = relptr_access(base, p->hdr.parent);

        if (p == NULL)
            return NULL;        /* we were passed the rightmost page */

        index = FreePageBtreeSearchInternal(p, first_page);
        if (index > 0)
        {
            Assert(p->u.internal_key[index].first_page == first_page);
            p = relptr_access(base, p->u.internal_key[index - 1].child);
            break;
        }
        Assert(index == 0);
        ++levels;
    }

    /* Descend left. */
    while (levels > 0)
    {
        Assert(p->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        p = relptr_access(base, p->u.internal_key[p->hdr.nused - 1].child);
        --levels;
    }
    Assert(p->hdr.magic == btp->hdr.magic);

    return p;
}

/*
 * Find the passed page's right sibling; that is, the page at the same level
 * of the tree whose keyspace immediately follows ours.
 */
static FreePageBtree *
FreePageBtreeFindRightSibling(char *base, FreePageBtree *btp)
{
    FreePageBtree *p = btp;
    int         levels = 0;

    /* Move up until we can move right. */
    for (;;)
    {
        Size        first_page;
        Size        index;

        first_page = FreePageBtreeFirstKey(p);
        p = relptr_access(base, p->hdr.parent);

        if (p == NULL)
            return NULL;        /* we were passed the rightmost page */

        index = FreePageBtreeSearchInternal(p, first_page);
        if (index < p->hdr.nused - 1)
        {
            Assert(p->u.internal_key[index].first_page == first_page);
            p = relptr_access(base, p->u.internal_key[index + 1].child);
            break;
        }
        Assert(index == p->hdr.nused - 1);
        ++levels;
    }

    /* Descend left. */
    while (levels > 0)
    {
        Assert(p->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        p = relptr_access(base, p->u.internal_key[0].child);
        --levels;
    }
    Assert(p->hdr.magic == btp->hdr.magic);

    return p;
}

/*
 * Get the first key on a btree page.
 */
static Size
FreePageBtreeFirstKey(FreePageBtree *btp)
{
    Assert(btp->hdr.nused > 0);

    if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        return btp->u.leaf_key[0].first_page;
    else
    {
        Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        return btp->u.internal_key[0].first_page;
    }
}

/*
 * Get a page from the btree recycle list for use as a btree page.
 */
static FreePageBtree *
FreePageBtreeGetRecycled(FreePageManager *fpm)
{
    char       *base = fpm_segment_base(fpm);
    FreePageSpanLeader *victim = relptr_access(base, fpm->btree_recycle);
    FreePageSpanLeader *newhead;

    Assert(victim != NULL);
    newhead = relptr_access(base, victim->next);
    if (newhead != NULL)
        relptr_copy(newhead->prev, victim->prev);
    relptr_store(base, fpm->btree_recycle, newhead);
    Assert(fpm_pointer_is_page_aligned(base, victim));
    fpm->btree_recycle_count--;
    return (FreePageBtree *) victim;
}

/*
 * Insert an item into an internal page.
 */
static void
FreePageBtreeInsertInternal(char *base, FreePageBtree *btp, Size index,
                            Size first_page, FreePageBtree *child)
{
    Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
    Assert(btp->hdr.nused <= FPM_ITEMS_PER_INTERNAL_PAGE);
    Assert(index <= btp->hdr.nused);
    memmove(&btp->u.internal_key[index + 1], &btp->u.internal_key[index],
            sizeof(FreePageBtreeInternalKey) * (btp->hdr.nused - index));
    btp->u.internal_key[index].first_page = first_page;
    relptr_store(base, btp->u.internal_key[index].child, child);
    ++btp->hdr.nused;
}

/*
 * Insert an item into a leaf page.
 */
static void
FreePageBtreeInsertLeaf(FreePageBtree *btp, Size index, Size first_page,
                        Size npages)
{
    Assert(btp->hdr.magic == FREE_PAGE_LEAF_MAGIC);
    Assert(btp->hdr.nused <= FPM_ITEMS_PER_LEAF_PAGE);
    Assert(index <= btp->hdr.nused);
    memmove(&btp->u.leaf_key[index + 1], &btp->u.leaf_key[index],
            sizeof(FreePageBtreeLeafKey) * (btp->hdr.nused - index));
    btp->u.leaf_key[index].first_page = first_page;
    btp->u.leaf_key[index].npages = npages;
    ++btp->hdr.nused;
}

