avlint32.go

     1  // Copyright 2022 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package abt
     6  
     7  import (
     8  	"fmt"
     9  	"strconv"
    10  	"strings"
    11  )
    12  
    13  const (
    14  	LEAF_HEIGHT = 1
    15  	ZERO_HEIGHT = 0
    16  	NOT_KEY32   = int32(-0x80000000)
    17  )
    18  
    19  // T is the exported applicative balanced tree data type.
    20  // A T can be used as a value; updates to one copy of the value
    21  // do not change other copies.
    22  type T struct {
    23  	root *node32
    24  	size int
    25  }
    26  
    27  // node32 is the internal tree node data type
    28  type node32 struct {
    29  	// Standard conventions hold for left = smaller, right = larger
    30  	left, right *node32
    31  	data        interface{}
    32  	key         int32
    33  	height_     int8
    34  }
    35  
    36  func makeNode(key int32) *node32 {
    37  	return &node32{key: key, height_: LEAF_HEIGHT}
    38  }
    39  
    40  // IsEmpty returns true iff t is empty.
    41  func (t *T) IsEmpty() bool {
    42  	return t.root == nil
    43  }
    44  
    45  // IsSingle returns true iff t is a singleton (leaf).
    46  func (t *T) IsSingle() bool {
    47  	return t.root != nil && t.root.isLeaf()
    48  }
    49  
    50  // VisitInOrder applies f to the key and data pairs in t,
    51  // with keys ordered from smallest to largest.
    52  func (t *T) VisitInOrder(f func(int32, interface{})) {
    53  	if t.root == nil {
    54  		return
    55  	}
    56  	t.root.visitInOrder(f)
    57  }
    58  
    59  func (n *node32) nilOrData() interface{} {
    60  	if n == nil {
    61  		return nil
    62  	}
    63  	return n.data
    64  }
    65  
    66  func (n *node32) nilOrKeyAndData() (k int32, d interface{}) {
    67  	if n == nil {
    68  		k = NOT_KEY32
    69  		d = nil
    70  	} else {
    71  		k = n.key
    72  		d = n.data
    73  	}
    74  	return
    75  }
    76  
    77  func (n *node32) height() int8 {
    78  	if n == nil {
    79  		return 0
    80  	}
    81  	return n.height_
    82  }
    83  
    84  // Find returns the data associated with x in the tree, or
    85  // nil if x is not in the tree.
    86  func (t *T) Find(x int32) interface{} {
    87  	return t.root.find(x).nilOrData()
    88  }
    89  
    90  // Insert either adds x to the tree if x was not previously
    91  // a key in the tree, or updates the data for x in the tree if
    92  // x was already a key in the tree.  The previous data associated
    93  // with x is returned, and is nil if x was not previously a
    94  // key in the tree.
    95  func (t *T) Insert(x int32, data interface{}) interface{} {
    96  	if x == NOT_KEY32 {
    97  		panic("Cannot use sentinel value -0x80000000 as key")
    98  	}
    99  	n := t.root
   100  	var newroot *node32
   101  	var o *node32
   102  	if n == nil {
   103  		n = makeNode(x)
   104  		newroot = n
   105  	} else {
   106  		newroot, n, o = n.aInsert(x)
   107  	}
   108  	var r interface{}
   109  	if o != nil {
   110  		r = o.data
   111  	} else {
   112  		t.size++
   113  	}
   114  	n.data = data
   115  	t.root = newroot
   116  	return r
   117  }
   118  
   119  func (t *T) Copy() *T {
   120  	u := *t
   121  	return &u
   122  }
   123  
   124  func (t *T) Delete(x int32) interface{} {
   125  	n := t.root
   126  	if n == nil {
   127  		return nil
   128  	}
   129  	d, s := n.aDelete(x)
   130  	if d == nil {
   131  		return nil
   132  	}
   133  	t.root = s
   134  	t.size--
   135  	return d.data
   136  }
   137  
   138  func (t *T) DeleteMin() (int32, interface{}) {
   139  	n := t.root
   140  	if n == nil {
   141  		return NOT_KEY32, nil
   142  	}
   143  	d, s := n.aDeleteMin()
   144  	if d == nil {
   145  		return NOT_KEY32, nil
   146  	}
   147  	t.root = s
   148  	t.size--
   149  	return d.key, d.data
   150  }
   151  
   152  func (t *T) DeleteMax() (int32, interface{}) {
   153  	n := t.root
   154  	if n == nil {
   155  		return NOT_KEY32, nil
   156  	}
   157  	d, s := n.aDeleteMax()
   158  	if d == nil {
   159  		return NOT_KEY32, nil
   160  	}
   161  	t.root = s
   162  	t.size--
   163  	return d.key, d.data
   164  }
   165  
   166  func (t *T) Size() int {
   167  	return t.size
   168  }
   169  
   170  // Intersection returns the intersection of t and u, where the result
   171  // data for any common keys is given by f(t's data, u's data) -- f need
   172  // not be symmetric.  If f returns nil, then the key and data are not
   173  // added to the result.  If f itself is nil, then whatever value was
   174  // already present in the smaller set is used.
