Adding upstream version 14.5.upstream/14.5upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 1045 insertions, 0 deletions

diff --git a/src/backend/lib/integerset.c b/src/backend/lib/integerset.c
new file mode 100644
index 0000000..278a91b
--- /dev/null
+++ b/src/backend/lib/integerset.c
@@ -0,0 +1,1045 @@
+/*-------------------------------------------------------------------------
+ *
+ * integerset.c
+ *	  Data structure to hold a large set of 64-bit integers efficiently
+ *
+ * IntegerSet provides an in-memory data structure to hold a set of
+ * arbitrary 64-bit integers.  Internally, the values are stored in a
+ * B-tree, with a special packed representation at the leaf level using
+ * the Simple-8b algorithm, which can pack clusters of nearby values
+ * very tightly.
+ *
+ * Memory consumption depends on the number of values stored, but also
+ * on how far the values are from each other.  In the best case, with
+ * long runs of consecutive integers, memory consumption can be as low as
+ * 0.1 bytes per integer.  In the worst case, if integers are more than
+ * 2^32 apart, it uses about 8 bytes per integer.  In typical use, the
+ * consumption per integer is somewhere between those extremes, depending
+ * on the range of integers stored, and how "clustered" they are.
+ *
+ *
+ * Interface
+ * ---------
+ *
+ *	intset_create			- Create a new, empty set
+ *	intset_add_member		- Add an integer to the set
+ *	intset_is_member		- Test if an integer is in the set
+ *	intset_begin_iterate	- Begin iterating through all integers in set
+ *	intset_iterate_next		- Return next set member, if any
+ *
+ * intset_create() creates the set in the current memory context.  Subsequent
+ * operations that add to the data structure will continue to allocate from
+ * that same context, even if it's not current anymore.
+ *
+ * Note that there is no function to free an integer set.  If you need to do
+ * that, create a dedicated memory context to hold it, and destroy the memory
+ * context instead.
+ *
+ *
+ * Limitations
+ * -----------
+ *
+ * - Values must be added in order.  (Random insertions would require
+ *   splitting nodes, which hasn't been implemented.)
+ *
+ * - Values cannot be added while iteration is in progress.
+ *
+ * - No support for removing values.
+ *
+ * None of these limitations are fundamental to the data structure, so they
+ * could be lifted if needed, by writing some new code.  But the current
+ * users of this facility don't need them.
+ *
+ *
+ * References
+ * ----------
+ *
+ * Simple-8b encoding is based on:
+ *
+ * Vo Ngoc Anh, Alistair Moffat, Index compression using 64-bit words,
+ *   Software - Practice & Experience, v.40 n.2, p.131-147, February 2010
+ *   (https://doi.org/10.1002/spe.948)
+ *
+ *
+ * Portions Copyright (c) 1996-2021, PostgreSQL Global Development Group
+ * Portions Copyright (c) 1994, Regents of the University of California
+ *
+ * IDENTIFICATION
+ *	  src/backend/lib/integerset.c
+ *
+ *-------------------------------------------------------------------------
+ */
+#include "postgres.h"
+
+#include "access/htup_details.h"
+#include "lib/integerset.h"
+#include "port/pg_bitutils.h"
+#include "utils/memutils.h"
+
+
+/*
+ * Maximum number of integers that can be encoded in a single Simple-8b
+ * codeword. (Defined here before anything else, so that we can size arrays
+ * using this.)
+ */
+#define SIMPLE8B_MAX_VALUES_PER_CODEWORD 240
+
+/*
+ * Parameters for shape of the in-memory B-tree.
+ *
+ * These set the size of each internal and leaf node.  They don't necessarily
+ * need to be the same, because the tree is just an in-memory structure.
+ * With the default 64, each node is about 1 kb.
+ *
+ * If you change these, you must recalculate MAX_TREE_LEVELS, too!
+ */
+#define MAX_INTERNAL_ITEMS	64
+#define MAX_LEAF_ITEMS	64
+
+/*
+ * Maximum height of the tree.
+ *
+ * MAX_TREE_ITEMS is calculated from the "fan-out" of the B-tree.  The
+ * theoretical maximum number of items that we can store in a set is 2^64,
+ * so MAX_TREE_LEVELS should be set so that:
+ *
+ *   MAX_LEAF_ITEMS * MAX_INTERNAL_ITEMS ^ (MAX_TREE_LEVELS - 1) >= 2^64.
