// Copyright 2013 Google Inc. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//
// A btree implementation of the STL set and map interfaces. A btree is both
// smaller and faster than STL set/map. The red-black tree implementation of
// STL set/map has an overhead of 3 pointers (left, right and parent) plus the
// node color information for each stored value. So a set<int32> consumes 20
// bytes for each value stored. This btree implementation stores multiple
// values on fixed size nodes (usually 256 bytes) and doesn't store child
// pointers for leaf nodes. The result is that a btree_set<int32> may use much
// less memory per stored value. For the random insertion benchmark in
// btree_test.cc, a btree_set<int32> with node-size of 256 uses 4.9 bytes per
// stored value.
//
// The packing of multiple values on to each node of a btree has another effect
// besides better space utilization: better cache locality due to fewer cache
// lines being accessed. Better cache locality translates into faster
// operations.
//
// CAVEATS
//
// Insertions and deletions on a btree can cause splitting, merging or
// rebalancing of btree nodes. And even without these operations, insertions
// and deletions on a btree will move values around within a node. In both
// cases, the result is that insertions and deletions can invalidate iterators
// pointing to values other than the one being inserted/deleted. This is
// notably different from STL set/map which takes care to not invalidate
// iterators on insert/erase except, of course, for iterators pointing to the
// value being erased.  A partial workaround when erasing is available:
// erase() returns an iterator pointing to the item just after the one that was
// erased (or end() if none exists).  See also safe_btree.

// PERFORMANCE
//
//   btree_bench --benchmarks=. 2>&1 | ./benchmarks.awk
//
// Run on pmattis-warp.nyc (4 X 2200 MHz CPUs); 2010/03/04-15:23:06
// Benchmark                 STL(ns) B-Tree(ns) @    <size>
// --------------------------------------------------------
// BM_set_int32_insert        1516      608  +59.89%  <256>    [40.0,  5.2]
// BM_set_int32_lookup        1160      414  +64.31%  <256>    [40.0,  5.2]
// BM_set_int32_fulllookup     960      410  +57.29%  <256>    [40.0,  4.4]
// BM_set_int32_delete        1741      528  +69.67%  <256>    [40.0,  5.2]
// BM_set_int32_queueaddrem   3078     1046  +66.02%  <256>    [40.0,  5.5]
// BM_set_int32_mixedaddrem   3600     1384  +61.56%  <256>    [40.0,  5.3]
// BM_set_int32_fifo           227      113  +50.22%  <256>    [40.0,  4.4]
// BM_set_int32_fwditer        158       26  +83.54%  <256>    [40.0,  5.2]
// BM_map_int32_insert        1551      636  +58.99%  <256>    [48.0, 10.5]
// BM_map_int32_lookup        1200      508  +57.67%  <256>    [48.0, 10.5]
// BM_map_int32_fulllookup     989      487  +50.76%  <256>    [48.0,  8.8]
// BM_map_int32_delete        1794      628  +64.99%  <256>    [48.0, 10.5]
// BM_map_int32_queueaddrem   3189     1266  +60.30%  <256>    [48.0, 11.6]
// BM_map_int32_mixedaddrem   3822     1623  +57.54%  <256>    [48.0, 10.9]
// BM_map_int32_fifo           151      134  +11.26%  <256>    [48.0,  8.8]
// BM_map_int32_fwditer        161       32  +80.12%  <256>    [48.0, 10.5]
// BM_set_int64_insert        1546      636  +58.86%  <256>    [40.0, 10.5]
// BM_set_int64_lookup        1200      512  +57.33%  <256>    [40.0, 10.5]
// BM_set_int64_fulllookup     971      487  +49.85%  <256>    [40.0,  8.8]
// BM_set_int64_delete        1745      616  +64.70%  <256>    [40.0, 10.5]
// BM_set_int64_queueaddrem   3163     1195  +62.22%  <256>    [40.0, 11.6]
// BM_set_int64_mixedaddrem   3760     1564  +58.40%  <256>    [40.0, 10.9]
// BM_set_int64_fifo           146      103  +29.45%  <256>    [40.0,  8.8]
// BM_set_int64_fwditer        162       31  +80.86%  <256>    [40.0, 10.5]
// BM_map_int64_insert        1551      720  +53.58%  <256>    [48.0, 20.7]
// BM_map_int64_lookup        1214      612  +49.59%  <256>    [48.0, 20.7]
// BM_map_int64_fulllookup     994      592  +40.44%  <256>    [48.0, 17.2]
// BM_map_int64_delete        1778      764  +57.03%  <256>    [48.0, 20.7]
// BM_map_int64_queueaddrem   3189     1547  +51.49%  <256>    [48.0, 20.9]
// BM_map_int64_mixedaddrem   3779     1887  +50.07%  <256>    [48.0, 21.6]
// BM_map_int64_fifo           147      145   +1.36%  <256>    [48.0, 17.2]
// BM_map_int64_fwditer        162       41  +74.69%  <256>    [48.0, 20.7]
// BM_set_string_insert       1989     1966   +1.16%  <256>    [64.0, 44.5]
// BM_set_string_lookup       1709     1600   +6.38%  <256>    [64.0, 44.5]
// BM_set_string_fulllookup   1573     1529   +2.80%  <256>    [64.0, 35.4]
// BM_set_string_delete       2520     1920  +23.81%  <256>    [64.0, 44.5]
// BM_set_string_queueaddrem  4706     4309   +8.44%  <256>    [64.0, 48.3]
// BM_set_string_mixedaddrem  5080     4654   +8.39%  <256>    [64.0, 46.7]
// BM_set_string_fifo          318      512  -61.01%  <256>    [64.0, 35.4]
// BM_set_string_fwditer       182       93  +48.90%  <256>    [64.0, 44.5]
// BM_map_string_insert       2600     2227  +14.35%  <256>    [72.0, 55.8]
// BM_map_string_lookup       2068     1730  +16.34%  <256>    [72.0, 55.8]
// BM_map_string_fulllookup   1859     1618  +12.96%  <256>    [72.0, 44.0]
// BM_map_string_delete       3168     2080  +34.34%  <256>    [72.0, 55.8]
// BM_map_string_queueaddrem  5840     4701  +19.50%  <256>    [72.0, 59.4]
// BM_map_string_mixedaddrem  6400     5200  +18.75%  <256>    [72.0, 57.8]
// BM_map_string_fifo          398      596  -49.75%  <256>    [72.0, 44.0]
// BM_map_string_fwditer       243      113  +53.50%  <256>    [72.0, 55.8]

#ifndef UTIL_BTREE_BTREE_H__
#define UTIL_BTREE_BTREE_H__

#ifndef _MSC_VER
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Warray-bounds"
#endif

#include <assert.h>
#include <stddef.h>
#include <string.h>
#include <sys/types.h>
#include <algorithm>
#include <functional>
#include <iostream>
#include <iterator>
#include <limits>
#include <new>
#include <ostream>
#include <string>
#include <type_traits>
#include <utility>
#include "tools/assertrx.h"

#ifndef NDEBUG_BTREE
#define NDEBUG_BTREE 1
#endif

namespace btree {

#if defined(_MSC_VER)
typedef intptr_t ssize_t;
#endif

// Inside a btree method, if we just call swap(), it will choose the
// btree::swap method, which we don't want. And we can't say ::swap
// because then MSVC won't pickup any std::swap() implementations. We
// can't just use std::swap() directly because then we don't get the
// specialization for types outside the std namespace. So the solution
// is to have a special swap helper function whose name doesn't
// collide with other swap functions defined by the btree classes.
template <typename T>
inline void btree_swap_helper(T &a, T &b) {
	using std::swap;
	swap(a, b);
}

// A template helper used to select A or B based on a condition.
template <bool cond, typename A, typename B>
struct if_ {
	typedef A type;
};

template <typename A, typename B>
struct if_<false, A, B> {
	typedef B type;
};

// Types small_ and big_ are promise that sizeof(small_) < sizeof(big_)
typedef char small_;

struct big_ {
	char dummy[2];
};

// A compile-time assertion.
template <bool>
struct CompileAssert {};

#define COMPILE_ASSERT(expr, msg) typedef CompileAssert<(bool(expr))> msg[bool(expr) ? 1 : -1]

// A helper type used to indicate that a key-compare-to functor has been
// provided. A user can specify a key-compare-to functor by doing:
//
//  struct MyStringComparer
//      : public util::btree::btree_key_compare_to_tag {
//    int operator()(const string &a, const string &b) const {
//      return a.compare(b);
//    }
//  };
//
// Note that the return type is an int and not a bool. There is a
// COMPILE_ASSERT which enforces this return type.
struct btree_key_compare_to_tag {};

// A helper class that indicates if the Compare parameter is derived from
// btree_key_compare_to_tag.
template <typename Compare>
struct btree_is_key_compare_to : public std::is_convertible<Compare, btree_key_compare_to_tag> {};

// A helper class to convert a boolean comparison into a three-way
// "compare-to" comparison that returns a negative value to indicate
// less-than, zero to indicate equality and a positive value to
// indicate greater-than. This helper class is specialized for
// less<string> and greater<string>. The btree_key_compare_to_adapter
// class is provided so that btree users automatically get the more
// efficient compare-to code when using common google string types
// with common comparison functors.
template <typename Compare>
struct btree_key_compare_to_adapter : Compare {
	btree_key_compare_to_adapter() {}
	btree_key_compare_to_adapter(const Compare &c) : Compare(c) {}
	btree_key_compare_to_adapter(const btree_key_compare_to_adapter<Compare> &c) : Compare(c) {}
	btree_key_compare_to_adapter<Compare> &operator=(const btree_key_compare_to_adapter<Compare> &c) {
		Compare::operator=(c);
		return *this;
	}
};

template <>
struct btree_key_compare_to_adapter<std::less<std::string> > : public btree_key_compare_to_tag {
	btree_key_compare_to_adapter() {}
	btree_key_compare_to_adapter(const std::less<std::string> &) {}
	btree_key_compare_to_adapter(const btree_key_compare_to_adapter<std::less<std::string> > &) {}
	int operator()(const std::string &a, const std::string &b) const { return a.compare(b); }
};

template <>
struct btree_key_compare_to_adapter<std::greater<std::string> > : public btree_key_compare_to_tag {
	btree_key_compare_to_adapter() {}
	btree_key_compare_to_adapter(const std::greater<std::string> &) {}
	btree_key_compare_to_adapter(const btree_key_compare_to_adapter<std::greater<std::string> > &) {}
	int operator()(const std::string &a, const std::string &b) const { return b.compare(a); }
};