/*
 * Put a page on the btree recycle list.
 */
static void
FreePageBtreeRecycle(FreePageManager *fpm, Size pageno)
{
    char       *base = fpm_segment_base(fpm);
    FreePageSpanLeader *head = relptr_access(base, fpm->btree_recycle);
    FreePageSpanLeader *span;

    span = (FreePageSpanLeader *) fpm_page_to_pointer(base, pageno);
    span->magic = FREE_PAGE_SPAN_LEADER_MAGIC;
    span->npages = 1;
    relptr_store(base, span->next, head);
    relptr_store(base, span->prev, (FreePageSpanLeader *) NULL);
    if (head != NULL)
        relptr_store(base, head->prev, span);
    relptr_store(base, fpm->btree_recycle, span);
    fpm->btree_recycle_count++;
}

/*
 * Remove an item from the btree at the given position on the given page.
 */
static void
FreePageBtreeRemove(FreePageManager *fpm, FreePageBtree *btp, Size index)
{
    Assert(btp->hdr.magic == FREE_PAGE_LEAF_MAGIC);
    Assert(index < btp->hdr.nused);

    /* When last item is removed, extirpate entire page from btree. */
    if (btp->hdr.nused == 1)
    {
        FreePageBtreeRemovePage(fpm, btp);
        return;
    }

    /* Physically remove the key from the page. */
    --btp->hdr.nused;
    if (index < btp->hdr.nused)
        memmove(&btp->u.leaf_key[index], &btp->u.leaf_key[index + 1],
                sizeof(FreePageBtreeLeafKey) * (btp->hdr.nused - index));

    /* If we just removed the first key, adjust ancestor keys. */
    if (index == 0)
        FreePageBtreeAdjustAncestorKeys(fpm, btp);

    /* Consider whether to consolidate this page with a sibling. */
    FreePageBtreeConsolidate(fpm, btp);
}

/*
 * Remove a page from the btree.  Caller is responsible for having relocated
 * any keys from this page that are still wanted.  The page is placed on the
 * recycled list.
 */
static void
FreePageBtreeRemovePage(FreePageManager *fpm, FreePageBtree *btp)
{
    char       *base = fpm_segment_base(fpm);
    FreePageBtree *parent;
    Size        index;
    Size        first_page;

    for (;;)
    {
        /* Find parent page. */
        parent = relptr_access(base, btp->hdr.parent);
        if (parent == NULL)
        {
            /* We are removing the root page. */
            relptr_store(base, fpm->btree_root, (FreePageBtree *) NULL);
            fpm->btree_depth = 0;
            Assert(fpm->singleton_first_page == 0);
            Assert(fpm->singleton_npages == 0);
            return;
        }

        /*
         * If the parent contains only one item, we need to remove it as well.
         */
        if (parent->hdr.nused > 1)
            break;
        FreePageBtreeRecycle(fpm, fpm_pointer_to_page(base, btp));
        btp = parent;
    }

    /* Find and remove the downlink. */
    first_page = FreePageBtreeFirstKey(btp);
    if (parent->hdr.magic == FREE_PAGE_LEAF_MAGIC)
    {
        index = FreePageBtreeSearchLeaf(parent, first_page);
        Assert(index < parent->hdr.nused);
        if (index < parent->hdr.nused - 1)
            memmove(&parent->u.leaf_key[index],
                    &parent->u.leaf_key[index + 1],
                    sizeof(FreePageBtreeLeafKey)
                    * (parent->hdr.nused - index - 1));
    }
    else
    {
        index = FreePageBtreeSearchInternal(parent, first_page);
        Assert(index < parent->hdr.nused);
        if (index < parent->hdr.nused - 1)
            memmove(&parent->u.internal_key[index],
                    &parent->u.internal_key[index + 1],
                    sizeof(FreePageBtreeInternalKey)
                    * (parent->hdr.nused - index - 1));
    }
    parent->hdr.nused--;
    Assert(parent->hdr.nused > 0);

    /* Recycle the page. */
    FreePageBtreeRecycle(fpm, fpm_pointer_to_page(base, btp));

    /* Adjust ancestor keys if needed. */
    if (index == 0)
        FreePageBtreeAdjustAncestorKeys(fpm, parent);

    /* Consider whether to consolidate the parent with a sibling. */
    FreePageBtreeConsolidate(fpm, parent);
}

/*
 * Search the btree for an entry for the given first page and initialize
 * *result with the results of the search.  result->page and result->index
 * indicate either the position of an exact match or the position at which
 * the new key should be inserted.  result->found is true for an exact match,
 * otherwise false.  result->split_pages will contain the number of additional
 * btree pages that will be needed when performing a split to insert a key.
 * Except as described above, the contents of fields in the result object are
 * undefined on return.
 */
static void
FreePageBtreeSearch(FreePageManager *fpm, Size first_page,
                    FreePageBtreeSearchResult *result)
{
    char       *base = fpm_segment_base(fpm);
    FreePageBtree *btp = relptr_access(base, fpm->btree_root);
    Size        index;

    result->split_pages = 1;