+ *
+ * In practice, we'll need far fewer levels, because you will run out of
+ * memory long before reaching that number, but let's be conservative.
+ */
+#define MAX_TREE_LEVELS		11
+
+/*
+ * Node structures, for the in-memory B-tree.
+ *
+ * An internal node holds a number of downlink pointers to leaf nodes, or
+ * to internal nodes on a lower level.  For each downlink, the key value
+ * corresponding to the lower level node is stored in a sorted array.  The
+ * stored key values are low keys.  In other words, if the downlink has value
+ * X, then all items stored on that child are >= X.
+ *
+ * Each leaf node holds a number of "items", with a varying number of
+ * integers packed into each item.  Each item consists of two 64-bit words:
+ * The first word holds the first integer stored in the item, in plain format.
+ * The second word contains between 0 and 240 more integers, packed using
+ * Simple-8b encoding.  By storing the first integer in plain, unpacked,
+ * format, we can use binary search to quickly find an item that holds (or
+ * would hold) a particular integer.  And by storing the rest in packed form,
+ * we still get pretty good memory density, if there are clusters of integers
+ * with similar values.
+ *
+ * Each leaf node also has a pointer to the next leaf node, so that the leaf
+ * nodes can be easily walked from beginning to end when iterating.
+ */
+typedef struct intset_node intset_node;
+typedef struct intset_leaf_node intset_leaf_node;
+typedef struct intset_internal_node intset_internal_node;
+
+/* Common structure of both leaf and internal nodes. */
+struct intset_node
+{
+	uint16		level;			/* tree level of this node */
+	uint16		num_items;		/* number of items in this node */
+};
+
+/* Internal node */
+struct intset_internal_node
+{
+	/* common header, must match intset_node */
+	uint16		level;			/* >= 1 on internal nodes */
+	uint16		num_items;
+
+	/*
+	 * 'values' is an array of key values, and 'downlinks' are pointers to
+	 * lower-level nodes, corresponding to the key values.
+	 */
+	uint64		values[MAX_INTERNAL_ITEMS];
+	intset_node *downlinks[MAX_INTERNAL_ITEMS];
+};
+
+/* Leaf node */
+typedef struct
+{
+	uint64		first;			/* first integer in this item */
+	uint64		codeword;		/* simple8b encoded differences from 'first' */
+} leaf_item;
+
+#define MAX_VALUES_PER_LEAF_ITEM	(1 + SIMPLE8B_MAX_VALUES_PER_CODEWORD)
+
+struct intset_leaf_node
+{
+	/* common header, must match intset_node */
+	uint16		level;			/* 0 on leafs */
+	uint16		num_items;
+
+	intset_leaf_node *next;		/* right sibling, if any */
+
+	leaf_item	items[MAX_LEAF_ITEMS];
+};
+
+/*
+ * We buffer insertions in a simple array, before packing and inserting them
+ * into the B-tree.  MAX_BUFFERED_VALUES sets the size of the buffer.  The
+ * encoder assumes that it is large enough that we can always fill a leaf
+ * item with buffered new items.  In other words, MAX_BUFFERED_VALUES must be
+ * larger than MAX_VALUES_PER_LEAF_ITEM.  For efficiency, make it much larger.
+ */
+#define MAX_BUFFERED_VALUES			(MAX_VALUES_PER_LEAF_ITEM * 2)
+
+/*
+ * IntegerSet is the top-level object representing the set.
+ *
+ * The integers are stored in an in-memory B-tree structure, plus an array
+ * for newly-added integers.  IntegerSet also tracks information about memory
+ * usage, as well as the current position when iterating the set with
+ * intset_begin_iterate / intset_iterate_next.
+ */
+struct IntegerSet
+{
+	/*
+	 * 'context' is the memory context holding this integer set and all its
+	 * tree nodes.
+	 *
+	 * 'mem_used' tracks the amount of memory used.  We don't do anything with
+	 * it in integerset.c itself, but the callers can ask for it with
+	 * intset_memory_usage().
+	 */
+	MemoryContext context;
+	uint64		mem_used;
+
+	uint64		num_entries;	/* total # of values in the set */
+	uint64		highest_value;	/* highest value stored in this set */
+
+	/*
+	 * B-tree to hold the packed values.
+	 *
+	 * 'rightmost_nodes' hold pointers to the rightmost node on each level.
+	 * rightmost_parent[0] is rightmost leaf, rightmost_parent[1] is its
+	 * parent, and so forth, all the way up to the root. These are needed when
+	 * adding new values. (Currently, we require that new values are added at
+	 * the end.)