   175  func (t *T) Intersection(u *T, f func(x, y interface{}) interface{}) *T {
   176  	if t.Size() == 0 || u.Size() == 0 {
   177  		return &T{}
   178  	}
   179  
   180  	// For faster execution and less allocation, prefer t smaller, iterate over t.
   181  	if t.Size() <= u.Size() {
   182  		v := t.Copy()
   183  		for it := t.Iterator(); !it.Done(); {
   184  			k, d := it.Next()
   185  			e := u.Find(k)
   186  			if e == nil {
   187  				v.Delete(k)
   188  				continue
   189  			}
   190  			if f == nil {
   191  				continue
   192  			}
   193  			if c := f(d, e); c != d {
   194  				if c == nil {
   195  					v.Delete(k)
   196  				} else {
   197  					v.Insert(k, c)
   198  				}
   199  			}
   200  		}
   201  		return v
   202  	}
   203  	v := u.Copy()
   204  	for it := u.Iterator(); !it.Done(); {
   205  		k, e := it.Next()
   206  		d := t.Find(k)
   207  		if d == nil {
   208  			v.Delete(k)
   209  			continue
   210  		}
   211  		if f == nil {
   212  			continue
   213  		}
   214  		if c := f(d, e); c != d {
   215  			if c == nil {
   216  				v.Delete(k)
   217  			} else {
   218  				v.Insert(k, c)
   219  			}
   220  		}
   221  	}
   222  
   223  	return v
   224  }
   225  
   226  // Union returns the union of t and u, where the result data for any common keys
   227  // is given by f(t's data, u's data) -- f need not be symmetric.  If f returns nil,
   228  // then the key and data are not added to the result.  If f itself is nil, then
   229  // whatever value was already present in the larger set is used.
   230  func (t *T) Union(u *T, f func(x, y interface{}) interface{}) *T {
   231  	if t.Size() == 0 {
   232  		return u
   233  	}
   234  	if u.Size() == 0 {
   235  		return t
   236  	}
   237  
   238  	if t.Size() >= u.Size() {
   239  		v := t.Copy()
   240  		for it := u.Iterator(); !it.Done(); {
   241  			k, e := it.Next()
   242  			d := t.Find(k)
   243  			if d == nil {
   244  				v.Insert(k, e)
   245  				continue
   246  			}
   247  			if f == nil {
   248  				continue
   249  			}
   250  			if c := f(d, e); c != d {
   251  				if c == nil {
   252  					v.Delete(k)
   253  				} else {
   254  					v.Insert(k, c)
   255  				}
   256  			}
   257  		}
   258  		return v
   259  	}
   260  
   261  	v := u.Copy()
   262  	for it := t.Iterator(); !it.Done(); {
   263  		k, d := it.Next()
   264  		e := u.Find(k)
   265  		if e == nil {
   266  			v.Insert(k, d)
   267  			continue
   268  		}
   269  		if f == nil {
   270  			continue
   271  		}
   272  		if c := f(d, e); c != d {
   273  			if c == nil {
   274  				v.Delete(k)
   275  			} else {
   276  				v.Insert(k, c)
   277  			}
   278  		}
   279  	}
   280  	return v
   281  }
   282  
   283  // Difference returns the difference of t and u, subject to the result
   284  // of f applied to data corresponding to equal keys.  If f returns nil
   285  // (or if f is nil) then the key+data are excluded, as usual.  If f
   286  // returns not-nil, then that key+data pair is inserted. instead.