    /* If the btree is empty, there's nothing to find. */
    if (btp == NULL)
    {
        result->page = NULL;
        result->found = false;
        return;
    }

    /* Descend until we hit a leaf. */
    while (btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC)
    {
        FreePageBtree *child;
        bool        found_exact;

        index = FreePageBtreeSearchInternal(btp, first_page);
        found_exact = index < btp->hdr.nused &&
            btp->u.internal_key[index].first_page == first_page;

        /*
         * If we found an exact match we descend directly.  Otherwise, we
         * descend into the child to the left if possible so that we can find
         * the insertion point at that child's high end.
         */
        if (!found_exact && index > 0)
            --index;

        /* Track required split depth for leaf insert. */
        if (btp->hdr.nused >= FPM_ITEMS_PER_INTERNAL_PAGE)
        {
            Assert(btp->hdr.nused == FPM_ITEMS_PER_INTERNAL_PAGE);
            result->split_pages++;
        }
        else
            result->split_pages = 0;

        /* Descend to appropriate child page. */
        Assert(index < btp->hdr.nused);
        child = relptr_access(base, btp->u.internal_key[index].child);
        Assert(relptr_access(base, child->hdr.parent) == btp);
        btp = child;
    }

    /* Track required split depth for leaf insert. */
    if (btp->hdr.nused >= FPM_ITEMS_PER_LEAF_PAGE)
    {
        Assert(btp->hdr.nused == FPM_ITEMS_PER_INTERNAL_PAGE);
        result->split_pages++;
    }
    else
        result->split_pages = 0;

    /* Search leaf page. */
    index = FreePageBtreeSearchLeaf(btp, first_page);

    /* Assemble results. */
    result->page = btp;
    result->index = index;
    result->found = index < btp->hdr.nused &&
        first_page == btp->u.leaf_key[index].first_page;
}

/*
 * Search an internal page for the first key greater than or equal to a given
 * page number.  Returns the index of that key, or one greater than the number
 * of keys on the page if none.
 */
static Size
FreePageBtreeSearchInternal(FreePageBtree *btp, Size first_page)
{
    Size        low = 0;
    Size        high = btp->hdr.nused;

    Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
    Assert(high > 0 && high <= FPM_ITEMS_PER_INTERNAL_PAGE);

    while (low < high)
    {
        Size        mid = (low + high) / 2;
        Size        val = btp->u.internal_key[mid].first_page;

        if (first_page == val)
            return mid;
        else if (first_page < val)
            high = mid;
        else
            low = mid + 1;
    }

    return low;
}

/*
 * Search a leaf page for the first key greater than or equal to a given
 * page number.  Returns the index of that key, or one greater than the number
 * of keys on the page if none.
 */
static Size
FreePageBtreeSearchLeaf(FreePageBtree *btp, Size first_page)
{
    Size        low = 0;
    Size        high = btp->hdr.nused;

    Assert(btp->hdr.magic == FREE_PAGE_LEAF_MAGIC);
    Assert(high > 0 && high <= FPM_ITEMS_PER_LEAF_PAGE);

    while (low < high)
    {
        Size        mid = (low + high) / 2;
        Size        val = btp->u.leaf_key[mid].first_page;

        if (first_page == val)
            return mid;
        else if (first_page < val)
            high = mid;
        else
            low = mid + 1;
    }

    return low;
}

/*
 * Allocate a new btree page and move half the keys from the provided page
 * to the new page.  Caller is responsible for making sure that there's a
 * page available from fpm->btree_recycle.  Returns a pointer to the new page,
 * to which caller must add a downlink.
 */
static FreePageBtree *
FreePageBtreeSplitPage(FreePageManager *fpm, FreePageBtree *btp)
{
    FreePageBtree *newsibling;

    newsibling = FreePageBtreeGetRecycled(fpm);
    newsibling->hdr.magic = btp->hdr.magic;
    newsibling->hdr.nused = btp->hdr.nused / 2;
    relptr_copy(newsibling->hdr.parent, btp->hdr.parent);
    btp->hdr.nused -= newsibling->hdr.nused;

    if (btp->hdr.magic == FREE_PAGE_LEAF_MAGIC)
        memcpy(&newsibling->u.leaf_key,
               &btp->u.leaf_key[btp->hdr.nused],
               sizeof(FreePageBtreeLeafKey) * newsibling->hdr.nused);
    else
    {
        Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
        memcpy(&newsibling->u.internal_key,
               &btp->u.internal_key[btp->hdr.nused],
               sizeof(FreePageBtreeInternalKey) * newsibling->hdr.nused);
        FreePageBtreeUpdateParentPointers(fpm_segment_base(fpm), newsibling);
    }

    return newsibling;
}

/*
 * When internal pages are split or merged, the parent pointers of their
 * children must be updated.
 */
static void
FreePageBtreeUpdateParentPointers(char *base, FreePageBtree *btp)
{
    Size        i;