+	 */
+	int			num_levels;		/* height of the tree */
+	intset_node *root;			/* root node */
+	intset_node *rightmost_nodes[MAX_TREE_LEVELS];
+	intset_leaf_node *leftmost_leaf;	/* leftmost leaf node */
+
+	/*
+	 * Holding area for new items that haven't been inserted to the tree yet.
+	 */
+	uint64		buffered_values[MAX_BUFFERED_VALUES];
+	int			num_buffered_values;
+
+	/*
+	 * Iterator support.
+	 *
+	 * 'iter_values' is an array of integers ready to be returned to the
+	 * caller; 'iter_num_values' is the length of that array, and
+	 * 'iter_valueno' is the next index.  'iter_node' and 'iter_itemno' point
+	 * to the leaf node, and item within the leaf node, to get the next batch
+	 * of values from.
+	 *
+	 * Normally, 'iter_values' points to 'iter_values_buf', which holds items
+	 * decoded from a leaf item.  But after we have scanned the whole B-tree,
+	 * we iterate through all the unbuffered values, too, by pointing
+	 * iter_values to 'buffered_values'.
+	 */
+	bool		iter_active;	/* is iteration in progress? */
+
+	const uint64 *iter_values;
+	int			iter_num_values;	/* number of elements in 'iter_values' */
+	int			iter_valueno;	/* next index into 'iter_values' */
+
+	intset_leaf_node *iter_node;	/* current leaf node */
+	int			iter_itemno;	/* next item in 'iter_node' to decode */
+
+	uint64		iter_values_buf[MAX_VALUES_PER_LEAF_ITEM];
+};
+
+/*
+ * Prototypes for internal functions.
+ */
+static void intset_update_upper(IntegerSet *intset, int level,
+								intset_node *child, uint64 child_key);
+static void intset_flush_buffered_values(IntegerSet *intset);
+
+static int	intset_binsrch_uint64(uint64 value, uint64 *arr, int arr_elems,
+								  bool nextkey);
+static int	intset_binsrch_leaf(uint64 value, leaf_item *arr, int arr_elems,
+								bool nextkey);
+
+static uint64 simple8b_encode(const uint64 *ints, int *num_encoded, uint64 base);
+static int	simple8b_decode(uint64 codeword, uint64 *decoded, uint64 base);
+static bool simple8b_contains(uint64 codeword, uint64 key, uint64 base);
+
+
+/*
+ * Create a new, initially empty, integer set.
+ *
+ * The integer set is created in the current memory context.
+ * We will do all subsequent allocations in the same context, too, regardless
+ * of which memory context is current when new integers are added to the set.
+ */
+IntegerSet *
+intset_create(void)
+{
+	IntegerSet *intset;
+
+	intset = (IntegerSet *) palloc(sizeof(IntegerSet));
+	intset->context = CurrentMemoryContext;
+	intset->mem_used = GetMemoryChunkSpace(intset);
+
+	intset->num_entries = 0;
+	intset->highest_value = 0;
+
+	intset->num_levels = 0;
+	intset->root = NULL;
+	memset(intset->rightmost_nodes, 0, sizeof(intset->rightmost_nodes));
+	intset->leftmost_leaf = NULL;
+
+	intset->num_buffered_values = 0;
+
+	intset->iter_active = false;
+	intset->iter_node = NULL;
+	intset->iter_itemno = 0;
+	intset->iter_valueno = 0;
+	intset->iter_num_values = 0;
+	intset->iter_values = NULL;
+
+	return intset;
+}
+
+/*
+ * Allocate a new node.
+ */
+static intset_internal_node *
+intset_new_internal_node(IntegerSet *intset)
+{
+	intset_internal_node *n;
+
+	n = (intset_internal_node *) MemoryContextAlloc(intset->context,
+													sizeof(intset_internal_node));
+	intset->mem_used += GetMemoryChunkSpace(n);
+
+	n->level = 0;				/* caller must set */
+	n->num_items = 0;
+
+	return n;
+}
+
+static intset_leaf_node *
+intset_new_leaf_node(IntegerSet *intset)
+{
+	intset_leaf_node *n;
+
+	n = (intset_leaf_node *) MemoryContextAlloc(intset->context,
+												sizeof(intset_leaf_node));
+	intset->mem_used += GetMemoryChunkSpace(n);
+
+	n->level = 0;
+	n->num_items = 0;
+	n->next = NULL;
+
+	return n;
+}
+
+/*
+ * Return the number of entries in the integer set.
