More benchmarks, more documentation

all_test.go

@@ -683,3 +683,84 @@ func TestEnumeratorPrev(t *testing.T) {
 
 	}
 }
+
+func BenchmarkSeekSeq(b *testing.B) {
+	t := TreeNew(cmp)
+	for i := 0; i < b.N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		t.Seek(i)
+	}
+}
+
+func BenchmarkSeekRnd(b *testing.B) {
+	r := TreeNew(cmp)
+	rng := rng()
+	a := make([]int, b.N)
+	for i := range a {
+		a[i] = rng.Next()
+	}
+	for _, v := range a {
+		r.Set(v, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for _, v := range a {
+		r.Seek(v)
+	}
+}
+
+func BenchmarkNext1e3(b *testing.B) {
+	const N = 1e3
+	t := TreeNew(cmp)
+	for i := 0; i < N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		en, err := t.SeekFirst()
+		if err != nil {
+			b.Fatal(err)
+		}
+		n := 0
+		for {
+			if _, _, err = en.Next(); err != nil {
+				break
+			}
+			n++
+		}
+		if n != N {
+			b.Fatal(n)
+		}
+	}
+}
+
+func BenchmarkPrev1e3(b *testing.B) {
+	const N = 1e3
+	t := TreeNew(cmp)
+	for i := 0; i < N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		en, err := t.SeekLast()
+		if err != nil {
+			b.Fatal(err)
+		}
+		n := 0
+		for {
+			if _, _, err = en.Prev(); err != nil {
+				break
+			}
+			n++
+		}
+		if n != N {
+			b.Fatal(n)
+		}
+	}
+}

btree.go

@@ -17,6 +17,47 @@
 // CAPS) and every value type occurrence is replaced by the word 'value'
 // (written in all CAPS). Then you have to replace theses strings with your
 // desired type(s), using any technique you're comfortable with.
+//
+// This is how, for example, 'example/int.go' was created:
+//
+//	$ mkdir example
+//	$
+//	$ # Note: the command bellow must be actually written using the words
+//	$ # 'key' and 'value' in all CAPS. The proper form is avoided in this
+//	$ # documentation to not confuse any text replacement mechanism.
+//	$
+//	$ make generic | sed -e 's/key/int/g' -e 's/value/int/g' > example/int.go
+//
+// No other changes to int.go are (strictly) necessary, it compiles just fine.
+// In a next step, benchmarks from all_test.go were copied into
+// example/bench_test.go without any changes.  A comparator is defined in
+// bench_test.go like this:
+//
+//	func cmp(a, b int) int {
+//		return a - b
+//	}
+//
+// Running the benchmarks on a machine with Intel X5450 CPU @ 3 GHz:
+//
+//	$ go test -bench . example/bench_test.go example/int.go
+//	testing: warning: no tests to run
+//	PASS
+//	BenchmarkSetSeq	 5000000	       590 ns/op
+//	BenchmarkSetRnd	 1000000	      1530 ns/op
+//	BenchmarkGetSeq	10000000	       373 ns/op
+//	BenchmarkGetRnd	 2000000	      1109 ns/op
+//	BenchmarkDelSeq	 5000000	       672 ns/op
+//	BenchmarkDelRnd	 1000000	      1275 ns/op
+//	BenchmarkSeekSeq	 5000000	       552 ns/op
+//	BenchmarkSeekRnd	 1000000	      1108 ns/op
+//	BenchmarkNext1e3	  200000	     13414 ns/op
+//	BenchmarkPrev1e3	  200000	     13215 ns/op
+//	ok  	command-line-arguments	51.372s
+//	$
+//
+// Note that the Next and Prev benchmarks enumerate 1000 items (KV pairs), so
+// getting the next or previous iterated item is performed in about 13 ns. This
+// is the nice O(1) property of B+trees, usually not found in other tree types.