    Assert(btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC);
    for (i = 0; i < btp->hdr.nused; ++i)
    {
        FreePageBtree *child;

        child = relptr_access(base, btp->u.internal_key[i].child);
        relptr_store(base, child->hdr.parent, btp);
    }
}

/*
 * Debugging dump of btree data.
 */
static void
FreePageManagerDumpBtree(FreePageManager *fpm, FreePageBtree *btp,
                         FreePageBtree *parent, int level, StringInfo buf)
{
    char       *base = fpm_segment_base(fpm);
    Size        pageno = fpm_pointer_to_page(base, btp);
    Size        index;
    FreePageBtree *check_parent;

    check_stack_depth();
    check_parent = relptr_access(base, btp->hdr.parent);
    appendStringInfo(buf, "  %zu@%d %c", pageno, level,
                     btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC ? 'i' : 'l');
    if (parent != check_parent)
        appendStringInfo(buf, " [actual parent %zu, expected %zu]",
                         fpm_pointer_to_page(base, check_parent),
                         fpm_pointer_to_page(base, parent));
    appendStringInfoChar(buf, ':');
    for (index = 0; index < btp->hdr.nused; ++index)
    {
        if (btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC)
            appendStringInfo(buf, " %zu->%zu",
                             btp->u.internal_key[index].first_page,
                             btp->u.internal_key[index].child.relptr_off / FPM_PAGE_SIZE);
        else
            appendStringInfo(buf, " %zu(%zu)",
                             btp->u.leaf_key[index].first_page,
                             btp->u.leaf_key[index].npages);
    }
    appendStringInfo(buf, "\n");

    if (btp->hdr.magic == FREE_PAGE_INTERNAL_MAGIC)
    {
        for (index = 0; index < btp->hdr.nused; ++index)
        {
            FreePageBtree *child;

            child = relptr_access(base, btp->u.internal_key[index].child);
            FreePageManagerDumpBtree(fpm, child, btp, level + 1, buf);
        }
    }
}

/*
 * Debugging dump of free-span data.
 */
static void
FreePageManagerDumpSpans(FreePageManager *fpm, FreePageSpanLeader *span,
                         Size expected_pages, StringInfo buf)
{
    char       *base = fpm_segment_base(fpm);

    while (span != NULL)
    {
        if (span->npages != expected_pages)
            appendStringInfo(buf, " %zu(%zu)", fpm_pointer_to_page(base, span),
                             span->npages);
        else
            appendStringInfo(buf, " %zu", fpm_pointer_to_page(base, span));
        span = relptr_access(base, span->next);
    }

    appendStringInfo(buf, "\n");
}

/*
 * This function allocates a run of pages of the given length from the free
 * page manager.
 */
static bool
FreePageManagerGetInternal(FreePageManager *fpm, Size npages, Size *first_page)
{
    char       *base = fpm_segment_base(fpm);
    FreePageSpanLeader *victim = NULL;
    FreePageSpanLeader *prev;
    FreePageSpanLeader *next;
    FreePageBtreeSearchResult result;
    Size        victim_page = 0;    /* placate compiler */
    Size        f;

    /*
     * Search for a free span.
     *
     * Right now, we use a simple best-fit policy here, but it's possible for
     * this to result in memory fragmentation if we're repeatedly asked to
     * allocate chunks just a little smaller than what we have available.
     * Hopefully, this is unlikely, because we expect most requests to be
     * single pages or superblock-sized chunks -- but no policy can be optimal
     * under all circumstances unless it has knowledge of future allocation
     * patterns.
     */
    for (f = Min(npages, FPM_NUM_FREELISTS) - 1; f < FPM_NUM_FREELISTS; ++f)
    {
        /* Skip empty freelists. */
        if (relptr_is_null(fpm->freelist[f]))
            continue;

        /*
         * All of the freelists except the last one contain only items of a
         * single size, so we just take the first one.  But the final free
         * list contains everything too big for any of the other lists, so we
         * need to search the list.
         */
        if (f < FPM_NUM_FREELISTS - 1)
            victim = relptr_access(base, fpm->freelist[f]);
        else
        {
            FreePageSpanLeader *candidate;

            candidate = relptr_access(base, fpm->freelist[f]);
            do
            {
                if (candidate->npages >= npages && (victim == NULL ||
                                                    victim->npages > candidate->npages))
                {
                    victim = candidate;
                    if (victim->npages == npages)
                        break;
                }
                candidate = relptr_access(base, candidate->next);
            } while (candidate != NULL);
        }
        break;
    }