// A helper class that allows a compare-to functor to behave like a plain
// compare functor. This specialization is used when we do not have a
// compare-to functor.
template <typename Key, typename Compare, bool HaveCompareTo>
struct btree_key_comparer {
	btree_key_comparer() {}
	btree_key_comparer(Compare c) : comp(c) {}
	template <typename K>
	static bool bool_compare(const Compare &comp, const Key &x, const K &y) {
		return comp(x, y);
	}
	template <typename K>
	bool operator()(const Key &x, const K &y) const {
		return bool_compare(comp, x, y);
	}
	Compare comp;
};

// A specialization of btree_key_comparer when a compare-to functor is
// present. We need a plain (boolean) comparison in some parts of the btree
// code, such as insert-with-hint.
template <typename Key, typename Compare>
struct btree_key_comparer<Key, Compare, true> {
	btree_key_comparer() {}
	btree_key_comparer(Compare c) : comp(c) {}
	static bool bool_compare(const Compare &comp, const Key &x, const Key &y) { return comp(x, y) < 0; }
	bool operator()(const Key &x, const Key &y) const { return bool_compare(comp, x, y); }
	Compare comp;
};

// A helper function to compare to keys using the specified compare
// functor. This dispatches to the appropriate btree_key_comparer comparison,
// depending on whether we have a compare-to functor or not (which depends on
// whether Compare is derived from btree_key_compare_to_tag).
template <typename Key, typename K2, typename Compare>
static bool btree_compare_keys(const Compare &comp, const Key &x, const K2 &y) {
	typedef btree_key_comparer<Key, Compare, btree_is_key_compare_to<Compare>::value> key_comparer;
	return key_comparer::bool_compare(comp, x, y);
}

template <typename Key, typename Compare, typename Alloc, int TargetNodeSize, int ValueSize>
struct btree_common_params {
	// If Compare is derived from btree_key_compare_to_tag then use it as the
	// key_compare type. Otherwise, use btree_key_compare_to_adapter<> which will
	// fall-back to Compare if we don't have an appropriate specialization.
	typedef typename if_<btree_is_key_compare_to<Compare>::value, Compare, btree_key_compare_to_adapter<Compare> >::type key_compare;
	// A type which indicates if we have a key-compare-to functor or a plain old
	// key-compare functor.
	typedef btree_is_key_compare_to<key_compare> is_key_compare_to;

	typedef Alloc allocator_type;
	typedef Key key_type;
	typedef ssize_t size_type;
	typedef ptrdiff_t difference_type;

	enum {
		kTargetNodeSize = TargetNodeSize,

		// Available space for values.  This is largest for leaf nodes,
		// which has overhead no fewer than two pointers.
		kNodeValueSpace = TargetNodeSize - 2 * sizeof(void *),
	};

	// This is an integral type large enough to hold as many
	// ValueSize-values as will fit a node of TargetNodeSize bytes.
	typedef typename if_<(kNodeValueSpace / ValueSize) >= 256, uint16_t, uint8_t>::type node_count_type;
};

// A parameters structure for holding the type parameters for a btree_map.
template <typename Key, typename Data, typename Compare, typename Alloc, int TargetNodeSize>
struct btree_map_params : public btree_common_params<Key, Compare, Alloc, TargetNodeSize, sizeof(Key) + sizeof(Data)> {
	typedef Data data_type;
	typedef Data mapped_type;
	typedef std::pair<const Key, data_type> value_type;
	typedef std::pair<Key, data_type> mutable_value_type;
	typedef value_type *pointer;
	typedef const value_type *const_pointer;
	typedef value_type &reference;
	typedef const value_type &const_reference;

	enum {
		kValueSize = sizeof(Key) + sizeof(data_type),
	};

	static const Key &key(const value_type &x) { return x.first; }
	static const Key &key(const mutable_value_type &x) { return x.first; }
	static void swap(mutable_value_type *a, mutable_value_type *b) {
		btree_swap_helper(a->first, b->first);
		btree_swap_helper(a->second, b->second);
	}
};

// A parameters structure for holding the type parameters for a btree_set.
template <typename Key, typename Compare, typename Alloc, int TargetNodeSize>
struct btree_set_params : public btree_common_params<Key, Compare, Alloc, TargetNodeSize, sizeof(Key)> {
	typedef std::false_type data_type;
	typedef std::false_type mapped_type;
	typedef Key value_type;
	typedef value_type mutable_value_type;
	typedef value_type *pointer;
	typedef const value_type *const_pointer;
	typedef value_type &reference;
	typedef const value_type &const_reference;

	enum {
		kValueSize = sizeof(Key),
	};

	static const Key &key(const value_type &x) { return x; }
	static void swap(mutable_value_type *a, mutable_value_type *b) { btree_swap_helper<mutable_value_type>(*a, *b); }
};

// An adapter class that converts a lower-bound compare into an upper-bound
// compare.
template <typename Compare>
struct btree_upper_bound_adapter : public Compare {
	btree_upper_bound_adapter(Compare c) : Compare(c) {}
	template <typename K1, typename K2>
	bool operator()(const K1 &a, const K2 &b) const {
		return !static_cast<const Compare &>(*this)(b, a);
	}
};

template <typename CompareTo>
struct btree_upper_bound_compare_to_adapter : public CompareTo {
	btree_upper_bound_compare_to_adapter(CompareTo c) : CompareTo(c) {}
	template <typename K1, typename K2>
	int operator()(const K1 &a, const K2 &b) const {
		return static_cast<const CompareTo &>(*this)(b, a);
	}
};

// Dispatch helper class for using linear search with plain compare.
template <typename N, typename Compare>
struct btree_linear_search_plain_compare {
	template <typename K>
	static int lower_bound(const K &k, const N &n, Compare comp) {
		return n.linear_search_plain_compare(k, 0, n.count(), comp);
	}
	template <typename K>
	static int upper_bound(const K &k, const N &n, Compare comp) {
		typedef btree_upper_bound_adapter<Compare> upper_compare;
		return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
	}
};

// Dispatch helper class for using linear search with compare-to
template <typename N, typename CompareTo>
struct btree_linear_search_compare_to {
	template <typename K>
	static int lower_bound(const K &k, const N &n, CompareTo comp) {
		return n.linear_search_compare_to(k, 0, n.count(), comp);
	}
	template <typename K>
	static int upper_bound(const K &k, const N &n, CompareTo comp) {
		typedef btree_upper_bound_adapter<btree_key_comparer<K, CompareTo, true> > upper_compare;
		return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
	}
};

// Dispatch helper class for using binary search with plain compare.
template <typename N, typename Compare>
struct btree_binary_search_plain_compare {
	template <typename K>
	static int lower_bound(const K &k, const N &n, Compare comp) {
		return n.binary_search_plain_compare(k, 0, n.count(), comp);
	}
	template <typename K>
	static int upper_bound(const K &k, const N &n, Compare comp) {
		typedef btree_upper_bound_adapter<Compare> upper_compare;
		return n.binary_search_plain_compare(k, 0, n.count(), upper_compare(comp));
	}
};

// Dispatch helper class for using binary search with compare-to.
template <typename N, typename CompareTo>
struct btree_binary_search_compare_to {
	template <typename K>
	static int lower_bound(const K &k, const N &n, CompareTo comp) {
		return n.binary_search_compare_to(k, 0, n.count(), comp);
	}
	template <typename K>
	static int upper_bound(const K &k, const N &n, CompareTo comp) {
		typedef btree_upper_bound_adapter<btree_key_comparer<K, CompareTo, true> > upper_compare;
		return n.linear_search_plain_compare(k, 0, n.count(), upper_compare(comp));
	}
};

// A node in the btree holding. The same node type is used for both internal
// and leaf nodes in the btree, though the nodes are allocated in such a way
// that the children array is only valid in internal nodes.
template <typename Params>
class btree_node {
public:
	typedef Params params_type;
	typedef btree_node<Params> self_type;
	typedef typename Params::key_type key_type;
	typedef typename Params::data_type data_type;
	typedef typename Params::value_type value_type;
	typedef typename Params::mutable_value_type mutable_value_type;
	typedef typename Params::pointer pointer;
	typedef typename Params::const_pointer const_pointer;
	typedef typename Params::reference reference;
	typedef typename Params::const_reference const_reference;
	typedef typename Params::key_compare key_compare;
	typedef typename Params::size_type size_type;
	typedef typename Params::difference_type difference_type;
	// Typedefs for the various types of node searches.
	typedef btree_linear_search_plain_compare<self_type, key_compare> linear_search_plain_compare_type;
	typedef btree_linear_search_compare_to<self_type, key_compare> linear_search_compare_to_type;
	typedef btree_binary_search_plain_compare<self_type, key_compare> binary_search_plain_compare_type;
	typedef btree_binary_search_compare_to<self_type, key_compare> binary_search_compare_to_type;
	// If we have a valid key-compare-to type, use linear_search_compare_to,
	// otherwise use linear_search_plain_compare.
	typedef typename if_<Params::is_key_compare_to::value, linear_search_compare_to_type, linear_search_plain_compare_type>::type
		linear_search_type;
	// If we have a valid key-compare-to type, use binary_search_compare_to,
	// otherwise use binary_search_plain_compare.
	typedef typename if_<Params::is_key_compare_to::value, binary_search_compare_to_type, binary_search_plain_compare_type>::type
		binary_search_type;
	// If the key is an integral or floating point type, use linear search which
	// is faster than binary search for such types. Might be wise to also
	// configure linear search based on node-size.
	typedef typename if_</*std::is_integral<key_type>::value ||*/ std::is_floating_point<key_type>::value, linear_search_type,
						 binary_search_type>::type search_type;

	struct base_fields {
		typedef typename Params::node_count_type field_type;