 package b
 
 import (
@@ -33,9 +74,10 @@ const (
 type (
 	// Cmp compares a and b. Return value is:
 	//
-	//	-1 if a <  b
-	//	 0 if a == b
-	//	+1 if a >  b
+	//	< 0 if a <  b
+	//	  0 if a == b
+	//	> 0 if a >  b
+	//
 	Cmp func(a, b interface{} /*K*/) int
 
 	d struct { // data page

example/bench_test.go

+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package b
+
+import (
+	"math"
+	"runtime/debug"
+	"testing"
+
+	"github.com/cznic/mathutil"
+)
+
+func cmp(a, b int) int {
+	return a - b
+}
+
+func rng() *mathutil.FC32 {
+	x, err := mathutil.NewFC32(math.MinInt32, math.MaxInt32, false)
+	if err != nil {
+		panic(err)
+	}
+
+	return x
+}
+
+func BenchmarkSetSeq(b *testing.B) {
+	r := TreeNew(cmp)
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		r.Set(i, i)
+	}
+}
+
+func BenchmarkSetRnd(b *testing.B) {
+	r := TreeNew(cmp)
+	rng := rng()
+	a := make([]int, b.N)
+	for i := range a {
+		a[i] = rng.Next()
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for _, v := range a {
+		r.Set(v, 0)
+	}
+}
+
+func BenchmarkGetSeq(b *testing.B) {
+	r := TreeNew(cmp)
+	for i := 0; i < b.N; i++ {
+		r.Set(i, i)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		r.Get(i)
+	}
+}
+
+func BenchmarkGetRnd(b *testing.B) {
+	r := TreeNew(cmp)
+	rng := rng()
+	a := make([]int, b.N)
+	for i := range a {
+		a[i] = rng.Next()
+	}
+	for _, v := range a {
+		r.Set(v, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for _, v := range a {
+		r.Get(v)
+	}
+}
+
+func BenchmarkDelSeq(b *testing.B) {
+	r := TreeNew(cmp)
+	for i := 0; i < b.N; i++ {
+		r.Set(i, i)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		r.Delete(i)
+	}
+}
+
+func BenchmarkDelRnd(b *testing.B) {
+	r := TreeNew(cmp)
+	rng := rng()
+	a := make([]int, b.N)
+	for i := range a {
+		a[i] = rng.Next()
+	}
+	for _, v := range a {
+		r.Set(v, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for _, v := range a {
+		r.Delete(v)
+	}
+}
+
+func BenchmarkSeekSeq(b *testing.B) {
+	t := TreeNew(cmp)
+	for i := 0; i < b.N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		t.Seek(i)
+	}
+}
+
+func BenchmarkSeekRnd(b *testing.B) {
+	r := TreeNew(cmp)
+	rng := rng()
+	a := make([]int, b.N)
+	for i := range a {
+		a[i] = rng.Next()
+	}
+	for _, v := range a {
+		r.Set(v, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for _, v := range a {
+		r.Seek(v)
+	}
+}
+
+func BenchmarkNext1e3(b *testing.B) {
+	const N = 1e3
+	t := TreeNew(cmp)
+	for i := 0; i < N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		en, err := t.SeekFirst()
+		if err != nil {
+			b.Fatal(err)
+		}
+		n := 0
+		for {
+			if _, _, err = en.Next(); err != nil {
+				break
+			}
+			n++
+		}
+		if n != N {
+			b.Fatal(n)
+		}
+	}
+}
+
+func BenchmarkPrev1e3(b *testing.B) {
+	const N = 1e3
+	t := TreeNew(cmp)
+	for i := 0; i < N; i++ {
+		t.Set(i, 0)
+	}
+	debug.FreeOSMemory()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		en, err := t.SeekLast()
+		if err != nil {
+			b.Fatal(err)
+		}
+		n := 0
+		for {
+			if _, _, err = en.Prev(); err != nil {
+				break
+			}
+			n++
+		}
+		if n != N {
+			b.Fatal(n)
+		}
+	}
+}

example/int.go

+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package b implements a B+tree.