    /* If we didn't find an allocatable span, return failure. */
    if (victim == NULL)
        return false;

    /* Remove span from free list. */
    Assert(victim->magic == FREE_PAGE_SPAN_LEADER_MAGIC);
    prev = relptr_access(base, victim->prev);
    next = relptr_access(base, victim->next);
    if (prev != NULL)
        relptr_copy(prev->next, victim->next);
    else
        relptr_copy(fpm->freelist[f], victim->next);
    if (next != NULL)
        relptr_copy(next->prev, victim->prev);
    victim_page = fpm_pointer_to_page(base, victim);

    /* Decide whether we might be invalidating contiguous_pages. */
    if (f == FPM_NUM_FREELISTS - 1 &&
        victim->npages == fpm->contiguous_pages)
    {
        /*
         * The victim span came from the oversized freelist, and had the same
         * size as the longest span.  There may or may not be another one of
         * the same size, so contiguous_pages must be recomputed just to be
         * safe.
         */
        fpm->contiguous_pages_dirty = true;
    }
    else if (f + 1 == fpm->contiguous_pages &&
             relptr_is_null(fpm->freelist[f]))
    {
        /*
         * The victim span came from a fixed sized freelist, and it was the
         * list for spans of the same size as the current longest span, and
         * the list is now empty after removing the victim.  So
         * contiguous_pages must be recomputed without a doubt.
         */
        fpm->contiguous_pages_dirty = true;
    }

    /*
     * If we haven't initialized the btree yet, the victim must be the single
     * span stored within the FreePageManager itself.  Otherwise, we need to
     * update the btree.
     */
    if (relptr_is_null(fpm->btree_root))
    {
        Assert(victim_page == fpm->singleton_first_page);
        Assert(victim->npages == fpm->singleton_npages);
        Assert(victim->npages >= npages);
        fpm->singleton_first_page += npages;
        fpm->singleton_npages -= npages;
        if (fpm->singleton_npages > 0)
            FreePagePushSpanLeader(fpm, fpm->singleton_first_page,
                                   fpm->singleton_npages);
    }
    else
    {
        /*
         * If the span we found is exactly the right size, remove it from the
         * btree completely.  Otherwise, adjust the btree entry to reflect the
         * still-unallocated portion of the span, and put that portion on the
         * appropriate free list.
         */
        FreePageBtreeSearch(fpm, victim_page, &result);
        Assert(result.found);
        if (victim->npages == npages)
            FreePageBtreeRemove(fpm, result.page, result.index);
        else
        {
            FreePageBtreeLeafKey *key;

            /* Adjust btree to reflect remaining pages. */
            Assert(victim->npages > npages);
            key = &result.page->u.leaf_key[result.index];
            Assert(key->npages == victim->npages);
            key->first_page += npages;
            key->npages -= npages;
            if (result.index == 0)
                FreePageBtreeAdjustAncestorKeys(fpm, result.page);

            /* Put the unallocated pages back on the appropriate free list. */
            FreePagePushSpanLeader(fpm, victim_page + npages,
                                   victim->npages - npages);
        }
    }

    /* Return results to caller. */
    *first_page = fpm_pointer_to_page(base, victim);
    return true;
}

/*
 * Put a range of pages into the btree and freelists, consolidating it with
 * existing free spans just before and/or after it.  If 'soft' is true,
 * only perform the insertion if it can be done without allocating new btree
 * pages; if false, do it always.  Returns 0 if the soft flag caused the
 * insertion to be skipped, or otherwise the size of the contiguous span
 * created by the insertion.  This may be larger than npages if we're able
 * to consolidate with an adjacent range.  *internal_pages_used is set to
 * true if the btree allocated pages for internal purposes, which might
 * invalidate the current largest run requiring it to be recomputed.
 */
static Size
FreePageManagerPutInternal(FreePageManager *fpm, Size first_page, Size npages,
                           bool soft)
{
    char       *base = fpm_segment_base(fpm);
    FreePageBtreeSearchResult result;
    FreePageBtreeLeafKey *prevkey = NULL;
    FreePageBtreeLeafKey *nextkey = NULL;
    FreePageBtree *np;
    Size        nindex;

    Assert(npages > 0);