+ */
+uint64
+intset_num_entries(IntegerSet *intset)
+{
+	return intset->num_entries;
+}
+
+/*
+ * Return the amount of memory used by the integer set.
+ */
+uint64
+intset_memory_usage(IntegerSet *intset)
+{
+	return intset->mem_used;
+}
+
+/*
+ * Add a value to the set.
+ *
+ * Values must be added in order.
+ */
+void
+intset_add_member(IntegerSet *intset, uint64 x)
+{
+	if (intset->iter_active)
+		elog(ERROR, "cannot add new values to integer set while iteration is in progress");
+
+	if (x <= intset->highest_value && intset->num_entries > 0)
+		elog(ERROR, "cannot add value to integer set out of order");
+
+	if (intset->num_buffered_values >= MAX_BUFFERED_VALUES)
+	{
+		/* Time to flush our buffer */
+		intset_flush_buffered_values(intset);
+		Assert(intset->num_buffered_values < MAX_BUFFERED_VALUES);
+	}
+
+	/* Add it to the buffer of newly-added values */
+	intset->buffered_values[intset->num_buffered_values] = x;
+	intset->num_buffered_values++;
+	intset->num_entries++;
+	intset->highest_value = x;
+}
+
+/*
+ * Take a batch of buffered values, and pack them into the B-tree.
+ */
+static void
+intset_flush_buffered_values(IntegerSet *intset)
+{
+	uint64	   *values = intset->buffered_values;
+	uint64		num_values = intset->num_buffered_values;
+	int			num_packed = 0;
+	intset_leaf_node *leaf;
+
+	leaf = (intset_leaf_node *) intset->rightmost_nodes[0];
+
+	/*
+	 * If the tree is completely empty, create the first leaf page, which is
+	 * also the root.
+	 */
+	if (leaf == NULL)
+	{
+		/*
+		 * This is the very first item in the set.
+		 *
+		 * Allocate root node. It's also a leaf.
+		 */
+		leaf = intset_new_leaf_node(intset);
+
+		intset->root = (intset_node *) leaf;
+		intset->leftmost_leaf = leaf;
+		intset->rightmost_nodes[0] = (intset_node *) leaf;
+		intset->num_levels = 1;
+	}
+
+	/*
+	 * If there are less than MAX_VALUES_PER_LEAF_ITEM values in the buffer,
+	 * stop.  In most cases, we cannot encode that many values in a single
+	 * value, but this way, the encoder doesn't have to worry about running
+	 * out of input.
+	 */
+	while (num_values - num_packed >= MAX_VALUES_PER_LEAF_ITEM)
+	{
+		leaf_item	item;
+		int			num_encoded;
+
+		/*
+		 * Construct the next leaf item, packing as many buffered values as
+		 * possible.
+		 */
+		item.first = values[num_packed];
+		item.codeword = simple8b_encode(&values[num_packed + 1],
+										&num_encoded,
+										item.first);
+
+		/*
+		 * Add the item to the node, allocating a new node if the old one is
+		 * full.
+		 */
+		if (leaf->num_items >= MAX_LEAF_ITEMS)
+		{
+			/* Allocate new leaf and link it to the tree */
+			intset_leaf_node *old_leaf = leaf;
+
+			leaf = intset_new_leaf_node(intset);
+			old_leaf->next = leaf;
+			intset->rightmost_nodes[0] = (intset_node *) leaf;
+			intset_update_upper(intset, 1, (intset_node *) leaf, item.first);
+		}
+		leaf->items[leaf->num_items++] = item;
+
+		num_packed += 1 + num_encoded;
+	}
+
+	/*
+	 * Move any remaining buffered values to the beginning of the array.
+	 */
+	if (num_packed < intset->num_buffered_values)
+	{
+		memmove(&intset->buffered_values[0],
+				&intset->buffered_values[num_packed],
+				(intset->num_buffered_values - num_packed) * sizeof(uint64));
+	}
+	intset->num_buffered_values -= num_packed;
+}
+
+/*
+ * Insert a downlink into parent node, after creating a new node.
+ *
+ * Recurses if the parent node is full, too.
+ */
+static void
+intset_update_upper(IntegerSet *intset, int level, intset_node *child,
+					uint64 child_key)
+{
+	intset_internal_node *parent;
+
+	Assert(level > 0);
+
+	/*
+	 * Create a new root node, if necessary.
