Use a pairing heap for the priority queue in kNN-GiST searches.

This performs slightly better, uses less memory, and needs slightly less
code in GiST, than the Red-Black tree previously used.

Reviewed by Peter Geoghegan

src/backend/lib/pairingheap.c

@@ -0,0 +1,274 @@
+/*-------------------------------------------------------------------------
+ *
+ * pairingheap.c
+ *	  A Pairing Heap implementation
+ *
+ * A pairing heap is a data structure that's useful for implementing
+ * priority queues. It is simple to implement, and provides amortized O(1)
+ * insert and find-min operations, and amortized O(log n) delete-min.
+ *
+ * The pairing heap was first described in this paper:
+ *
+ *	Michael L. Fredman, Robert Sedgewick, Daniel D. Sleator, and Robert E.
+ *	 Tarjan. 1986.
+ *	The pairing heap: a new form of self-adjusting heap.
+ *	Algorithmica 1, 1 (January 1986), pages 111-129. DOI: 10.1007/BF01840439
+ *
+ * Portions Copyright (c) 2012-2014, PostgreSQL Global Development Group
+ *
+ * IDENTIFICATION
+ *	  src/backend/lib/pairingheap.c
+ *
+ *-------------------------------------------------------------------------
+ */
+
+#include "postgres.h"
+
+#include "lib/pairingheap.h"
+
+static pairingheap_node *merge(pairingheap *heap, pairingheap_node *a,
+	  pairingheap_node *b);
+static pairingheap_node *merge_children(pairingheap *heap,
+			   pairingheap_node *children);
+
+/*
+ * pairingheap_allocate
+ *
+ * Returns a pointer to a newly-allocated heap, with the heap property defined
+ * by the given comparator function, which will be invoked with the additional
+ * argument specified by 'arg'.
+ */
+pairingheap *
+pairingheap_allocate(pairingheap_comparator compare, void *arg)
+{
+	pairingheap *heap;
+
+	heap = (pairingheap *) palloc(sizeof(pairingheap));
+	heap->ph_compare = compare;
+	heap->ph_arg = arg;
+
+	heap->ph_root = NULL;
+
+	return heap;
+}
+
+/*
+ * pairingheap_free
+ *
+ * Releases memory used by the given pairingheap.
+ *
+ * Note: The nodes in the heap are not freed!
+ */
+void
+pairingheap_free(pairingheap *heap)
+{
+	pfree(heap);
+}
+
+/*
+ * A helper function to merge two subheaps into one.
+ *
+ * The subheap with smaller value is put as a child of the other one (assuming
+ * a max-heap).
+ */
+static pairingheap_node *
+merge(pairingheap *heap, pairingheap_node *a, pairingheap_node *b)
+{
+	if (a == NULL)
+		return b;
+	if (b == NULL)
+		return a;
+
+	/* swap 'a' and 'b' so that 'a' is the one with larger value */
+	if (heap->ph_compare(a, b, heap->ph_arg) < 0)
+	{
+		pairingheap_node *tmp;
+
+		tmp = a;
+		a = b;
+		b = tmp;
+	}
+
+	/* and put 'b' as a child of 'a' */
+	if (a->first_child)
+		a->first_child->prev_or_parent = b;
+	b->prev_or_parent = a;
+	b->next_sibling = a->first_child;
+	a->first_child = b;
+
+	return a;
+}
+
+/*
+ * pairingheap_add
+ *
+ * Adds the given node to the heap in O(1) time.
+ */
+void
+pairingheap_add(pairingheap *heap, pairingheap_node *node)
+{
+	node->first_child = NULL;
+
+	/* Link the new node as a new tree */
+	heap->ph_root = merge(heap, heap->ph_root, node);
+}
+
+/*
+ * pairingheap_first
+ *
+ * Returns a pointer to the first (root, topmost) node in the heap without
+ * modifying the heap. The caller must ensure that this routine is not used on
+ * an empty heap. Always O(1).
+ */
+pairingheap_node *
+pairingheap_first(pairingheap *heap)
+{
+	Assert(!pairingheap_is_empty(heap));
+
+	return heap->ph_root;
+}