    /* We can store a single free span without initializing the btree. */
    if (fpm->btree_depth == 0)
    {
        if (fpm->singleton_npages == 0)
        {
            /* Don't have a span yet; store this one. */
            fpm->singleton_first_page = first_page;
            fpm->singleton_npages = npages;
            FreePagePushSpanLeader(fpm, first_page, npages);
            return fpm->singleton_npages;
        }
        else if (fpm->singleton_first_page + fpm->singleton_npages ==
                 first_page)
        {
            /* New span immediately follows sole existing span. */
            fpm->singleton_npages += npages;
            FreePagePopSpanLeader(fpm, fpm->singleton_first_page);
            FreePagePushSpanLeader(fpm, fpm->singleton_first_page,
                                   fpm->singleton_npages);
            return fpm->singleton_npages;
        }
        else if (first_page + npages == fpm->singleton_first_page)
        {
            /* New span immediately precedes sole existing span. */
            FreePagePopSpanLeader(fpm, fpm->singleton_first_page);
            fpm->singleton_first_page = first_page;
            fpm->singleton_npages += npages;
            FreePagePushSpanLeader(fpm, fpm->singleton_first_page,
                                   fpm->singleton_npages);
            return fpm->singleton_npages;
        }
        else
        {
            /* Not contiguous; we need to initialize the btree. */
            Size        root_page;
            FreePageBtree *root;

            if (!relptr_is_null(fpm->btree_recycle))
                root = FreePageBtreeGetRecycled(fpm);
            else if (FreePageManagerGetInternal(fpm, 1, &root_page))
                root = (FreePageBtree *) fpm_page_to_pointer(base, root_page);
            else
            {
                /* We'd better be able to get a page from the existing range. */
                elog(FATAL, "free page manager btree is corrupt");
            }

            /* Create the btree and move the preexisting range into it. */
            root->hdr.magic = FREE_PAGE_LEAF_MAGIC;
            root->hdr.nused = 1;
            relptr_store(base, root->hdr.parent, (FreePageBtree *) NULL);
            root->u.leaf_key[0].first_page = fpm->singleton_first_page;
            root->u.leaf_key[0].npages = fpm->singleton_npages;
            relptr_store(base, fpm->btree_root, root);
            fpm->singleton_first_page = 0;
            fpm->singleton_npages = 0;
            fpm->btree_depth = 1;

            /*
             * Corner case: it may be that the btree root took the very last
             * free page.  In that case, the sole btree entry covers a zero
             * page run, which is invalid.  Overwrite it with the entry we're
             * trying to insert and get out.
             */
            if (root->u.leaf_key[0].npages == 0)
            {
                root->u.leaf_key[0].first_page = first_page;
                root->u.leaf_key[0].npages = npages;
                FreePagePushSpanLeader(fpm, first_page, npages);
                return npages;
            }

            /* Fall through to insert the new key. */
        }
    }

    /* Search the btree. */
    FreePageBtreeSearch(fpm, first_page, &result);
    Assert(!result.found);
    if (result.index > 0)
        prevkey = &result.page->u.leaf_key[result.index - 1];
    if (result.index < result.page->hdr.nused)
    {
        np = result.page;
        nindex = result.index;
        nextkey = &result.page->u.leaf_key[result.index];
    }
    else
    {
        np = FreePageBtreeFindRightSibling(base, result.page);
        nindex = 0;
        if (np != NULL)
            nextkey = &np->u.leaf_key[0];
    }

    /* Consolidate with the previous entry if possible. */
    if (prevkey != NULL && prevkey->first_page + prevkey->npages >= first_page)
    {
        bool        remove_next = false;
        Size        result;

        Assert(prevkey->first_page + prevkey->npages == first_page);
        prevkey->npages = (first_page - prevkey->first_page) + npages;

        /* Check whether we can *also* consolidate with the following entry. */
        if (nextkey != NULL &&
            prevkey->first_page + prevkey->npages >= nextkey->first_page)
        {
            Assert(prevkey->first_page + prevkey->npages ==
                   nextkey->first_page);
            prevkey->npages = (nextkey->first_page - prevkey->first_page)
                + nextkey->npages;
            FreePagePopSpanLeader(fpm, nextkey->first_page);
            remove_next = true;
        }

        /* Put the span on the correct freelist and save size. */
        FreePagePopSpanLeader(fpm, prevkey->first_page);
        FreePagePushSpanLeader(fpm, prevkey->first_page, prevkey->npages);
        result = prevkey->npages;

        /*
         * If we consolidated with both the preceding and following entries,
         * we must remove the following entry.  We do this last, because
         * removing an element from the btree may invalidate pointers we hold
         * into the current data structure.
         *
         * NB: The btree is technically in an invalid state a this point
         * because we've already updated prevkey to cover the same key space
         * as nextkey.  FreePageBtreeRemove() shouldn't notice that, though.
         */
        if (remove_next)
            FreePageBtreeRemove(fpm, np, nindex);

        return result;
    }

    /* Consolidate with the next entry if possible. */
    if (nextkey != NULL && first_page + npages >= nextkey->first_page)
    {
        Size        newpages;