+//
+// Keys and their associated values are interface{} typed, similarly to all of
+// the containers in the standard library.
+//
+// Semiautomatic production of a type specific variant of this package is
+// supported via
+//
+//	$ make generic
+//
+// Performing it will write to stdout a version of the btree.go file where
+// every key type occurrence is replaced by the word 'key' (written in all
+// CAPS) and every value type occurrence is replaced by the word 'value'
+// (written in all CAPS). Then you have to replace theses strings with your
+// desired type(s), using any technique you're comfortable with.
+//
+// This is how, for example, 'example/int.go' was created:
+//
+//	$ mkdir example
+//	$ make generic | sed -e 's/int/int/g' -e 's/int/int/g' > example/int.go
+//
+package b
+
+import (
+	"io"
+)
+
+//TODO check vs orig initialize/finalize
+
+const (
+	kx = 128 //TODO benchmark tune this number if using custom key/value type(s).
+	kd = 64  //TODO benchmark tune this number if using custom key/value type(s).
+)
+
+type (
+	// Cmp compares a and b. Return value is:
+	//
+	//	-1 if a <  b
+	//	 0 if a == b
+	//	+1 if a >  b
+	Cmp func(a, b int) int
+
+	d struct { // data page
+		c int
+		d [2*kd + 1]de
+		n *d
+		p *d
+	}
+
+	de struct { // d element
+		k int
+		v int
+	}
+
+	// Enumerator captures the state of enumerating a tree. It is returned
+	// from the Seek* methods. The enumerator is aware of any mutations
+	// made to the tree in the process of enumerating it and automatically
+	// resumes the enumeration at the proper key, if possible.
+	//
+	// However, once an Enumerator returns io.EOF to signal "no more
+	// items", it does no more attempt to "resync" on tree mutation(s).  In
+	// other words, io.EOF from an Enumaretor is "sticky" (idempotent).
+	Enumerator struct {
+		err error
+		hit bool
+		i   int
+		k   int
+		q   *d
+		t   *Tree
+		ver int64
+	}
+
+	// Tree is a B+tree.
+	Tree struct {
+		c     int
+		cmp   Cmp
+		first *d
+		last  *d
+		r     interface{}
+		ver   int64
+	}
+
+	xe struct { // x element
+		ch  interface{}
+		sep *d
+	}
+
+	x struct { // index page
+		c int
+		x [2*kx + 2]xe
+	}
+)
+
+var ( // R/O zero values
+	zd  d
+	zde de
+	zx  x
+	zxe xe
+)
+
+func clr(q interface{}) {
+	switch x := q.(type) {
+	case *x:
+		for i := 0; i <= x.c; i++ { // Ch0 Sep0 ... Chn-1 Sepn-1 Chn
+			clr(x.x[i].ch)
+		}
+		*x = zx // GC
+	case *d:
+		*x = zd // GC
+	}
+}
+
+// -------------------------------------------------------------------------- x
+
+func newX(ch0 interface{}) *x {
+	r := &x{}
+	r.x[0].ch = ch0
+	return r
+}
+
+func (q *x) extract(i int) {
+	q.c--
+	if i < q.c {
+		copy(q.x[i:], q.x[i+1:q.c+1])
+		q.x[q.c].ch = q.x[q.c+1].ch
+		q.x[q.c].sep = nil // GC
+		q.x[q.c+1] = zxe   // GC
+	}
+}
+
+func (q *x) insert(i int, d *d, ch interface{}) *x {
+	c := q.c
+	if i < c {
+		q.x[c+1].ch = q.x[c].ch
+		copy(q.x[i+2:], q.x[i+1:c])
+		q.x[i+1].sep = q.x[i].sep
+	}
+	c++
+	q.c = c
+	q.x[i].sep = d
+	q.x[i+1].ch = ch
+	return q
+}
+
+func (q *x) siblings(i int) (l, r *d) {
+	if i >= 0 {
+		if i > 0 {
+			l = q.x[i-1].ch.(*d)
+		}
+		if i < q.c {
+			r = q.x[i+1].ch.(*d)
+		}
+	}
+	return
+}
+
+// -------------------------------------------------------------------------- d
+
+func (l *d) mvL(r *d, c int) {
+	copy(l.d[l.c:], r.d[:c])
+	copy(r.d[:], r.d[c:r.c])
+	l.c += c
+	r.c -= c
+}
+
+func (l *d) mvR(r *d, c int) {
+	copy(r.d[c:], r.d[:r.c])
+	copy(r.d[:c], l.d[l.c-c:])
+	r.c += c
+	l.c -= c
+}
+
+// ----------------------------------------------------------------------- Tree
+
+// TreeNew returns a newly created, empty Tree. The compare function is used
+// for key collation.