+
+/*
+ * pairingheap_remove_first
+ *
+ * Removes the first (root, topmost) node in the heap and returns a pointer to
+ * it after rebalancing the heap. The caller must ensure that this routine is
+ * not used on an empty heap. O(log n) amortized.
+ */
+pairingheap_node *
+pairingheap_remove_first(pairingheap *heap)
+{
+	pairingheap_node *result;
+	pairingheap_node *children;
+
+	Assert(!pairingheap_is_empty(heap));
+
+	/* Remove the root, and form a new heap of its children. */
+	result = heap->ph_root;
+	children = result->first_child;
+
+	heap->ph_root = merge_children(heap, children);
+
+	return result;
+}
+
+/*
+ * Remove 'node' from the heap. O(log n) amortized.
+ */
+void
+pairingheap_remove(pairingheap *heap, pairingheap_node *node)
+{
+	pairingheap_node *children;
+	pairingheap_node *replacement;
+	pairingheap_node *next_sibling;
+	pairingheap_node **prev_ptr;
+
+	/*
+	 * If the removed node happens to be the root node, do it with
+	 * pairingheap_remove_first().
+	 */
+	if (node == heap->ph_root)
+	{
+		(void) pairingheap_remove_first(heap);
+		return;
+	}
+
+	/*
+	 * Before we modify anything, remember the removed node's first_child and
+	 * next_sibling pointers.
+	 */
+	children = node->first_child;
+	next_sibling = node->next_sibling;
+
+	/*
+	 * Also find the pointer to the removed node in its previous sibling, or
+	 * if this is the first child of its parent, in its parent.
+	 */
+	if (node->prev_or_parent->first_child == node)
+		prev_ptr = &node->prev_or_parent->first_child;
+	else
+		prev_ptr = &node->prev_or_parent->next_sibling;
+	Assert(*prev_ptr == node);
+
+	/*
+	 * If this node has children, make a new subheap of the children and link
+	 * the subheap in place of the removed node. Otherwise just unlink this
+	 * node.
+	 */
+	if (children)
+	{
+		replacement = merge_children(heap, children);
+
+		replacement->prev_or_parent = node->prev_or_parent;
+		replacement->next_sibling = node->next_sibling;
+		*prev_ptr = replacement;
+		if (next_sibling)
+			next_sibling->prev_or_parent = replacement;
+	}
+	else
+	{
+		*prev_ptr = next_sibling;
+		if (next_sibling)
+			next_sibling->prev_or_parent = node->prev_or_parent;
+	}
+}
+
+/*
+ * Merge a list of subheaps into a single heap.
+ *
+ * This implements the basic two-pass merging strategy, first forming pairs
+ * from left to right, and then merging the pairs.
+ */
+static pairingheap_node *
+merge_children(pairingheap *heap, pairingheap_node *children)
+{
+	pairingheap_node *curr,
+			   *next;
+	pairingheap_node *pairs;
+	pairingheap_node *newroot;
+
+	if (children == NULL || children->next_sibling == NULL)
+		return children;
+
+	/* Walk the subheaps from left to right, merging in pairs */
+	next = children;
+	pairs = NULL;
+	for (;;)
+	{
+		curr = next;
+
+		if (curr == NULL)
+			break;
+
+		if (curr->next_sibling == NULL)
+		{
+			/* last odd node at the end of list */
+			curr->next_sibling = pairs;
+			pairs = curr;
+			break;
+		}
+
+		next = curr->next_sibling->next_sibling;
+
+		/* merge this and the next subheap, and add to 'pairs' list. */
+
+		curr = merge(heap, curr, curr->next_sibling);
+		curr->next_sibling = pairs;
+		pairs = curr;
+	}
+
+	/*
+	 * Merge all the pairs together to form a single heap.
+	 */
+	newroot = pairs;
+	next = pairs->next_sibling;
+	while (next)
+	{
+		curr = next;
+		next = curr->next_sibling;
+
+		newroot = merge(heap, newroot, curr);
+	}
+
+	return newroot;
+}