		// A boolean indicating whether the node is a leaf or not.
		bool leaf;
		// The position of the node in the node's parent.
		field_type position;
		// The maximum number of values the node can hold.
		field_type max_count;
		// The count of the number of values in the node.
		field_type count;
		// A pointer to the node's parent.
		btree_node *parent;
	};

	enum {
		kValueSize = params_type::kValueSize,
		kTargetNodeSize = params_type::kTargetNodeSize,

		// Compute how many values we can fit onto a leaf node.
		kNodeTargetValues = (kTargetNodeSize - sizeof(base_fields)) / kValueSize,
		// We need a minimum of 3 values per internal node in order to perform
		// splitting (1 value for the two nodes involved in the split and 1 value
		// propagated to the parent as the delimiter for the split).
		kNodeValues = kNodeTargetValues >= 3 ? kNodeTargetValues : 3,

		kExactMatch = 1 << 30,
		kMatchMask = kExactMatch - 1,
	};

	struct leaf_fields : public base_fields {
		// The array of values. Only the first count of these values have been
		// constructed and are valid.
		mutable_value_type values[kNodeValues];
	};

	struct internal_fields : public leaf_fields {
		// The array of child pointers. The keys in children_[i] are all less than
		// key(i). The keys in children_[i + 1] are all greater than key(i). There
		// are always count + 1 children.
		btree_node *children[kNodeValues + 1];
	};

	struct root_fields : public internal_fields {
		btree_node *rightmost;
		size_type size;
	};

public:
	// Getter/setter for whether this is a leaf node or not. This value doesn't
	// change after the node is created.
	bool leaf() const { return fields_.leaf; }

	// Getter for the position of this node in its parent.
	int position() const { return fields_.position; }
	void set_position(int v) { fields_.position = v; }

	// Getter/setter for the number of values stored in this node.
	int count() const { return fields_.count; }
	void set_count(int v) { fields_.count = v; }
	int max_count() const { return fields_.max_count; }

	// Getter for the parent of this node.
	btree_node *parent() const { return fields_.parent; }
	// Getter for whether the node is the root of the tree. The parent of the
	// root of the tree is the leftmost node in the tree which is guaranteed to
	// be a leaf.
	bool is_root() const { return parent()->leaf(); }
	void make_root() {
		assertrx(parent()->is_root());
		fields_.parent = fields_.parent->parent();
	}

	// Getter for the rightmost root node field. Only valid on the root node.
	btree_node *rightmost() const { return fields_.rightmost; }
	btree_node **mutable_rightmost() { return &fields_.rightmost; }

	// Getter for the size root node field. Only valid on the root node.
	size_type size() const { return fields_.size; }
	size_type *mutable_size() { return &fields_.size; }

	// Getters for the key/value at position i in the node.
	const key_type &key(int i) const { return params_type::key(fields_.values[i]); }
	reference value(int i) { return reinterpret_cast<reference>(mutable_reference(i)); }
	const_reference value(int i) const { return reinterpret_cast<const_reference>(mutable_reference(i)); }
	mutable_value_type *mutable_value(int i) { return &fields_.values[i]; }

	// Swap value i in this node with value j in node x.
	void value_swap(int i, btree_node *x, int j) { params_type::swap(mutable_value(i), x->mutable_value(j)); }

	// Getters/setter for the child at position i in the node.
	btree_node *child(int i) const { return fields_.children[i]; }
	btree_node **mutable_child(int i) { return &fields_.children[i]; }
	void set_child(int i, btree_node *c) {
		*mutable_child(i) = c;
		c->fields_.parent = this;
		c->fields_.position = i;
	}

	// Returns the position of the first value whose key is not less than k.
	template <typename Compare, typename K>
	int lower_bound(const K &k, const Compare &comp) const {
		return search_type::lower_bound(k, *this, comp);
	}
	// Returns the position of the first value whose key is greater than k.
	template <typename Compare, typename K>
	int upper_bound(const K &k, const Compare &comp) const {
		return search_type::upper_bound(k, *this, comp);
	}

	// Returns the position of the first value whose key is not less than k using
	// linear search performed using plain compare.
	template <typename Compare, typename K>
	int linear_search_plain_compare(const K &k, int s, int e, const Compare &comp) const {
		while (s < e) {
			if (!btree_compare_keys(comp, key(s), k)) {
				break;
			}
			++s;
		}
		return s;
	}

	// Returns the position of the first value whose key is not less than k using
	// linear search performed using compare-to.
	template <typename Compare, typename K>
	int linear_search_compare_to(const K &k, int s, int e, const Compare &comp) const {
		while (s < e) {
			int c = comp(key(s), k);
			if (c == 0) {
				return s | kExactMatch;
			} else if (c > 0) {
				break;
			}
			++s;
		}
		return s;
	}

	// Returns the position of the first value whose key is not less than k using
	// binary search performed using plain compare.
	template <typename Compare, typename K>
	int binary_search_plain_compare(const K &k, int s, int e, const Compare &comp) const {
		while (s != e) {
			int mid = (s + e) / 2;
			if (btree_compare_keys(comp, key(mid), k)) {
				s = mid + 1;
			} else {
				e = mid;
			}
		}
		return s;
	}

	// Returns the position of the first value whose key is not less than k using
	// binary search performed using compare-to.
	template <typename CompareTo, typename K>
	int binary_search_compare_to(const K &k, int s, int e, const CompareTo &comp) const {
		while (s != e) {
			int mid = (s + e) / 2;
			int c = comp(key(mid), k);
			if (c < 0) {
				s = mid + 1;
			} else if (c > 0) {
				e = mid;
			} else {
				// Need to return the first value whose key is not less than k, which
				// requires continuing the binary search. Note that we are guaranteed
				// that the result is an exact match because if "key(mid-1) < k" the
				// call to binary_search_compare_to() will return "mid".
				s = binary_search_compare_to(k, s, mid, comp);
				return s | kExactMatch;
			}
		}
		return s;
	}

	// Inserts the value x at position i, shifting all existing values and
	// children at positions >= i to the right by 1.
	void insert_value(int i, const value_type &x);

	// Removes the value at position i, shifting all existing values and children
	// at positions > i to the left by 1.
	void remove_value(int i);

	// Rebalances a node with its right sibling.
	void rebalance_right_to_left(btree_node *sibling, int to_move);
	void rebalance_left_to_right(btree_node *sibling, int to_move);

	// Splits a node, moving a portion of the node's values to its right sibling.
	void split(btree_node *sibling, int insert_position);

	// Merges a node with its right sibling, moving all of the values and the
	// delimiting key in the parent node onto itself.
	void merge(btree_node *sibling);

	// Swap the contents of "this" and "src".
	void swap(btree_node *src);

	// Node allocation/deletion routines.
	static btree_node *init_leaf(leaf_fields *f, btree_node *parent, int max_count) {
		btree_node *n = reinterpret_cast<btree_node *>(f);
		f->leaf = 1;
		f->position = 0;
		f->max_count = max_count;
		f->count = 0;
		f->parent = parent;
		if (!NDEBUG_BTREE) {
			memset(&f->values, 0, max_count * sizeof(value_type));
		}
		return n;
	}
	static btree_node *init_internal(internal_fields *f, btree_node *parent) {
		btree_node *n = init_leaf(f, parent, kNodeValues);
		f->leaf = 0;
		if (!NDEBUG_BTREE) {
			memset(f->children, 0, sizeof(f->children));
		}
		return n;
	}
	static btree_node *init_root(root_fields *f, btree_node *parent) {
		btree_node *n = init_internal(f, parent);
		f->rightmost = parent;
		f->size = parent->count();
		return n;
	}
	void destroy() {
		for (int i = 0; i < count(); ++i) {
			value_destroy(i);
		}
	}

private:
	void value_init(int i) { new (&fields_.values[i]) mutable_value_type; }
	void value_init(int i, const value_type &x) { new (&fields_.values[i]) mutable_value_type(x); }
	void value_destroy(int i) { fields_.values[i].~mutable_value_type(); }
	mutable_value_type &mutable_reference(int i) { return fields_.values[i]; }
	const mutable_value_type &mutable_reference(int i) const { return fields_.values[i]; }

private:
	root_fields fields_;

private:
	btree_node(const btree_node &);
	void operator=(const btree_node &);
};

template <typename Node, typename Reference, typename Pointer>
struct btree_iterator {
	typedef typename Node::key_type key_type;
	typedef typename Node::size_type size_type;
	typedef typename Node::difference_type difference_type;
	typedef typename Node::params_type params_type;

	typedef Node node_type;
	typedef typename std::remove_const<Node>::type normal_node;
	typedef const Node const_node;
	typedef typename params_type::value_type value_type;
	typedef typename params_type::pointer normal_pointer;
	typedef typename params_type::reference normal_reference;
	typedef typename params_type::const_pointer const_pointer;
	typedef typename params_type::const_reference const_reference;

	typedef Pointer pointer;
	typedef Reference reference;
	typedef std::bidirectional_iterator_tag iterator_category;

	typedef btree_iterator<normal_node, normal_reference, normal_pointer> iterator;
	typedef btree_iterator<const_node, const_reference, const_pointer> const_iterator;
	typedef btree_iterator<Node, Reference, Pointer> self_type;

	btree_iterator() noexcept : node(NULL), position(-1) {}
	btree_iterator(Node *n, int p) noexcept : node(n), position(p) {}
	btree_iterator(const iterator &x) noexcept : node(x.node), position(x.position) {}
	btree_iterator &operator=(const iterator &x) noexcept {
		if (reinterpret_cast<iterator *>(this) != &x) {
			node = x.node;
			position = x.position;
		}
		return *this;
	}