        /* Compute new size for span. */
        Assert(first_page + npages == nextkey->first_page);
        newpages = (nextkey->first_page - first_page) + nextkey->npages;

        /* Put span on correct free list. */
        FreePagePopSpanLeader(fpm, nextkey->first_page);
        FreePagePushSpanLeader(fpm, first_page, newpages);

        /* Update key in place. */
        nextkey->first_page = first_page;
        nextkey->npages = newpages;

        /* If reducing first key on page, ancestors might need adjustment. */
        if (nindex == 0)
            FreePageBtreeAdjustAncestorKeys(fpm, np);

        return nextkey->npages;
    }

    /* Split leaf page and as many of its ancestors as necessary. */
    if (result.split_pages > 0)
    {
        /*
         * NB: We could consider various coping strategies here to avoid a
         * split; most obviously, if np != result.page, we could target that
         * page instead.   More complicated shuffling strategies could be
         * possible as well; basically, unless every single leaf page is 100%
         * full, we can jam this key in there if we try hard enough.  It's
         * unlikely that trying that hard is worthwhile, but it's possible we
         * might need to make more than no effort.  For now, we just do the
         * easy thing, which is nothing.
         */

        /* If this is a soft insert, it's time to give up. */
        if (soft)
            return 0;

        /* Check whether we need to allocate more btree pages to split. */
        if (result.split_pages > fpm->btree_recycle_count)
        {
            Size        pages_needed;
            Size        recycle_page;
            Size        i;

            /*
             * Allocate the required number of pages and split each one in
             * turn.  This should never fail, because if we've got enough
             * spans of free pages kicking around that we need additional
             * storage space just to remember them all, then we should
             * certainly have enough to expand the btree, which should only
             * ever use a tiny number of pages compared to the number under
             * management.  If it does, something's badly screwed up.
             */
            pages_needed = result.split_pages - fpm->btree_recycle_count;
            for (i = 0; i < pages_needed; ++i)
            {
                if (!FreePageManagerGetInternal(fpm, 1, &recycle_page))
                    elog(FATAL, "free page manager btree is corrupt");
                FreePageBtreeRecycle(fpm, recycle_page);
            }

            /*
             * The act of allocating pages to recycle may have invalidated the
             * results of our previous btree reserch, so repeat it. (We could
             * recheck whether any of our split-avoidance strategies that were
             * not viable before now are, but it hardly seems worthwhile, so
             * we don't bother. Consolidation can't be possible now if it
             * wasn't previously.)
             */
            FreePageBtreeSearch(fpm, first_page, &result);

            /*
             * The act of allocating pages for use in constructing our btree
             * should never cause any page to become more full, so the new
             * split depth should be no greater than the old one, and perhaps
             * less if we fortuitously allocated a chunk that freed up a slot
             * on the page we need to update.
             */
            Assert(result.split_pages <= fpm->btree_recycle_count);
        }

        /* If we still need to perform a split, do it. */
        if (result.split_pages > 0)
        {
            FreePageBtree *split_target = result.page;
            FreePageBtree *child = NULL;
            Size        key = first_page;

            for (;;)
            {
                FreePageBtree *newsibling;
                FreePageBtree *parent;

                /* Identify parent page, which must receive downlink. */
                parent = relptr_access(base, split_target->hdr.parent);

                /* Split the page - downlink not added yet. */
                newsibling = FreePageBtreeSplitPage(fpm, split_target);

                /*
                 * At this point in the loop, we're always carrying a pending
                 * insertion.  On the first pass, it's the actual key we're
                 * trying to insert; on subsequent passes, it's the downlink
                 * that needs to be added as a result of the split performed
                 * during the previous loop iteration.  Since we've just split
                 * the page, there's definitely room on one of the two
                 * resulting pages.
                 */
                if (child == NULL)
                {
                    Size        index;
                    FreePageBtree *insert_into;

                    insert_into = key < newsibling->u.leaf_key[0].first_page ?
                        split_target : newsibling;
                    index = FreePageBtreeSearchLeaf(insert_into, key);
                    FreePageBtreeInsertLeaf(insert_into, index, key, npages);
                    if (index == 0 && insert_into == split_target)
                        FreePageBtreeAdjustAncestorKeys(fpm, split_target);
                }
                else
                {
                    Size        index;
                    FreePageBtree *insert_into;

                    insert_into =
                        key < newsibling->u.internal_key[0].first_page ?
                        split_target : newsibling;
                    index = FreePageBtreeSearchInternal(insert_into, key);
                    FreePageBtreeInsertInternal(base, insert_into, index,
                                                key, child);
                    relptr_store(base, child->hdr.parent, insert_into);
                    if (index == 0 && insert_into == split_target)
                        FreePageBtreeAdjustAncestorKeys(fpm, split_target);
                }