+	 */
+	if (level >= intset->num_levels)
+	{
+		intset_node *oldroot = intset->root;
+		uint64		downlink_key;
+
+		/* MAX_TREE_LEVELS should be more than enough, this shouldn't happen */
+		if (intset->num_levels == MAX_TREE_LEVELS)
+			elog(ERROR, "could not expand integer set, maximum number of levels reached");
+		intset->num_levels++;
+
+		/*
+		 * Get the first value on the old root page, to be used as the
+		 * downlink.
+		 */
+		if (intset->root->level == 0)
+			downlink_key = ((intset_leaf_node *) oldroot)->items[0].first;
+		else
+			downlink_key = ((intset_internal_node *) oldroot)->values[0];
+
+		parent = intset_new_internal_node(intset);
+		parent->level = level;
+		parent->values[0] = downlink_key;
+		parent->downlinks[0] = oldroot;
+		parent->num_items = 1;
+
+		intset->root = (intset_node *) parent;
+		intset->rightmost_nodes[level] = (intset_node *) parent;
+	}
+
+	/*
+	 * Place the downlink on the parent page.
+	 */
+	parent = (intset_internal_node *) intset->rightmost_nodes[level];
+
+	if (parent->num_items < MAX_INTERNAL_ITEMS)
+	{
+		parent->values[parent->num_items] = child_key;
+		parent->downlinks[parent->num_items] = child;
+		parent->num_items++;
+	}
+	else
+	{
+		/*
+		 * Doesn't fit.  Allocate new parent, with the downlink as the first
+		 * item on it, and recursively insert the downlink to the new parent
+		 * to the grandparent.
+		 */
+		parent = intset_new_internal_node(intset);
+		parent->level = level;
+		parent->values[0] = child_key;
+		parent->downlinks[0] = child;
+		parent->num_items = 1;
+
+		intset->rightmost_nodes[level] = (intset_node *) parent;
+
+		intset_update_upper(intset, level + 1, (intset_node *) parent, child_key);
+	}
+}
+
+/*
+ * Does the set contain the given value?
+ */
+bool
+intset_is_member(IntegerSet *intset, uint64 x)
+{
+	intset_node *node;
+	intset_leaf_node *leaf;
+	int			level;
+	int			itemno;
+	leaf_item  *item;
+
+	/*
+	 * The value might be in the buffer of newly-added values.
+	 */
+	if (intset->num_buffered_values > 0 && x >= intset->buffered_values[0])
+	{
+		int			itemno;
+
+		itemno = intset_binsrch_uint64(x,
+									   intset->buffered_values,
+									   intset->num_buffered_values,
+									   false);
+		if (itemno >= intset->num_buffered_values)
+			return false;
+		else
+			return (intset->buffered_values[itemno] == x);
+	}
+
+	/*
+	 * Start from the root, and walk down the B-tree to find the right leaf
+	 * node.
+	 */
+	if (!intset->root)
+		return false;
+	node = intset->root;
+	for (level = intset->num_levels - 1; level > 0; level--)
+	{
+		intset_internal_node *n = (intset_internal_node *) node;
+
+		Assert(node->level == level);
+
+		itemno = intset_binsrch_uint64(x, n->values, n->num_items, true);
+		if (itemno == 0)
+			return false;
+		node = n->downlinks[itemno - 1];
+	}
+	Assert(node->level == 0);
+	leaf = (intset_leaf_node *) node;
+
+	/*
+	 * Binary search to find the right item on the leaf page
+	 */
+	itemno = intset_binsrch_leaf(x, leaf->items, leaf->num_items, true);
+	if (itemno == 0)
+		return false;
+	item = &leaf->items[itemno - 1];
+
+	/* Is this a match to the first value on the item? */
+	if (item->first == x)
+		return true;
+	Assert(x > item->first);
+
+	/* Is it in the packed codeword? */
+	if (simple8b_contains(item->codeword, x, item->first))
+		return true;
+
+	return false;
+}
+
+/*
+ * Begin in-order scan through all the values.
+ *
+ * While the iteration is in-progress, you cannot add new values to the set.
+ */
+void
+intset_begin_iterate(IntegerSet *intset)
+{
+	/* Note that we allow an iteration to be abandoned midway */
+	intset->iter_active = true;
+	intset->iter_node = intset->leftmost_leaf;
+	intset->iter_itemno = 0;
+	intset->iter_valueno = 0;
+	intset->iter_num_values = 0;
+	intset->iter_values = intset->iter_values_buf;
+}
+
+/*
+ * Returns the next integer, when iterating.