	// Increment/decrement the iterator.
	void increment() noexcept {
		if (node->leaf() && ++position < node->count()) {
			return;
		}
		increment_slow();
	}
	void increment_by(int count) noexcept;
	void increment_slow() noexcept;

	void decrement() noexcept {
		if (node->leaf() && --position >= 0) {
			return;
		}
		decrement_slow();
	}
	void decrement_slow() noexcept;

	bool operator==(const const_iterator &x) const noexcept { return node == x.node && position == x.position; }
	bool operator!=(const const_iterator &x) const noexcept { return node != x.node || position != x.position; }

	// Accessors for the key/value the iterator is pointing at.
	const key_type &key() const noexcept { return node->key(position); }
	reference operator*() const noexcept { return node->value(position); }
	pointer operator->() const noexcept { return &node->value(position); }

	self_type &operator++() noexcept {
		increment();
		return *this;
	}
	self_type &operator--() noexcept {
		decrement();
		return *this;
	}
	self_type operator++(int) noexcept {
		self_type tmp = *this;
		++*this;
		return tmp;
	}
	self_type operator--(int) noexcept {
		self_type tmp = *this;
		--*this;
		return tmp;
	}

	// The node in the tree the iterator is pointing at.
	Node *node;
	// The position within the node of the tree the iterator is pointing at.
	int position;
};

// Dispatch helper class for using btree::internal_locate with plain compare.
struct btree_internal_locate_plain_compare {
	template <typename K, typename T, typename Iter>
	static std::pair<Iter, int> dispatch(const K &k, const T &t, Iter iter) {
		return t.internal_locate_plain_compare(k, iter);
	}
};

// Dispatch helper class for using btree::internal_locate with compare-to.
struct btree_internal_locate_compare_to {
	template <typename K, typename T, typename Iter>
	static std::pair<Iter, int> dispatch(const K &k, const T &t, Iter iter) {
		return t.internal_locate_compare_to(k, iter);
	}
};

template <typename Params>
class btree : public Params::key_compare {
	typedef btree<Params> self_type;
	typedef btree_node<Params> node_type;
	typedef typename node_type::base_fields base_fields;
	typedef typename node_type::leaf_fields leaf_fields;
	typedef typename node_type::internal_fields internal_fields;
	typedef typename node_type::root_fields root_fields;
	typedef typename Params::is_key_compare_to is_key_compare_to;

	friend struct btree_internal_locate_plain_compare;
	friend struct btree_internal_locate_compare_to;
	typedef typename if_<is_key_compare_to::value, btree_internal_locate_compare_to, btree_internal_locate_plain_compare>::type
		internal_locate_type;

	enum {
		kNodeValues = node_type::kNodeValues,
		kMinNodeValues = kNodeValues / 2,
		kValueSize = node_type::kValueSize,
		kExactMatch = node_type::kExactMatch,
		kMatchMask = node_type::kMatchMask,
	};

	// A helper class to get the empty base class optimization for 0-size
	// allocators. Base is internal_allocator_type.
	// (e.g. empty_base_handle<internal_allocator_type, node_type*>). If Base is
	// 0-size, the compiler doesn't have to reserve any space for it and
	// sizeof(empty_base_handle) will simply be sizeof(Data). Google [empty base
	// class optimization] for more details.
	template <typename Base, typename Data>
	struct empty_base_handle : public Base {
		empty_base_handle(const Base &b, const Data &d) : Base(b), data(d) {}
		Data data;
	};

	struct node_stats {
		node_stats(ssize_t l, ssize_t i) : leaf_nodes(l), internal_nodes(i) {}

		node_stats &operator+=(const node_stats &x) {
			leaf_nodes += x.leaf_nodes;
			internal_nodes += x.internal_nodes;
			return *this;
		}

		ssize_t leaf_nodes;
		ssize_t internal_nodes;
	};

public:
	typedef Params params_type;
	typedef typename Params::key_type key_type;
	typedef typename Params::data_type data_type;
	typedef typename Params::mapped_type mapped_type;
	typedef typename Params::value_type value_type;
	typedef typename Params::key_compare key_compare;
	typedef typename Params::pointer pointer;
	typedef typename Params::const_pointer const_pointer;
	typedef typename Params::reference reference;
	typedef typename Params::const_reference const_reference;
	typedef typename Params::size_type size_type;
	typedef typename Params::difference_type difference_type;
	typedef btree_iterator<node_type, reference, pointer> iterator;
	typedef typename iterator::const_iterator const_iterator;
	typedef std::reverse_iterator<const_iterator> const_reverse_iterator;
	typedef std::reverse_iterator<iterator> reverse_iterator;

	typedef typename Params::allocator_type allocator_type;

	typedef typename std::allocator_traits<allocator_type>::template rebind_alloc<char> internal_allocator_type;

public:
	// Default constructor.
	btree(const key_compare &comp, const allocator_type &alloc);

	// Copy constructor.
	btree(const self_type &x);

	// Destructor.
	~btree() { clear(); }

	// Iterator routines.
	iterator begin() { return iterator(leftmost(), 0); }
	const_iterator begin() const { return const_iterator(leftmost(), 0); }
	iterator end() { return iterator(rightmost(), rightmost() ? rightmost()->count() : 0); }
	const_iterator end() const { return const_iterator(rightmost(), rightmost() ? rightmost()->count() : 0); }
	reverse_iterator rbegin() { return reverse_iterator(end()); }
	const_reverse_iterator rbegin() const { return const_reverse_iterator(end()); }
	reverse_iterator rend() { return reverse_iterator(begin()); }
	const_reverse_iterator rend() const { return const_reverse_iterator(begin()); }

	// Finds the first element whose key is not less than key.
	template <typename K>
	iterator lower_bound(const K &key) {
		return internal_end(internal_lower_bound(key, iterator(root(), 0)));
	}
	template <typename K>
	const_iterator lower_bound(const K &key) const {
		return internal_end(internal_lower_bound(key, const_iterator(root(), 0)));
	}

	// Finds the first element whose key is greater than key.
	template <typename K>
	iterator upper_bound(const K &key) {
		return internal_end(internal_upper_bound(key, iterator(root(), 0)));
	}
	template <typename K>
	const_iterator upper_bound(const K &key) const {
		return internal_end(internal_upper_bound(key, const_iterator(root(), 0)));
	}

	// Finds the range of values which compare equal to key. The first member of
	// the returned pair is equal to lower_bound(key). The second member pair of
	// the pair is equal to upper_bound(key).

	template <typename K>
	std::pair<iterator, iterator> equal_range(const K &key) {
		return std::make_pair(lower_bound(key), upper_bound(key));
	}
	template <typename K>
	std::pair<const_iterator, const_iterator> equal_range(const K &key) const {
		return std::make_pair(lower_bound(key), upper_bound(key));
	}

	// Inserts a value into the btree only if it does not already exist. The
	// boolean return value indicates whether insertion succeeded or failed. The
	// ValuePointer type is used to avoid instatiating the value unless the key
	// is being inserted. Value is not dereferenced if the key already exists in
	// the btree. See btree_map::operator[].
	template <typename ValuePointer>
	std::pair<iterator, bool> insert_unique(const key_type &key, ValuePointer value);

	// Inserts a value into the btree only if it does not already exist. The
	// boolean return value indicates whether insertion succeeded or failed.
	std::pair<iterator, bool> insert_unique(const value_type &v) { return insert_unique(params_type::key(v), &v); }

	// Insert with hint. Check to see if the value should be placed immediately
	// before position in the tree. If it does, then the insertion will take
	// amortized constant time. If not, the insertion will take amortized
	// logarithmic time as if a call to insert_unique(v) were made.
	iterator insert_unique(iterator position, const value_type &v);

	// Insert a range of values into the btree.
	template <typename InputIterator>
	void insert_unique(InputIterator b, InputIterator e);

	// Inserts a value into the btree. The ValuePointer type is used to avoid
	// instatiating the value unless the key is being inserted. Value is not
	// dereferenced if the key already exists in the btree. See
	// btree_map::operator[].
	template <typename ValuePointer>
	iterator insert_multi(const key_type &key, ValuePointer value);

	// Inserts a value into the btree.
	iterator insert_multi(const value_type &v) { return insert_multi(params_type::key(v), &v); }

	// Insert with hint. Check to see if the value should be placed immediately
	// before position in the tree. If it does, then the insertion will take
	// amortized constant time. If not, the insertion will take amortized
	// logarithmic time as if a call to insert_multi(v) were made.
	iterator insert_multi(iterator position, const value_type &v);

	// Insert a range of values into the btree.
	template <typename InputIterator>
	void insert_multi(InputIterator b, InputIterator e);

	void assign(const self_type &x);

	// Erase the specified iterator from the btree. The iterator must be valid
	// (i.e. not equal to end()).  Return an iterator pointing to the node after
	// the one that was erased (or end() if none exists).
	iterator erase(iterator iter);

	// Erases range. Returns the number of keys erased.
	int erase(iterator begin, iterator end);

	// Erases the specified key from the btree. Returns 1 if an element was
	// erased and 0 otherwise.
	int erase_unique(const key_type &key);

	// Erases all of the entries matching the specified key from the
	// btree. Returns the number of elements erased.
	int erase_multi(const key_type &key);

	// Finds the iterator corresponding to a key or returns end() if the key is
	// not present.
	template <typename K>
	iterator find_unique(const K &key) {
		return internal_end(internal_find_unique(key, iterator(root(), 0)));
	}
	template <typename K>
	const_iterator find_unique(const K &key) const {
		return internal_end(internal_find_unique(key, const_iterator(root(), 0)));
	}
	template <typename K>
	iterator find_multi(const K &key) {
		return internal_end(internal_find_multi(key, iterator(root(), 0)));
	}
	template <typename K>
	const_iterator find_multi(const K &key) const {
		return internal_end(internal_find_multi(key, const_iterator(root(), 0)));
	}