                /* If the page we just split has no parent, split the root. */
                if (parent == NULL)
                {
                    FreePageBtree *newroot;

                    newroot = FreePageBtreeGetRecycled(fpm);
                    newroot->hdr.magic = FREE_PAGE_INTERNAL_MAGIC;
                    newroot->hdr.nused = 2;
                    relptr_store(base, newroot->hdr.parent,
                                 (FreePageBtree *) NULL);
                    newroot->u.internal_key[0].first_page =
                        FreePageBtreeFirstKey(split_target);
                    relptr_store(base, newroot->u.internal_key[0].child,
                                 split_target);
                    relptr_store(base, split_target->hdr.parent, newroot);
                    newroot->u.internal_key[1].first_page =
                        FreePageBtreeFirstKey(newsibling);
                    relptr_store(base, newroot->u.internal_key[1].child,
                                 newsibling);
                    relptr_store(base, newsibling->hdr.parent, newroot);
                    relptr_store(base, fpm->btree_root, newroot);
                    fpm->btree_depth++;

                    break;
                }

                /* If the parent page isn't full, insert the downlink. */
                key = newsibling->u.internal_key[0].first_page;
                if (parent->hdr.nused < FPM_ITEMS_PER_INTERNAL_PAGE)
                {
                    Size        index;

                    index = FreePageBtreeSearchInternal(parent, key);
                    FreePageBtreeInsertInternal(base, parent, index,
                                                key, newsibling);
                    relptr_store(base, newsibling->hdr.parent, parent);
                    if (index == 0)
                        FreePageBtreeAdjustAncestorKeys(fpm, parent);
                    break;
                }

                /* The parent also needs to be split, so loop around. */
                child = newsibling;
                split_target = parent;
            }

            /*
             * The loop above did the insert, so just need to update the free
             * list, and we're done.
             */
            FreePagePushSpanLeader(fpm, first_page, npages);

            return npages;
        }
    }

    /* Physically add the key to the page. */
    Assert(result.page->hdr.nused < FPM_ITEMS_PER_LEAF_PAGE);
    FreePageBtreeInsertLeaf(result.page, result.index, first_page, npages);

    /* If new first key on page, ancestors might need adjustment. */
    if (result.index == 0)
        FreePageBtreeAdjustAncestorKeys(fpm, result.page);

    /* Put it on the free list. */
    FreePagePushSpanLeader(fpm, first_page, npages);

    return npages;
}

/*
 * Remove a FreePageSpanLeader from the linked-list that contains it, either
 * because we're changing the size of the span, or because we're allocating it.
 */
static void
FreePagePopSpanLeader(FreePageManager *fpm, Size pageno)
{
    char       *base = fpm_segment_base(fpm);
    FreePageSpanLeader *span;
    FreePageSpanLeader *next;
    FreePageSpanLeader *prev;

    span = (FreePageSpanLeader *) fpm_page_to_pointer(base, pageno);

    next = relptr_access(base, span->next);
    prev = relptr_access(base, span->prev);
    if (next != NULL)
        relptr_copy(next->prev, span->prev);
    if (prev != NULL)
        relptr_copy(prev->next, span->next);
    else
    {
        Size        f = Min(span->npages, FPM_NUM_FREELISTS) - 1;

        Assert(fpm->freelist[f].relptr_off == pageno * FPM_PAGE_SIZE);
        relptr_copy(fpm->freelist[f], span->next);
    }
}

/*
 * Initialize a new FreePageSpanLeader and put it on the appropriate free list.
 */
static void
FreePagePushSpanLeader(FreePageManager *fpm, Size first_page, Size npages)
{
    char       *base = fpm_segment_base(fpm);
    Size        f = Min(npages, FPM_NUM_FREELISTS) - 1;
    FreePageSpanLeader *head = relptr_access(base, fpm->freelist[f]);
    FreePageSpanLeader *span;

    span = (FreePageSpanLeader *) fpm_page_to_pointer(base, first_page);
    span->magic = FREE_PAGE_SPAN_LEADER_MAGIC;
    span->npages = npages;
    relptr_store(base, span->next, head);
    relptr_store(base, span->prev, (FreePageSpanLeader *) NULL);
    if (head != NULL)
        relptr_store(base, head->prev, span);
    relptr_store(base, fpm->freelist[f], span);
}