+ *
+ * intset_begin_iterate() must be called first.  intset_iterate_next() returns
+ * the next value in the set.  Returns true, if there was another value, and
+ * stores the value in *next.  Otherwise, returns false.
+ */
+bool
+intset_iterate_next(IntegerSet *intset, uint64 *next)
+{
+	Assert(intset->iter_active);
+	for (;;)
+	{
+		/* Return next iter_values[] entry if any */
+		if (intset->iter_valueno < intset->iter_num_values)
+		{
+			*next = intset->iter_values[intset->iter_valueno++];
+			return true;
+		}
+
+		/* Decode next item in current leaf node, if any */
+		if (intset->iter_node &&
+			intset->iter_itemno < intset->iter_node->num_items)
+		{
+			leaf_item  *item;
+			int			num_decoded;
+
+			item = &intset->iter_node->items[intset->iter_itemno++];
+
+			intset->iter_values_buf[0] = item->first;
+			num_decoded = simple8b_decode(item->codeword,
+										  &intset->iter_values_buf[1],
+										  item->first);
+			intset->iter_num_values = num_decoded + 1;
+			intset->iter_valueno = 0;
+			continue;
+		}
+
+		/* No more items on this leaf, step to next node */
+		if (intset->iter_node)
+		{
+			intset->iter_node = intset->iter_node->next;
+			intset->iter_itemno = 0;
+			continue;
+		}
+
+		/*
+		 * We have reached the end of the B-tree.  But we might still have
+		 * some integers in the buffer of newly-added values.
+		 */
+		if (intset->iter_values == (const uint64 *) intset->iter_values_buf)
+		{
+			intset->iter_values = intset->buffered_values;
+			intset->iter_num_values = intset->num_buffered_values;
+			intset->iter_valueno = 0;
+			continue;
+		}
+
+		break;
+	}
+
+	/* No more results. */
+	intset->iter_active = false;
+	*next = 0;					/* prevent uninitialized-variable warnings */
+	return false;
+}
+
+/*
+ * intset_binsrch_uint64() -- search a sorted array of uint64s
+ *
+ * Returns the first position with key equal or less than the given key.
+ * The returned position would be the "insert" location for the given key,
+ * that is, the position where the new key should be inserted to.
+ *
+ * 'nextkey' affects the behavior on equal keys.  If true, and there is an
+ * equal key in the array, this returns the position immediately after the
+ * equal key.  If false, this returns the position of the equal key itself.
+ */
+static int
+intset_binsrch_uint64(uint64 item, uint64 *arr, int arr_elems, bool nextkey)
+{
+	int			low,
+				high,
+				mid;
+
+	low = 0;
+	high = arr_elems;
+	while (high > low)
+	{
+		mid = low + (high - low) / 2;
+
+		if (nextkey)
+		{
+			if (item >= arr[mid])
+				low = mid + 1;
+			else
+				high = mid;
+		}
+		else
+		{
+			if (item > arr[mid])
+				low = mid + 1;
+			else
+				high = mid;
+		}
+	}
+
+	return low;
+}
+
+/* same, but for an array of leaf items */
+static int
+intset_binsrch_leaf(uint64 item, leaf_item *arr, int arr_elems, bool nextkey)
+{
+	int			low,
+				high,
+				mid;
+
+	low = 0;
+	high = arr_elems;
+	while (high > low)
+	{
+		mid = low + (high - low) / 2;
+
+		if (nextkey)
+		{
+			if (item >= arr[mid].first)
+				low = mid + 1;
+			else
+				high = mid;
+		}
+		else
+		{
+			if (item > arr[mid].first)
+				low = mid + 1;
+			else
+				high = mid;
+		}
+	}
+
+	return low;
+}
+
+/*
+ * Simple-8b encoding.
+ *
+ * The simple-8b algorithm packs between 1 and 240 integers into 64-bit words,
+ * called "codewords".  The number of integers packed into a single codeword
+ * depends on the integers being packed; small integers are encoded using
+ * fewer bits than large integers.  A single codeword can store a single
+ * 60-bit integer, or two 30-bit integers, for example.
+ *
+ * Since we're storing a unique, sorted, set of integers, we actually encode
+ * the *differences* between consecutive integers.  That way, clusters of
+ * integers that are close to each other are packed efficiently, regardless
+ * of their absolute values.