+func TreeNew(cmp Cmp) *Tree {
+	return &Tree{cmp: cmp}
+}
+
+// Clear removes all K/V pairs from the tree.
+func (t *Tree) Clear() {
+	if t.r == nil {
+		return
+	}
+
+	clr(t.r)
+	t.c, t.first, t.last, t.r = 0, nil, nil, nil
+	t.ver++
+}
+
+func (t *Tree) cat(p *x, q, r *d, pi int) {
+	q.mvL(r, r.c)
+	if r.n != nil {
+		r.n.p = q
+	} else {
+		t.last = q
+	}
+	q.n = r.n //TODO recycle r
+	if p.c > 1 {
+		p.extract(pi)
+		p.x[pi].ch = q
+	} else { //TODO recycle r
+		t.r = q
+	}
+}
+
+func (t *Tree) catX(p, q, r *x, pi int) {
+	q.x[q.c].sep = p.x[pi].sep
+	copy(q.x[q.c+1:], r.x[:r.c])
+	q.c += r.c + 1
+	q.x[q.c].ch = r.x[r.c].ch //TODO recycle r
+	if p.c > 1 {
+		p.c--
+		pc := p.c
+		if pi < pc {
+			p.x[pi].sep = p.x[pi+1].sep
+			copy(p.x[pi+1:], p.x[pi+2:pc+1])
+			p.x[pc].ch = p.x[pc+1].ch
+			p.x[pc].sep = nil  // GC
+			p.x[pc+1].ch = nil // GC
+		}
+		return
+	}
+
+	t.r = q //TODO recycle r
+}
+
+//Delete removes the k's KV pair, if it exists, in which case Delete returns
+//true.
+func (t *Tree) Delete(k int) (ok bool) {
+	pi := -1
+	var p *x
+	q := t.r
+	if q == nil {
+		return
+	}
+
+	for {
+		var i int
+		i, ok = t.find(q, k)
+		if ok {
+			switch x := q.(type) {
+			case *x:
+				dp := x.x[i].sep
+				switch {
+				case dp.c > kd:
+					t.extract(dp, 0)
+				default:
+					if x.c < kx && q != t.r {
+						t.underflowX(p, &x, pi, &i)
+					}
+					pi = i + 1
+					p = x
+					q = x.x[pi].ch
+					ok = false
+					continue
+				}
+			case *d:
+				t.extract(x, i)
+				t.ver++
+				if x.c >= kd {
+					return
+				}
+
+				if q != t.r {
+					t.underflow(p, x, pi)
+				} else if t.c == 0 {
+					t.Clear()
+				}
+			}
+			return
+		}
+
+		switch x := q.(type) {
+		case *x:
+			if x.c < kx && q != t.r {
+				t.underflowX(p, &x, pi, &i)
+			}
+			pi = i
+			p = x
+			q = x.x[i].ch
+		case *d:
+			return
+		}
+	}
+}
+
+func (t *Tree) extract(q *d, i int) { // (r int) {
+	//r = q.d[i].v // prepared for Extract
+	q.c--
+	if i < q.c {
+		copy(q.d[i:], q.d[i+1:q.c+1])
+	}
+	q.d[q.c] = zde // GC
+	t.c--
+	return
+}
+
+func (t *Tree) find(q interface{}, k int) (i int, ok bool) {
+	var mk int
+	l := 0
+	switch x := q.(type) {
+	case *x:
+		h := x.c - 1
+		for l <= h {
+			m := (l + h) >> 1
+			mk = x.x[m].sep.d[0].k
+			switch cmp := t.cmp(k, mk); {
+			case cmp > 0:
+				l = m + 1
+			case cmp == 0:
+				return m, true
+			default:
+				h = m - 1
+			}
+		}
+	case *d:
+		h := x.c - 1
+		for l <= h {
+			m := (l + h) >> 1
+			mk = x.d[m].k
+			switch cmp := t.cmp(k, mk); {
+			case cmp > 0:
+				l = m + 1
+			case cmp == 0:
+				return m, true
+			default:
+				h = m - 1
+			}
+		}
+	}
+	return l, false
+}
+
+// First returns the first item of the tree in the key collating order, or
+// (nil, nil) if the tree is empty.