	// Returns a count of the number of times the key appears in the btree.
	template <typename K>
	size_type count_unique(const K &key) const {
		const_iterator begin = internal_find_unique(key, const_iterator(root(), 0));
		if (!begin.node) {
			// The key doesn't exist in the tree.
			return 0;
		}
		return 1;
	}
	// Returns a count of the number of times the key appears in the btree.
	template <typename K>
	size_type count_multi(const K &key) const {
		return distance(lower_bound(key), upper_bound(key));
	}

	// Clear the btree, deleting all of the values it contains.
	void clear();

	// Swap the contents of *this and x.
	void swap(self_type &x);

	// Assign the contents of x to *this.
	self_type &operator=(const self_type &x) {
		if (&x == this) {
			// Don't copy onto ourselves.
			return *this;
		}
		assign(x);
		return *this;
	}

	key_compare *mutable_key_comp() { return this; }
	const key_compare &key_comp() const { return *this; }
	template <typename Key, typename K2>
	bool compare_keys(const Key &x, const K2 &y) const {
		return btree_compare_keys(key_comp(), x, y);
	}

	// Dump the btree to the specified ostream. Requires that operator<< is
	// defined for Key and Value.
	void dump(std::ostream &os) const {
		if (root() != NULL) {
			internal_dump(os, root(), 0);
		}
	}

	// Verifies the structure of the btree.
	void verify() const;

	// Size routines. Note that empty() is slightly faster than doing size()==0.
	size_type size() const {
		if (empty()) return 0;
		if (root()->leaf()) return root()->count();
		return root()->size();
	}
	size_type max_size() const { return std::numeric_limits<size_type>::max(); }
	bool empty() const { return root() == NULL; }

	// The height of the btree. An empty tree will have height 0.
	size_type height() const {
		size_type h = 0;
		if (root()) {
			// Count the length of the chain from the leftmost node up to the
			// root. We actually count from the root back around to the level below
			// the root, but the calculation is the same because of the circularity
			// of that traversal.
			const node_type *n = root();
			do {
				++h;
				n = n->parent();
			} while (n != root());
		}
		return h;
	}

	// The number of internal, leaf and total nodes used by the btree.
	size_type leaf_nodes() const { return internal_stats(root()).leaf_nodes; }
	size_type internal_nodes() const { return internal_stats(root()).internal_nodes; }
	size_type nodes() const {
		node_stats stats = internal_stats(root());
		return stats.leaf_nodes + stats.internal_nodes;
	}

	// The total number of bytes used by the btree.
	size_type bytes_used() const {
		node_stats stats = internal_stats(root());
		if (stats.leaf_nodes == 1 && stats.internal_nodes == 0) {
			return sizeof(*this) + sizeof(base_fields) + root()->max_count() * sizeof(value_type);
		} else {
			return sizeof(*this) + sizeof(root_fields) - sizeof(internal_fields) + stats.leaf_nodes * sizeof(leaf_fields) +
				   stats.internal_nodes * sizeof(internal_fields);
		}
	}

	// The average number of bytes used per value stored in the btree.
	static double average_bytes_per_value() {
		// Returns the number of bytes per value on a leaf node that is 75%
		// full. Experimentally, this matches up nicely with the computed number of
		// bytes per value in trees that had their values inserted in random order.
		return sizeof(leaf_fields) / (kNodeValues * 0.75);
	}

	// The fullness of the btree. Computed as the number of elements in the btree
	// divided by the maximum number of elements a tree with the current number
	// of nodes could hold. A value of 1 indicates perfect space
	// utilization. Smaller values indicate space wastage.
	double fullness() const { return double(size()) / (nodes() * kNodeValues); }
	// The overhead of the btree structure in bytes per node. Computed as the
	// total number of bytes used by the btree minus the number of bytes used for
	// storing elements divided by the number of elements.
	double overhead() const {
		if (empty()) {
			return 0.0;
		}
		return (bytes_used() - size() * kValueSize) / double(size());
	}

private:
	// Internal accessor routines.
	node_type *root() { return root_.data; }
	const node_type *root() const { return root_.data; }
	node_type **mutable_root() { return &root_.data; }

	// The rightmost node is stored in the root node.
	node_type *rightmost() { return (!root() || root()->leaf()) ? root() : root()->rightmost(); }
	const node_type *rightmost() const { return (!root() || root()->leaf()) ? root() : root()->rightmost(); }
	node_type **mutable_rightmost() { return root()->mutable_rightmost(); }

	// The leftmost node is stored as the parent of the root node.
	node_type *leftmost() { return root() ? root()->parent() : NULL; }
	const node_type *leftmost() const { return root() ? root()->parent() : NULL; }

	// The size of the tree is stored in the root node.
	size_type *mutable_size() { return root()->mutable_size(); }

	// Allocator routines.
	internal_allocator_type *mutable_internal_allocator() { return static_cast<internal_allocator_type *>(&root_); }
	const internal_allocator_type &internal_allocator() const { return *static_cast<const internal_allocator_type *>(&root_); }

	// Node creation/deletion routines.
	node_type *new_internal_node(node_type *parent) {
		internal_fields *p = reinterpret_cast<internal_fields *>(mutable_internal_allocator()->allocate(sizeof(internal_fields)));
		return node_type::init_internal(p, parent);
	}
	node_type *new_internal_root_node() {
		root_fields *p = reinterpret_cast<root_fields *>(mutable_internal_allocator()->allocate(sizeof(root_fields)));
		return node_type::init_root(p, root()->parent());
	}
	node_type *new_leaf_node(node_type *parent) {
		leaf_fields *p = reinterpret_cast<leaf_fields *>(mutable_internal_allocator()->allocate(sizeof(leaf_fields)));
		return node_type::init_leaf(p, parent, kNodeValues);
	}
	node_type *new_leaf_root_node(int max_count) {
		leaf_fields *p =
			reinterpret_cast<leaf_fields *>(mutable_internal_allocator()->allocate(sizeof(base_fields) + max_count * sizeof(value_type)));
		return node_type::init_leaf(p, reinterpret_cast<node_type *>(p), max_count);
	}
	void delete_internal_node(node_type *node) {
		node->destroy();
		assertrx(node != root());
		mutable_internal_allocator()->deallocate(reinterpret_cast<char *>(node), sizeof(internal_fields));
	}
	void delete_internal_root_node() {
		root()->destroy();
		mutable_internal_allocator()->deallocate(reinterpret_cast<char *>(root()), sizeof(root_fields));
	}
	void delete_leaf_node(node_type *node) {
		node->destroy();
		mutable_internal_allocator()->deallocate(reinterpret_cast<char *>(node),
												 sizeof(base_fields) + node->max_count() * sizeof(value_type));
	}

	// Rebalances or splits the node iter points to.
	void rebalance_or_split(iterator *iter);

	// Merges the values of left, right and the delimiting key on their parent
	// onto left, removing the delimiting key and deleting right.
	void merge_nodes(node_type *left, node_type *right);

	// Tries to merge node with its left or right sibling, and failing that,
	// rebalance with its left or right sibling. Returns true if a merge
	// occurred, at which point it is no longer valid to access node. Returns
	// false if no merging took place.
	bool try_merge_or_rebalance(iterator *iter);

	// Tries to shrink the height of the tree by 1.
	void try_shrink();

	iterator internal_end(iterator iter) { return iter.node ? iter : end(); }
	const_iterator internal_end(const_iterator iter) const { return iter.node ? iter : end(); }

	// Inserts a value into the btree immediately before iter. Requires that
	// key(v) <= iter.key() and (--iter).key() <= key(v).
	iterator internal_insert(iterator iter, const value_type &v);

	// Returns an iterator pointing to the first value >= the value "iter" is
	// pointing at. Note that "iter" might be pointing to an invalid location as
	// iter.position == iter.node->count(). This routine simply moves iter up in
	// the tree to a valid location.
	template <typename IterType>
	static IterType internal_last(IterType iter);

	// Returns an iterator pointing to the leaf position at which key would
	// reside in the tree. We provide 2 versions of internal_locate. The first
	// version (internal_locate_plain_compare) always returns 0 for the second
	// field of the pair. The second version (internal_locate_compare_to) is for
	// the key-compare-to specialization and returns either kExactMatch (if the
	// key was found in the tree) or -kExactMatch (if it wasn't) in the second
	// field of the pair. The compare_to specialization allows the caller to
	// avoid a subsequent comparison to determine if an exact match was made,
	// speeding up string keys.
	template <typename IterType, typename K>
	std::pair<IterType, int> internal_locate(const K &key, IterType iter) const;
	template <typename IterType, typename K>
	std::pair<IterType, int> internal_locate_plain_compare(const K &key, IterType iter) const;
	template <typename IterType, typename K>
	std::pair<IterType, int> internal_locate_compare_to(const K &key, IterType iter) const;

	// Internal routine which implements lower_bound().
	template <typename IterType, typename K>
	IterType internal_lower_bound(const K &key, IterType iter) const;

	// Internal routine which implements upper_bound().
	template <typename IterType, typename K>
	IterType internal_upper_bound(const K &key, IterType iter) const;

	// Internal routine which implements find_unique().
	template <typename IterType, typename K>
	IterType internal_find_unique(const K &key, IterType iter) const;

	// Internal routine which implements find_multi().
	template <typename IterType, typename K>
	IterType internal_find_multi(const K &key, IterType iter) const;

	// Deletes a node and all of its children.
	void internal_clear(node_type *node);

	// Dumps a node and all of its children to the specified ostream.
	void internal_dump(std::ostream &os, const node_type *node, int level) const;

	// Verifies the tree structure of node.
	int internal_verify(const node_type *node, const key_type *lo, const key_type *hi) const;

	node_stats internal_stats(const node_type *node) const {
		if (!node) {
			return node_stats(0, 0);
		}
		if (node->leaf()) {
			return node_stats(1, 0);
		}
		node_stats res(0, 1);
		for (int i = 0; i <= node->count(); ++i) {
			res += internal_stats(node->child(i));
		}
		return res;
	}

private:
	empty_base_handle<internal_allocator_type, node_type *> root_;

private:
	// A never instantiated helper function that returns big_ if we have a
	// key-compare-to functor or if R is bool and small_ otherwise.
	template <typename R>
	static typename if_<if_<is_key_compare_to::value, std::is_same<R, int>, std::is_same<R, bool> >::type::value, big_, small_>::type
		key_compare_checker(R);