+ *
+ * In Simple-8b, each codeword consists of a 4-bit selector, which indicates
+ * how many integers are encoded in the codeword, and the encoded integers are
+ * packed into the remaining 60 bits.  The selector allows for 16 different
+ * ways of using the remaining 60 bits, called "modes".  The number of integers
+ * packed into a single codeword in each mode is listed in the simple8b_modes
+ * table below.  For example, consider the following codeword:
+ *
+ *      20-bit integer       20-bit integer       20-bit integer
+ * 1101 00000000000000010010 01111010000100100000 00000000000000010100
+ * ^
+ * selector
+ *
+ * The selector 1101 is 13 in decimal.  From the modes table below, we see
+ * that it means that the codeword encodes three 20-bit integers.  In decimal,
+ * those integers are 18, 500000 and 20.  Because we encode deltas rather than
+ * absolute values, the actual values that they represent are 18, 500018 and
+ * 500038.
+ *
+ * Modes 0 and 1 are a bit special; they encode a run of 240 or 120 zeroes
+ * (which means 240 or 120 consecutive integers, since we're encoding the
+ * deltas between integers), without using the rest of the codeword bits
+ * for anything.
+ *
+ * Simple-8b cannot encode integers larger than 60 bits.  Values larger than
+ * that are always stored in the 'first' field of a leaf item, never in the
+ * packed codeword.  If there is a sequence of integers that are more than
+ * 2^60 apart, the codeword will go unused on those items.  To represent that,
+ * we use a magic EMPTY_CODEWORD codeword value.
+ */
+static const struct simple8b_mode
+{
+	uint8		bits_per_int;
+	uint8		num_ints;
+}			simple8b_modes[17] =
+
+{
+	{0, 240},					/* mode  0: 240 zeroes */
+	{0, 120},					/* mode  1: 120 zeroes */
+	{1, 60},					/* mode  2: sixty 1-bit integers */
+	{2, 30},					/* mode  3: thirty 2-bit integers */
+	{3, 20},					/* mode  4: twenty 3-bit integers */
+	{4, 15},					/* mode  5: fifteen 4-bit integers */
+	{5, 12},					/* mode  6: twelve 5-bit integers */
+	{6, 10},					/* mode  7: ten 6-bit integers */
+	{7, 8},						/* mode  8: eight 7-bit integers (four bits
+								 * are wasted) */
+	{8, 7},						/* mode  9: seven 8-bit integers (four bits
+								 * are wasted) */
+	{10, 6},					/* mode 10: six 10-bit integers */
+	{12, 5},					/* mode 11: five 12-bit integers */
+	{15, 4},					/* mode 12: four 15-bit integers */
+	{20, 3},					/* mode 13: three 20-bit integers */
+	{30, 2},					/* mode 14: two 30-bit integers */
+	{60, 1},					/* mode 15: one 60-bit integer */
+
+	{0, 0}						/* sentinel value */
+};
+
+/*
+ * EMPTY_CODEWORD is a special value, used to indicate "no values".
+ * It is used if the next value is too large to be encoded with Simple-8b.
+ *
+ * This value looks like a mode-0 codeword, but we can distinguish it
+ * because a regular mode-0 codeword would have zeroes in the unused bits.
+ */
+#define EMPTY_CODEWORD		UINT64CONST(0x0FFFFFFFFFFFFFFF)
+
+/*
+ * Encode a number of integers into a Simple-8b codeword.
+ *
+ * (What we actually encode are deltas between successive integers.
+ * "base" is the value before ints[0].)
+ *
+ * The input array must contain at least SIMPLE8B_MAX_VALUES_PER_CODEWORD
+ * elements, ensuring that we can produce a full codeword.
+ *
+ * Returns the encoded codeword, and sets *num_encoded to the number of
+ * input integers that were encoded.  That can be zero, if the first delta
+ * is too large to be encoded.
+ */
+static uint64
+simple8b_encode(const uint64 *ints, int *num_encoded, uint64 base)
+{
+	int			selector;
+	int			nints;
+	int			bits;
+	uint64		diff;
+	uint64		last_val;
+	uint64		codeword;
+	int			i;
+
+	Assert(ints[0] > base);
+
+	/*
+	 * Select the "mode" to use for this codeword.
+	 *
+	 * In each iteration, check if the next value can be represented in the
+	 * current mode we're considering.  If it's too large, then step up the
+	 * mode to a wider one, and repeat.  If it fits, move on to the next
+	 * integer.  Repeat until the codeword is full, given the current mode.