+func (t *Tree) First() (k int, v int) {
+	if q := t.first; q != nil {
+		q := &q.d[0]
+		k, v = q.k, q.v
+	}
+	return
+}
+
+// Get returns the value associated with k and true if it exists. Otherwise Get
+// returns (nil, false).
+func (t *Tree) Get(k int) (v int, ok bool) {
+	q := t.r
+	if q == nil {
+		return
+	}
+
+	for {
+		var i int
+		if i, ok = t.find(q, k); ok {
+			switch x := q.(type) {
+			case *x:
+				return x.x[i].sep.d[0].v, true
+			case *d:
+				return x.d[i].v, true
+			}
+		}
+		switch x := q.(type) {
+		case *x:
+			q = x.x[i].ch
+		default:
+			return
+		}
+	}
+}
+
+func (t *Tree) insert(q *d, i int, k int, v int) *d {
+	c := q.c
+	if i < c {
+		copy(q.d[i+1:], q.d[i:c])
+	}
+	c++
+	q.c = c
+	q.d[i].k, q.d[i].v = k, v
+	t.c++
+	return q
+}
+
+// Last returns the last item of the tree in the key collating order, or (nil,
+// nil) if the tree is empty.
+func (t *Tree) Last() (k int, v int) {
+	if q := t.last; q != nil {
+		q := &q.d[q.c-1]
+		k, v = q.k, q.v
+	}
+	return
+}
+
+// Len returns the number of items in the tree.
+func (t *Tree) Len() int {
+	return t.c
+}
+
+func (t *Tree) overflow(p *x, q *d, pi, i int, k int, v int) {
+	l, r := p.siblings(pi)
+
+	if l != nil && l.c < 2*kd {
+		l.mvL(q, 1)
+		t.insert(q, i-1, k, v)
+		return
+	}
+
+	if r != nil && r.c < 2*kd {
+		if i < 2*kd {
+			q.mvR(r, 1)
+			t.insert(q, i, k, v)
+		} else {
+			t.insert(r, 0, k, v)
+		}
+		return
+	}
+
+	t.split(p, q, pi, i, k, v)
+}
+
+// Seek returns an Enumerator positioned on a an item such that k >= item's
+// key. ok reports if k == item.key The Enumerator's position is possibly
+// after the last item in the tree.
+func (t *Tree) Seek(k int) (e *Enumerator, ok bool) {
+	q := t.r
+	if q == nil {
+		e = &Enumerator{nil, false, 0, k, nil, t, t.ver}
+		return
+	}
+
+	for {
+		var i int
+		if i, ok = t.find(q, k); ok {
+			switch x := q.(type) {
+			case *x:
+				e = &Enumerator{nil, ok, 0, k, x.x[i].sep, t, t.ver}
+				return
+			case *d:
+				e = &Enumerator{nil, ok, i, k, x, t, t.ver}
+				return
+			}
+		}
+		switch x := q.(type) {
+		case *x:
+			q = x.x[i].ch
+		case *d:
+			e = &Enumerator{nil, ok, i, k, x, t, t.ver}
+			return
+		}
+	}
+}
+
+// SeekFirst returns an enumerator positioned on the first KV pair in the tree,
+// if any. For an empty tree, err == io.EOF is returned and e will be nil.