	// A never instantiated helper function that returns the key comparison
	// functor.
	static key_compare key_compare_helper();

	// Verify that key_compare returns a bool. This is similar to the way
	// is_convertible in base/type_traits.h works. Note that key_compare_checker
	// is never actually invoked. The compiler will select which
	// key_compare_checker() to instantiate and then figure out the size of the
	// return type of key_compare_checker() at compile time which we then check
	// against the sizeof of big_.
	// static_assert(sizeof(key_compare_checker(key_compare_helper()(key_type(), key_type()))) == sizeof(big_),
	//			  "key_comparison_function_must_return_bool");

	// Note: We insist on kTargetValues, which is computed from
	// Params::kTargetNodeSize, must fit the base_fields::field_type.
	static_assert(kNodeValues < (1 << (8 * sizeof(typename base_fields::field_type))), "target_node_size_too_large");

	// Test the assumption made in setting kNodeValueSpace.
	static_assert(sizeof(base_fields) >= 2 * sizeof(void *), "node_space_assumption_incorrect");
};

////
// btree_node methods
template <typename P>
inline void btree_node<P>::insert_value(int i, const value_type &x) {
	assertrx(i <= count());
	value_init(count(), x);
	for (int j = count(); j > i; --j) {
		value_swap(j, this, j - 1);
	}
	set_count(count() + 1);

	if (!leaf()) {
		++i;
		for (int j = count(); j > i; --j) {
			*mutable_child(j) = child(j - 1);
			child(j)->set_position(j);
		}
		*mutable_child(i) = NULL;
	}
}

template <typename P>
inline void btree_node<P>::remove_value(int i) {
	if (!leaf()) {
		assertrx(child(i + 1)->count() == 0);
		for (int j = i + 1; j < count(); ++j) {
			*mutable_child(j) = child(j + 1);
			child(j)->set_position(j);
		}
		*mutable_child(count()) = NULL;
	}

	set_count(count() - 1);
	for (; i < count(); ++i) {
		value_swap(i, this, i + 1);
	}
	value_destroy(i);
}

template <typename P>
void btree_node<P>::rebalance_right_to_left(btree_node *src, int to_move) {
	assertrx(parent() == src->parent());
	assertrx(position() + 1 == src->position());
	assertrx(src->count() >= count());
	assertrx(to_move >= 1);
	assertrx(to_move <= src->count());

	// Make room in the left node for the new values.
	for (int i = 0; i < to_move; ++i) {
		value_init(i + count());
	}

	// Move the delimiting value to the left node and the new delimiting value
	// from the right node.
	value_swap(count(), parent(), position());
	parent()->value_swap(position(), src, to_move - 1);

	// Move the values from the right to the left node.
	for (int i = 1; i < to_move; ++i) {
		value_swap(count() + i, src, i - 1);
	}
	// Shift the values in the right node to their correct position.
	for (int i = to_move; i < src->count(); ++i) {
		src->value_swap(i - to_move, src, i);
	}
	for (int i = 1; i <= to_move; ++i) {
		src->value_destroy(src->count() - i);
	}

	if (!leaf()) {
		// Move the child pointers from the right to the left node.
		for (int i = 0; i < to_move; ++i) {
			set_child(1 + count() + i, src->child(i));
		}
		for (int i = 0; i <= src->count() - to_move; ++i) {
			assertrx(i + to_move <= src->max_count());
			src->set_child(i, src->child(i + to_move));
			*src->mutable_child(i + to_move) = NULL;
		}
	}

	// Fixup the counts on the src and dest nodes.
	set_count(count() + to_move);
	src->set_count(src->count() - to_move);
}

template <typename P>
void btree_node<P>::rebalance_left_to_right(btree_node *dest, int to_move) {
	assertrx(parent() == dest->parent());
	assertrx(position() + 1 == dest->position());
	assertrx(count() >= dest->count());
	assertrx(to_move >= 1);
	assertrx(to_move <= count());

	// Make room in the right node for the new values.
	for (int i = 0; i < to_move; ++i) {
		dest->value_init(i + dest->count());
	}
	for (int i = dest->count() - 1; i >= 0; --i) {
		dest->value_swap(i, dest, i + to_move);
	}

	// Move the delimiting value to the right node and the new delimiting value
	// from the left node.
	dest->value_swap(to_move - 1, parent(), position());
	parent()->value_swap(position(), this, count() - to_move);
	value_destroy(count() - to_move);

	// Move the values from the left to the right node.
	for (int i = 1; i < to_move; ++i) {
		value_swap(count() - to_move + i, dest, i - 1);
		value_destroy(count() - to_move + i);
	}

	if (!leaf()) {
		// Move the child pointers from the left to the right node.
		for (int i = dest->count(); i >= 0; --i) {
			dest->set_child(i + to_move, dest->child(i));
			*dest->mutable_child(i) = NULL;
		}
		for (int i = 1; i <= to_move; ++i) {
			dest->set_child(i - 1, child(count() - to_move + i));
			*mutable_child(count() - to_move + i) = NULL;
		}
	}

	// Fixup the counts on the src and dest nodes.
	set_count(count() - to_move);
	dest->set_count(dest->count() + to_move);
}

template <typename P>
void btree_node<P>::split(btree_node *dest, int insert_position) {
	assertrx(dest->count() == 0);

	// We bias the split based on the position being inserted. If we're
	// inserting at the beginning of the left node then bias the split to put
	// more values on the right node. If we're inserting at the end of the
	// right node then bias the split to put more values on the left node.
	if (insert_position == 0) {
		dest->set_count(count() - 1);
	} else if (insert_position == max_count()) {
		dest->set_count(0);
	} else {
		dest->set_count(count() / 2);
	}
	set_count(count() - dest->count());
	assertrx(count() >= 1);

	// Move values from the left sibling to the right sibling.
	for (int i = 0; i < dest->count(); ++i) {
		dest->value_init(i);
		value_swap(count() + i, dest, i);
		value_destroy(count() + i);
	}

	// The split key is the largest value in the left sibling.
	set_count(count() - 1);
	parent()->insert_value(position(), value_type());
	value_swap(count(), parent(), position());
	value_destroy(count());
	parent()->set_child(position() + 1, dest);

	if (!leaf()) {
		for (int i = 0; i <= dest->count(); ++i) {
			assertrx(child(count() + i + 1) != NULL);
			dest->set_child(i, child(count() + i + 1));
			*mutable_child(count() + i + 1) = NULL;
		}
	}
}

template <typename P>
void btree_node<P>::merge(btree_node *src) {
	assertrx(parent() == src->parent());
	assertrx(position() + 1 == src->position());

	// Move the delimiting value to the left node.
	value_init(count());
	value_swap(count(), parent(), position());

	// Move the values from the right to the left node.
	for (int i = 0; i < src->count(); ++i) {
		value_init(1 + count() + i);
		value_swap(1 + count() + i, src, i);
		src->value_destroy(i);
	}

	if (!leaf()) {
		// Move the child pointers from the right to the left node.
		for (int i = 0; i <= src->count(); ++i) {
			set_child(1 + count() + i, src->child(i));
			*src->mutable_child(i) = NULL;
		}
	}

	// Fixup the counts on the src and dest nodes.
	set_count(1 + count() + src->count());
	src->set_count(0);

	// Remove the value on the parent node.
	parent()->remove_value(position());
}

template <typename P>
void btree_node<P>::swap(btree_node *x) {
	assertrx(leaf() == x->leaf());

	// Swap the values.
	for (int i = count(); i < x->count(); ++i) {
		value_init(i);
	}
	for (int i = x->count(); i < count(); ++i) {
		x->value_init(i);
	}
	int n = std::max(count(), x->count());
	for (int i = 0; i < n; ++i) {
		value_swap(i, x, i);
	}
	for (int i = count(); i < x->count(); ++i) {
		x->value_destroy(i);
	}
	for (int i = x->count(); i < count(); ++i) {
		value_destroy(i);
	}

	if (!leaf()) {
		// Swap the child pointers.
		for (int i = 0; i <= n; ++i) {
			btree_swap_helper(*mutable_child(i), *x->mutable_child(i));
		}
		for (int i = 0; i <= count(); ++i) {
			x->child(i)->fields_.parent = x;
		}
		for (int i = 0; i <= x->count(); ++i) {
			child(i)->fields_.parent = this;
		}
	}

	// Swap the counts.
	btree_swap_helper(fields_.count, x->fields_.count);
}

////
// btree_iterator methods
template <typename N, typename R, typename P>
void btree_iterator<N, R, P>::increment_slow() noexcept {
	if (node->leaf()) {
		assertrx(position >= node->count());
		self_type save(*this);
		while (position == node->count() && !node->is_root()) {
			assertrx(node->parent()->child(node->position()) == node);
			position = node->position();
			node = node->parent();
		}
		if (position == node->count()) {
			*this = save;
		}
	} else {
		assertrx(position < node->count());
		node = node->child(position + 1);
		while (!node->leaf()) {
			node = node->child(0);
		}
		position = 0;
	}
}

template <typename N, typename R, typename P>
void btree_iterator<N, R, P>::increment_by(int count) noexcept {
	while (count > 0) {
		if (node->leaf()) {
			int rest = node->count() - position;
			position += std::min(rest, count);
			count = count - rest;
			if (position < node->count()) {
				return;
			}
		} else {
			--count;
		}
		increment_slow();
	}
}

template <typename N, typename R, typename P>
void btree_iterator<N, R, P>::decrement_slow() noexcept {
	if (node->leaf()) {
		assertrx(position <= -1);
		self_type save(*this);
		while (position < 0 && !node->is_root()) {
			assertrx(node->parent()->child(node->position()) == node);
			position = node->position() - 1;
			node = node->parent();
		}
		if (position < 0) {
			*this = save;
		}
	} else {
		assertrx(position >= 0);
		node = node->child(position);
		while (!node->leaf()) {
			node = node->child(node->count());
		}
		position = node->count() - 1;
	}
}