+	 *
+	 * Note that we don't have any way to represent unused slots in the
+	 * codeword, so we require each codeword to be "full".  It is always
+	 * possible to produce a full codeword unless the very first delta is too
+	 * large to be encoded.  For example, if the first delta is small but the
+	 * second is too large to be encoded, we'll end up using the last "mode",
+	 * which has nints == 1.
+	 */
+	selector = 0;
+	nints = simple8b_modes[0].num_ints;
+	bits = simple8b_modes[0].bits_per_int;
+	diff = ints[0] - base - 1;
+	last_val = ints[0];
+	i = 0;						/* number of deltas we have accepted */
+	for (;;)
+	{
+		if (diff >= (UINT64CONST(1) << bits))
+		{
+			/* too large, step up to next mode */
+			selector++;
+			nints = simple8b_modes[selector].num_ints;
+			bits = simple8b_modes[selector].bits_per_int;
+			/* we might already have accepted enough deltas for this mode */
+			if (i >= nints)
+				break;
+		}
+		else
+		{
+			/* accept this delta; then done if codeword is full */
+			i++;
+			if (i >= nints)
+				break;
+			/* examine next delta */
+			Assert(ints[i] > last_val);
+			diff = ints[i] - last_val - 1;
+			last_val = ints[i];
+		}
+	}
+
+	if (nints == 0)
+	{
+		/*
+		 * The first delta is too large to be encoded with Simple-8b.
+		 *
+		 * If there is at least one not-too-large integer in the input, we
+		 * will encode it using mode 15 (or a more compact mode).  Hence, we
+		 * can only get here if the *first* delta is >= 2^60.
+		 */
+		Assert(i == 0);
+		*num_encoded = 0;
+		return EMPTY_CODEWORD;
+	}
+
+	/*
+	 * Encode the integers using the selected mode.  Note that we shift them
+	 * into the codeword in reverse order, so that they will come out in the
+	 * correct order in the decoder.
+	 */
+	codeword = 0;
+	if (bits > 0)
+	{
+		for (i = nints - 1; i > 0; i--)
+		{
+			diff = ints[i] - ints[i - 1] - 1;
+			codeword |= diff;
+			codeword <<= bits;
+		}
+		diff = ints[0] - base - 1;
+		codeword |= diff;
+	}
+
+	/* add selector to the codeword, and return */
+	codeword |= (uint64) selector << 60;
+
+	*num_encoded = nints;
+	return codeword;
+}
+
+/*
+ * Decode a codeword into an array of integers.
+ * Returns the number of integers decoded.
+ */
+static int
+simple8b_decode(uint64 codeword, uint64 *decoded, uint64 base)
+{
+	int			selector = (codeword >> 60);
+	int			nints = simple8b_modes[selector].num_ints;
+	int			bits = simple8b_modes[selector].bits_per_int;
+	uint64		mask = (UINT64CONST(1) << bits) - 1;
+	uint64		curr_value;
+
+	if (codeword == EMPTY_CODEWORD)
+		return 0;
+
+	curr_value = base;
+	for (int i = 0; i < nints; i++)
+	{
+		uint64		diff = codeword & mask;
+
+		curr_value += 1 + diff;
+		decoded[i] = curr_value;
+		codeword >>= bits;
+	}
+	return nints;
+}
+
+/*
+ * This is very similar to simple8b_decode(), but instead of decoding all
+ * the values to an array, it just checks if the given "key" is part of
+ * the codeword.
+ */
+static bool
+simple8b_contains(uint64 codeword, uint64 key, uint64 base)
+{
+	int			selector = (codeword >> 60);
+	int			nints = simple8b_modes[selector].num_ints;
+	int			bits = simple8b_modes[selector].bits_per_int;
+
+	if (codeword == EMPTY_CODEWORD)
+		return false;
+
+	if (bits == 0)
+	{
+		/* Special handling for 0-bit cases. */
+		return (key - base) <= nints;
+	}
+	else
+	{
+		uint64		mask = (UINT64CONST(1) << bits) - 1;
+		uint64		curr_value;
+
+		curr_value = base;
+		for (int i = 0; i < nints; i++)
+		{
+			uint64		diff = codeword & mask;
+
+			curr_value += 1 + diff;
+
+			if (curr_value >= key)
+			{
+				if (curr_value == key)
+					return true;
+				else
+					return false;
+			}
+
+			codeword >>= bits;
+		}
+	}
+	return false;
+}