+func (t *Tree) SeekFirst() (e *Enumerator, err error) {
+	q := t.first
+	if q == nil {
+		return nil, io.EOF
+	}
+
+	return &Enumerator{nil, true, 0, q.d[0].k, q, t, t.ver}, nil
+}
+
+// SeekLast returns an enumerator positioned on the last KV pair in the tree,
+// if any. For an empty tree, err == io.EOF is returned and e will be nil.
+func (t *Tree) SeekLast() (e *Enumerator, err error) {
+	q := t.last
+	if q == nil {
+		return nil, io.EOF
+	}
+
+	return &Enumerator{nil, true, q.c - 1, q.d[q.c-1].k, q, t, t.ver}, nil
+}
+
+// Set sets the value associated with k.
+func (t *Tree) Set(k int, v int) {
+	pi := -1
+	var p *x
+	q := t.r
+	if q != nil {
+		for {
+			i, ok := t.find(q, k)
+			if ok {
+				switch x := q.(type) {
+				case *x:
+					x.x[i].sep.d[0].v = v
+				case *d:
+					x.d[i].v = v
+				}
+				return
+			}
+
+			switch x := q.(type) {
+			case *x:
+				if x.c > 2*kx {
+					t.splitX(p, &x, pi, &i)
+				}
+				pi = i
+				p = x
+				q = x.x[i].ch
+			case *d:
+				switch {
+				case x.c < 2*kd:
+					t.insert(x, i, k, v)
+				default:
+					t.overflow(p, x, pi, i, k, v)
+				}
+				t.ver++
+				return
+			}
+		}
+	}
+
+	z := t.insert(&d{}, 0, k, v)
+	t.r, t.first, t.last = z, z, z
+	return
+}
+
+func (t *Tree) split(p *x, q *d, pi, i int, k int, v int) {
+	r := &d{}
+	if q.n != nil {
+		r.n = q.n
+		r.n.p = r
+	} else {
+		t.last = r
+	}
+	q.n = r
+	r.p = q
+
+	copy(r.d[:], q.d[kd:2*kd])
+	for i := range q.d[kd:] {
+		q.d[kd+i] = zde
+	}
+	q.c = kd
+	r.c = kd
+	if pi >= 0 {
+		p.insert(pi, r, r)
+	} else {
+		t.r = newX(q).insert(0, r, r)
+	}
+	if i > kd {
+		t.insert(r, i-kd, k, v)
+		return
+	}
+
+	t.insert(q, i, k, v)
+}
+
+func (t *Tree) splitX(p *x, pp **x, pi int, i *int) {
+	q := *pp
+	r := &x{}
+	copy(r.x[:], q.x[kx+1:])
+	q.c = kx
+	r.c = kx
+	if pi >= 0 {
+		p.insert(pi, q.x[kx].sep, r)
+	} else {
+		t.r = newX(q).insert(0, q.x[kx].sep, r)
+	}
+	q.x[kx].sep = nil
+	for i := range q.x[kx+1:] {
+		q.x[kx+i+1] = zxe
+	}
+	if *i > kx {
+		*pp = r
+		*i -= kx + 1
+	}
+}
+
+func (t *Tree) underflow(p *x, q *d, pi int) {
+	l, r := p.siblings(pi)
+
+	if l != nil && l.c+q.c >= 2*kd {
+		l.mvR(q, 1)
+	} else if r != nil && q.c+r.c >= 2*kd {
+		q.mvL(r, 1)
+		r.d[r.c] = zde // GC
+	} else if l != nil {
+		t.cat(p, l, q, pi-1)
+	} else {
+		t.cat(p, q, r, pi)
+	}
+}
+
+func (t *Tree) underflowX(p *x, pp **x, pi int, i *int) {
+	var l, r *x
+	q := *pp
+
+	if pi >= 0 {
+		if pi > 0 {
+			l = p.x[pi-1].ch.(*x)
+		}
+		if pi < p.c {
+			r = p.x[pi+1].ch.(*x)