////
// btree methods
template <typename P>
btree<P>::btree(const key_compare &comp, const allocator_type &alloc) : key_compare(comp), root_(alloc, NULL) {}

template <typename P>
btree<P>::btree(const self_type &x) : key_compare(x.key_comp()), root_(x.internal_allocator(), NULL) {
	assign(x);
}

template <typename P>
template <typename ValuePointer>
std::pair<typename btree<P>::iterator, bool> btree<P>::insert_unique(const key_type &key, ValuePointer value) {
	if (empty()) {
		*mutable_root() = new_leaf_root_node(1);
	}

	std::pair<iterator, int> res = internal_locate(key, iterator(root(), 0));
	iterator &iter = res.first;
	if (res.second == kExactMatch) {
		// The key already exists in the tree, do nothing.
		return std::make_pair(internal_last(iter), false);
	} else if (!res.second) {
		iterator last = internal_last(iter);
		if (last.node && !compare_keys(key, last.key())) {
			// The key already exists in the tree, do nothing.
			return std::make_pair(last, false);
		}
	}

	return std::make_pair(internal_insert(iter, *value), true);
}

template <typename P>
inline typename btree<P>::iterator btree<P>::insert_unique(iterator position, const value_type &v) {
	if (!empty()) {
		const key_type &key = params_type::key(v);
		if (position == end() || compare_keys(key, position.key())) {
			iterator prev = position;
			if (position == begin() || compare_keys((--prev).key(), key)) {
				// prev.key() < key < position.key()
				return internal_insert(position, v);
			}
		} else if (compare_keys(position.key(), key)) {
			iterator next = position;
			++next;
			if (next == end() || compare_keys(key, next.key())) {
				// position.key() < key < next.key()
				return internal_insert(next, v);
			}
		} else {
			// position.key() == key
			return position;
		}
	}
	return insert_unique(v).first;
}

template <typename P>
template <typename InputIterator>
void btree<P>::insert_unique(InputIterator b, InputIterator e) {
	for (; b != e; ++b) {
		insert_unique(end(), *b);
	}
}

template <typename P>
template <typename ValuePointer>
typename btree<P>::iterator btree<P>::insert_multi(const key_type &key, ValuePointer value) {
	if (empty()) {
		*mutable_root() = new_leaf_root_node(1);
	}

	iterator iter = internal_upper_bound(key, iterator(root(), 0));
	if (!iter.node) {
		iter = end();
	}
	return internal_insert(iter, *value);
}

template <typename P>
typename btree<P>::iterator btree<P>::insert_multi(iterator position, const value_type &v) {
	if (!empty()) {
		const key_type &key = params_type::key(v);
		if (position == end() || !compare_keys(position.key(), key)) {
			iterator prev = position;
			if (position == begin() || !compare_keys(key, (--prev).key())) {
				// prev.key() <= key <= position.key()
				return internal_insert(position, v);
			}
		} else {
			iterator next = position;
			++next;
			if (next == end() || !compare_keys(next.key(), key)) {
				// position.key() < key <= next.key()
				return internal_insert(next, v);
			}
		}
	}
	return insert_multi(v);
}

template <typename P>
template <typename InputIterator>
void btree<P>::insert_multi(InputIterator b, InputIterator e) {
	for (; b != e; ++b) {
		insert_multi(end(), *b);
	}
}

template <typename P>
void btree<P>::assign(const self_type &x) {
	clear();

	*mutable_key_comp() = x.key_comp();
	*mutable_internal_allocator() = x.internal_allocator();

	// Assignment can avoid key comparisons because we know the order of the
	// values is the same order we'll store them in.
	for (const_iterator iter = x.begin(); iter != x.end(); ++iter) {
		if (empty()) {
			insert_multi(*iter);
		} else {
			// If the btree is not empty, we can just insert the new value at the end
			// of the tree!
			internal_insert(end(), *iter);
		}
	}
}

template <typename P>
typename btree<P>::iterator btree<P>::erase(iterator iter) {
	bool internal_delete = false;
	if (!iter.node->leaf()) {
		// Deletion of a value on an internal node. Swap the key with the largest
		// value of our left child. This is easy, we just decrement iter.
		iterator tmp_iter(iter--);
		assertrx(iter.node->leaf());
		assertrx(!compare_keys(tmp_iter.key(), iter.key()));
		iter.node->value_swap(iter.position, tmp_iter.node, tmp_iter.position);
		internal_delete = true;
		--*mutable_size();
	} else if (!root()->leaf()) {
		--*mutable_size();
	}

	// Delete the key from the leaf.
	iter.node->remove_value(iter.position);

	// We want to return the next value after the one we just erased. If we
	// erased from an internal node (internal_delete == true), then the next
	// value is ++(++iter). If we erased from a leaf node (internal_delete ==
	// false) then the next value is ++iter. Note that ++iter may point to an
	// internal node and the value in the internal node may move to a leaf node
	// (iter.node) when rebalancing is performed at the leaf level.

	// Merge/rebalance as we walk back up the tree.
	iterator res(iter);
	for (;;) {
		if (iter.node == root()) {
			try_shrink();
			if (empty()) {
				return end();
			}
			break;
		}
		if (iter.node->count() >= kMinNodeValues) {
			break;
		}
		bool merged = try_merge_or_rebalance(&iter);
		if (iter.node->leaf()) {
			res = iter;
		}
		if (!merged) {
			break;
		}
		iter.node = iter.node->parent();
	}

	// Adjust our return value. If we're pointing at the end of a node, advance
	// the iterator.
	if (res.position == res.node->count()) {
		res.position = res.node->count() - 1;
		++res;
	}
	// If we erased from an internal node, advance the iterator.
	if (internal_delete) {
		++res;
	}
	return res;
}

template <typename P>
int btree<P>::erase(iterator begin, iterator end) {
	int count = distance(begin, end);
	for (int i = 0; i < count; i++) {
		begin = erase(begin);
	}
	return count;
}

template <typename P>
int btree<P>::erase_unique(const key_type &key) {
	iterator iter = internal_find_unique(key, iterator(root(), 0));
	if (!iter.node) {
		// The key doesn't exist in the tree, return nothing done.
		return 0;
	}
	erase(iter);
	return 1;
}

template <typename P>
int btree<P>::erase_multi(const key_type &key) {
	iterator begin = internal_lower_bound(key, iterator(root(), 0));
	if (!begin.node) {
		// The key doesn't exist in the tree, return nothing done.
		return 0;
	}
	// Delete all of the keys between begin and upper_bound(key).
	iterator end = internal_end(internal_upper_bound(key, iterator(root(), 0)));
	return erase(begin, end);
}

template <typename P>
void btree<P>::clear() {
	if (root() != NULL) {
		internal_clear(root());
	}
	*mutable_root() = NULL;
}

template <typename P>
void btree<P>::swap(self_type &x) {
	std::swap(static_cast<key_compare &>(*this), static_cast<key_compare &>(x));
	std::swap(root_, x.root_);
}

template <typename P>
void btree<P>::verify() const {
	if (root() != NULL) {
		assertrx(size() == internal_verify(root(), NULL, NULL));
		assertrx(leftmost() == (++const_iterator(root(), -1)).node);
		assertrx(rightmost() == (--const_iterator(root(), root()->count())).node);
		assertrx(leftmost()->leaf());
		assertrx(rightmost()->leaf());
	} else {
		assertrx(size() == 0);
		assertrx(leftmost() == NULL);
		assertrx(rightmost() == NULL);
	}
}

template <typename P>
void btree<P>::rebalance_or_split(iterator *iter) {
	node_type *&node = iter->node;
	int &insert_position = iter->position;
	assertrx(node->count() == node->max_count());

	// First try to make room on the node by rebalancing.
	node_type *parent = node->parent();
	if (node != root()) {
		if (node->position() > 0) {
			// Try rebalancing with our left sibling.
			node_type *left = parent->child(node->position() - 1);
			if (left->count() < left->max_count()) {
				// We bias rebalancing based on the position being inserted. If we're
				// inserting at the end of the right node then we bias rebalancing to
				// fill up the left node.
				int to_move = (left->max_count() - left->count()) / (1 + (insert_position < left->max_count()));
				to_move = std::max(1, to_move);

				if (((insert_position - to_move) >= 0) || ((left->count() + to_move) < left->max_count())) {
					left->rebalance_right_to_left(node, to_move);

					assertrx(node->max_count() - node->count() == to_move);
					insert_position = insert_position - to_move;
					if (insert_position < 0) {
						insert_position = insert_position + left->count() + 1;
						node = left;
					}

					assertrx(node->count() < node->max_count());
					return;
				}
			}
		}

		if (node->position() < parent->count()) {
			// Try rebalancing with our right sibling.
			node_type *right = parent->child(node->position() + 1);
			if (right->count() < right->max_count()) {
				// We bias rebalancing based on the position being inserted. If we're
				// inserting at the beginning of the left node then we bias rebalancing
				// to fill up the right node.
				int to_move = (right->max_count() - right->count()) / (1 + (insert_position > 0));
				to_move = std::max(1, to_move);

				if ((insert_position <= (node->count() - to_move)) || ((right->count() + to_move) < right->max_count())) {
					node->rebalance_left_to_right(right, to_move);

					if (insert_position > node->count()) {
						insert_position = insert_position - node->count() - 1;
						node = right;
					}

					assertrx(node->count() < node->max_count());
					return;
				}
			}
		}

		// Rebalancing failed, make sure there is room on the parent node for a new
		// value.
		if (parent->count() == parent->max_count()) {
			iterator parent_iter(node->parent(), node->position());
			rebalance_or_split(&parent_iter);
		}
	} else {
		// Rebalancing not possible because this is the root node.
		if (root()->leaf()) {
			// The root node is currently a leaf node: create a new root node and set
			// the current root node as the child of the new root.
			parent = new_internal_root_node();
			parent->set_child(0, root());
			*mutable_root() = parent;
			assertrx(*mutable_rightmost() == parent->child(0));
		} else {
			// The root node is an internal node. We do not want to create a new root
			// node because the root node is special and holds the size of the tree
			// and a pointer to the rightmost node. So we create a new internal node
			// and move all of the items on the current root into the new node.
			parent = new_internal_node(parent);
			parent->set_child(0, parent);
			parent->swap(root());
			node = parent;
		}
	}