+		}
+	}
+
+	if l != nil && l.c > kx {
+		q.x[q.c+1].ch = q.x[q.c].ch
+		copy(q.x[1:], q.x[:q.c])
+		q.x[0].ch = l.x[l.c].ch
+		q.x[0].sep = p.x[pi-1].sep
+		q.c++
+		*i++
+		l.c--
+		p.x[pi-1].sep = l.x[l.c].sep
+		return
+	}
+
+	if r != nil && r.c > kx {
+		q.x[q.c].sep = p.x[pi].sep
+		q.c++
+		q.x[q.c].ch = r.x[0].ch
+		p.x[pi].sep = r.x[0].sep
+		copy(r.x[:], r.x[1:r.c])
+		r.c--
+		rc := r.c
+		r.x[rc].ch = r.x[rc+1].ch
+		r.x[rc].sep = nil
+		r.x[rc+1].ch = nil
+		return
+	}
+
+	if l != nil {
+		*i += l.c + 1
+		t.catX(p, l, q, pi-1)
+		*pp = l
+		return
+	}
+
+	t.catX(p, q, r, pi)
+}
+
+// ----------------------------------------------------------------- Enumerator
+
+// Next returns the currently enumerated item, if it exists and moves to the
+// next item in the key collation order. If there is no item to return, err ==
+// io.EOF is returned.
+func (e *Enumerator) Next() (k int, v int, err error) {
+	if err = e.err; err != nil {
+		return
+	}
+
+	if e.ver != e.t.ver {
+		f, hit := e.t.Seek(e.k)
+		if !e.hit && hit {
+			if err = f.next(); err != nil {
+				return
+			}
+		}
+
+		*e = *f
+	}
+	if e.q == nil {
+		e.err, err = io.EOF, io.EOF
+		return
+	}
+
+	if e.i >= e.q.c {
+		if err = e.next(); err != nil {
+			return
+		}
+	}
+
+	i := e.q.d[e.i]
+	k, v = i.k, i.v
+	e.k, e.hit = k, false
+	e.next()
+	return
+}
+
+func (e *Enumerator) next() error {
+	if e.q == nil {
+		e.err = io.EOF
+		return io.EOF
+	}
+
+	switch {
+	case e.i < e.q.c-1:
+		e.i++
+	default:
+		if e.q, e.i = e.q.n, 0; e.q == nil {
+			e.err = io.EOF
+		}
+	}
+	return e.err
+}
+
+// Prev returns the currently enumerated item, if it exists and moves to the
+// previous item in the key collation order. If there is no item to return, err
+// == io.EOF is returned.
+func (e *Enumerator) Prev() (k int, v int, err error) {
+	if err = e.err; err != nil {
+		return
+	}
+
+	if e.ver != e.t.ver {
+		f, hit := e.t.Seek(e.k)
+		if !e.hit && hit {
+			if err = f.prev(); err != nil {
+				return
+			}
+		}
+
+		*e = *f
+	}
+	if e.q == nil {
+		e.err, err = io.EOF, io.EOF
+		return
+	}
+
+	if e.i >= e.q.c {
+		if err = e.next(); err != nil {
+			return
+		}
+	}
+
+	i := e.q.d[e.i]
+	k, v = i.k, i.v
+	e.k, e.hit = k, false
+	e.prev()
+	return
+}
+
+func (e *Enumerator) prev() error {
+	if e.q == nil {
+		e.err = io.EOF
+		return io.EOF
+	}
+
+	switch {
+	case e.i > 0:
+		e.i--
+	default:
+		if e.q = e.q.p; e.q == nil {
+			e.err = io.EOF
+			break
+		}
+
+		e.i = e.q.c - 1
+	}
+	return e.err
+}