	// Split the node.
	node_type *split_node;
	if (node->leaf()) {
		split_node = new_leaf_node(parent);
		node->split(split_node, insert_position);
		if (rightmost() == node) {
			*mutable_rightmost() = split_node;
		}
	} else {
		split_node = new_internal_node(parent);
		node->split(split_node, insert_position);
	}

	if (insert_position > node->count()) {
		insert_position = insert_position - node->count() - 1;
		node = split_node;
	}
}

template <typename P>
void btree<P>::merge_nodes(node_type *left, node_type *right) {
	left->merge(right);
	if (right->leaf()) {
		if (rightmost() == right) {
			*mutable_rightmost() = left;
		}
		delete_leaf_node(right);
	} else {
		delete_internal_node(right);
	}
}

template <typename P>
bool btree<P>::try_merge_or_rebalance(iterator *iter) {
	node_type *parent = iter->node->parent();
	if (iter->node->position() > 0) {
		// Try merging with our left sibling.
		node_type *left = parent->child(iter->node->position() - 1);
		if ((1 + left->count() + iter->node->count()) <= left->max_count()) {
			iter->position += 1 + left->count();
			merge_nodes(left, iter->node);
			iter->node = left;
			return true;
		}
	}
	if (iter->node->position() < parent->count()) {
		// Try merging with our right sibling.
		node_type *right = parent->child(iter->node->position() + 1);
		if ((1 + iter->node->count() + right->count()) <= right->max_count()) {
			merge_nodes(iter->node, right);
			return true;
		}
		// Try rebalancing with our right sibling. We don't perform rebalancing if
		// we deleted the first element from iter->node and the node is not
		// empty. This is a small optimization for the common pattern of deleting
		// from the front of the tree.
		if ((right->count() > kMinNodeValues) && ((iter->node->count() == 0) || (iter->position > 0))) {
			int to_move = (right->count() - iter->node->count()) / 2;
			to_move = std::min(to_move, right->count() - 1);
			iter->node->rebalance_right_to_left(right, to_move);
			return false;
		}
	}
	if (iter->node->position() > 0) {
		// Try rebalancing with our left sibling. We don't perform rebalancing if
		// we deleted the last element from iter->node and the node is not
		// empty. This is a small optimization for the common pattern of deleting
		// from the back of the tree.
		node_type *left = parent->child(iter->node->position() - 1);
		if ((left->count() > kMinNodeValues) && ((iter->node->count() == 0) || (iter->position < iter->node->count()))) {
			int to_move = (left->count() - iter->node->count()) / 2;
			to_move = std::min(to_move, left->count() - 1);
			left->rebalance_left_to_right(iter->node, to_move);
			iter->position += to_move;
			return false;
		}
	}
	return false;
}

template <typename P>
void btree<P>::try_shrink() {
	if (root()->count() > 0) {
		return;
	}
	// Deleted the last item on the root node, shrink the height of the tree.
	if (root()->leaf()) {
		assertrx(size() == 0);
		delete_leaf_node(root());
		*mutable_root() = NULL;
	} else {
		node_type *child = root()->child(0);
		if (child->leaf()) {
			// The child is a leaf node so simply make it the root node in the tree.
			child->make_root();
			delete_internal_root_node();
			*mutable_root() = child;
		} else {
			// The child is an internal node. We want to keep the existing root node
			// so we move all of the values from the child node into the existing
			// (empty) root node.
			child->swap(root());
			delete_internal_node(child);
		}
	}
}

template <typename P>
template <typename IterType>
inline IterType btree<P>::internal_last(IterType iter) {
	while (iter.node && iter.position == iter.node->count()) {
		iter.position = iter.node->position();
		iter.node = iter.node->parent();
		if (iter.node->leaf()) {
			iter.node = NULL;
		}
	}
	return iter;
}

template <typename P>
inline typename btree<P>::iterator btree<P>::internal_insert(iterator iter, const value_type &v) {
	if (!iter.node->leaf()) {
		// We can't insert on an internal node. Instead, we'll insert after the
		// previous value which is guaranteed to be on a leaf node.
		--iter;
		++iter.position;
	}
	if (iter.node->count() == iter.node->max_count()) {
		// Make room in the leaf for the new item.
		if (iter.node->max_count() < kNodeValues) {
			// Insertion into the root where the root is smaller that the full node
			// size. Simply grow the size of the root node.
			assertrx(iter.node == root());
			iter.node = new_leaf_root_node(std::min<int>(kNodeValues, 2 * iter.node->max_count()));
			iter.node->swap(root());
			delete_leaf_node(root());
			*mutable_root() = iter.node;
		} else {
			rebalance_or_split(&iter);
			++*mutable_size();
		}
	} else if (!root()->leaf()) {
		++*mutable_size();
	}
	iter.node->insert_value(iter.position, v);
	return iter;
}

template <typename P>
template <typename IterType, typename K>
inline std::pair<IterType, int> btree<P>::internal_locate(const K &key, IterType iter) const {
	return internal_locate_type::dispatch(key, *this, iter);
}

template <typename P>
template <typename IterType, typename K>
inline std::pair<IterType, int> btree<P>::internal_locate_plain_compare(const K &key, IterType iter) const {
	for (;;) {
		iter.position = iter.node->lower_bound(key, key_comp());
		if (iter.node->leaf()) {
			break;
		}
		iter.node = iter.node->child(iter.position);
	}
	return std::make_pair(iter, 0);
}

template <typename P>
template <typename IterType, typename K>
inline std::pair<IterType, int> btree<P>::internal_locate_compare_to(const K &key, IterType iter) const {
	for (;;) {
		int res = iter.node->lower_bound(key, key_comp());
		iter.position = res & kMatchMask;
		if (res & kExactMatch) {
			return std::make_pair(iter, static_cast<int>(kExactMatch));
		}
		if (iter.node->leaf()) {
			break;
		}
		iter.node = iter.node->child(iter.position);
	}
	return std::make_pair(iter, -kExactMatch);
}

template <typename P>
template <typename IterType, typename K>
IterType btree<P>::internal_lower_bound(const K &key, IterType iter) const {
	if (iter.node) {
		for (;;) {
			iter.position = iter.node->lower_bound(key, key_comp()) & kMatchMask;
			if (iter.node->leaf()) {
				break;
			}
			iter.node = iter.node->child(iter.position);
		}
		iter = internal_last(iter);
	}
	return iter;
}

template <typename P>
template <typename IterType, typename K>
IterType btree<P>::internal_upper_bound(const K &key, IterType iter) const {
	if (iter.node) {
		for (;;) {
			iter.position = iter.node->upper_bound(key, key_comp());
			if (iter.node->leaf()) {
				break;
			}
			iter.node = iter.node->child(iter.position);
		}
		iter = internal_last(iter);
	}
	return iter;
}

template <typename P>
template <typename IterType, typename K>
IterType btree<P>::internal_find_unique(const K &key, IterType iter) const {
	if (iter.node) {
		std::pair<IterType, int> res = internal_locate(key, iter);
		if (res.second == kExactMatch) {
			return res.first;
		}
		if (!res.second) {
			iter = internal_last(res.first);
			if (iter.node && !compare_keys(key, iter.key())) {
				return iter;
			}
		}
	}
	return IterType(NULL, 0);
}

template <typename P>
template <typename IterType, typename K>
IterType btree<P>::internal_find_multi(const K &key, IterType iter) const {
	if (iter.node) {
		iter = internal_lower_bound(key, iter);
		if (iter.node) {
			iter = internal_last(iter);
			if (iter.node && !compare_keys(key, iter.key())) {
				return iter;
			}
		}
	}
	return IterType(NULL, 0);
}

template <typename P>
void btree<P>::internal_clear(node_type *node) {
	if (!node->leaf()) {
		for (int i = 0; i <= node->count(); ++i) {
			internal_clear(node->child(i));
		}
		if (node == root()) {
			delete_internal_root_node();
		} else {
			delete_internal_node(node);
		}
	} else {
		delete_leaf_node(node);
	}
}

template <typename P>
void btree<P>::internal_dump(std::ostream &os, const node_type *node, int level) const {
	for (int i = 0; i < node->count(); ++i) {
		if (!node->leaf()) {
			internal_dump(os, node->child(i), level + 1);
		}
		for (int j = 0; j < level; ++j) {
			os << "  ";
		}
		os << node->key(i) << " [" << level << "]\n";
	}
	if (!node->leaf()) {
		internal_dump(os, node->child(node->count()), level + 1);
	}
}

template <typename P>
int btree<P>::internal_verify(const node_type *node, const key_type *lo, const key_type *hi) const {
	assertrx(node->count() > 0);
	assertrx(node->count() <= node->max_count());
	if (lo) {
		assertrx(!compare_keys(node->key(0), *lo));
	}
	if (hi) {
		assertrx(!compare_keys(*hi, node->key(node->count() - 1)));
	}
	for (int i = 1; i < node->count(); ++i) {
		assertrx(!compare_keys(node->key(i), node->key(i - 1)));
	}
	int count = node->count();
	if (!node->leaf()) {
		for (int i = 0; i <= node->count(); ++i) {
			assertrx(node->child(i) != NULL);
			assertrx(node->child(i)->parent() == node);
			assertrx(node->child(i)->position() == i);
			count += internal_verify(node->child(i), (i == 0) ? lo : &node->key(i - 1), (i == node->count()) ? hi : &node->key(i));
		}
	}
	return count;
}

}  // namespace btree

#ifndef _MSC_VER
#pragma GCC diagnostic pop
#endif

#endif	// UTIL_BTREE_BTREE_H__