   287  func (t *T) Difference(u *T, f func(x, y interface{}) interface{}) *T {
   288  	if t.Size() == 0 {
   289  		return &T{}
   290  	}
   291  	if u.Size() == 0 {
   292  		return t
   293  	}
   294  	v := t.Copy()
   295  	for it := t.Iterator(); !it.Done(); {
   296  		k, d := it.Next()
   297  		e := u.Find(k)
   298  		if e != nil {
   299  			if f == nil {
   300  				v.Delete(k)
   301  				continue
   302  			}
   303  			c := f(d, e)
   304  			if c == nil {
   305  				v.Delete(k)
   306  				continue
   307  			}
   308  			if c != d {
   309  				v.Insert(k, c)
   310  			}
   311  		}
   312  	}
   313  	return v
   314  }
   315  
   316  func (t *T) Iterator() Iterator {
   317  	return Iterator{it: t.root.iterator()}
   318  }
   319  
   320  func (t *T) Equals(u *T) bool {
   321  	if t == u {
   322  		return true
   323  	}
   324  	if t.Size() != u.Size() {
   325  		return false
   326  	}
   327  	return t.root.equals(u.root)
   328  }
   329  
   330  func (t *T) String() string {
   331  	var b strings.Builder
   332  	first := true
   333  	for it := t.Iterator(); !it.Done(); {
   334  		k, v := it.Next()
   335  		if first {
   336  			first = false
   337  		} else {
   338  			b.WriteString("; ")
   339  		}
   340  		b.WriteString(strconv.FormatInt(int64(k), 10))
   341  		b.WriteString(":")
   342  		fmt.Fprint(&b, v)
   343  	}
   344  	return b.String()
   345  }
   346  
   347  func (t *node32) equals(u *node32) bool {
   348  	if t == u {
   349  		return true
   350  	}
   351  	it, iu := t.iterator(), u.iterator()
   352  	for !it.done() && !iu.done() {
   353  		nt := it.next()
   354  		nu := iu.next()
   355  		if nt == nu {
   356  			continue
   357  		}
   358  		if nt.key != nu.key {
   359  			return false
   360  		}
   361  		if nt.data != nu.data {
   362  			return false
   363  		}
   364  	}
   365  	return it.done() == iu.done()
   366  }
   367  
   368  func (t *T) Equiv(u *T, eqv func(x, y interface{}) bool) bool {
   369  	if t == u {
   370  		return true
   371  	}
   372  	if t.Size() != u.Size() {
   373  		return false
   374  	}
   375  	return t.root.equiv(u.root, eqv)
   376  }
   377  
   378  func (t *node32) equiv(u *node32, eqv func(x, y interface{}) bool) bool {
   379  	if t == u {
   380  		return true
   381  	}
   382  	it, iu := t.iterator(), u.iterator()
   383  	for !it.done() && !iu.done() {
   384  		nt := it.next()
   385  		nu := iu.next()
   386  		if nt == nu {
   387  			continue
   388  		}
   389  		if nt.key != nu.key {
   390  			return false
   391  		}
   392  		if !eqv(nt.data, nu.data) {
   393  			return false
   394  		}
   395  	}
   396  	return it.done() == iu.done()
   397  }
   398  
   399  type iterator struct {
   400  	parents []*node32
   401  }
   402  
   403  type Iterator struct {
   404  	it iterator
   405  }
   406  
   407  func (it *Iterator) Next() (int32, interface{}) {
   408  	x := it.it.next()
   409  	if x == nil {
   410  		return NOT_KEY32, nil
   411  	}
   412  	return x.key, x.data
   413  }
   414  
   415  func (it *Iterator) Done() bool {
   416  	return len(it.it.parents) == 0
   417  }
   418  
   419  func (t *node32) iterator() iterator {
   420  	if t == nil {
   421  		return iterator{}
   422  	}
   423  	it := iterator{parents: make([]*node32, 0, int(t.height()))}
   424  	it.leftmost(t)
   425  	return it
   426  }
   427  
   428  func (it *iterator) leftmost(t *node32) {
   429  	for t != nil {
   430  		it.parents = append(it.parents, t)
   431  		t = t.left
   432  	}
   433  }
   434  
   435  func (it *iterator) done() bool {
   436  	return len(it.parents) == 0
   437  }
   438  
   439  func (it *iterator) next() *node32 {
   440  	l := len(it.parents)
   441  	if l == 0 {
   442  		return nil
   443  	}
   444  	x := it.parents[l-1] // return value
   445  	if x.right != nil {
   446  		it.leftmost(x.right)
   447  		return x
   448  	}
   449  	// discard visited top of parents
   450  	l--
   451  	it.parents = it.parents[:l]
   452  	y := x // y is known visited/returned
   453  	for l > 0 && y == it.parents[l-1].right {
   454  		y = it.parents[l-1]
   455  		l--
   456  		it.parents = it.parents[:l]
   457  	}
   458  
   459  	return x
   460  }
   461  
   462  // Min returns the minimum element of t.
   463  // If t is empty, then (NOT_KEY32, nil) is returned.
   464  func (t *T) Min() (k int32, d interface{}) {
   465  	return t.root.min().nilOrKeyAndData()
   466  }
   467  
   468  // Max returns the maximum element of t.
   469  // If t is empty, then (NOT_KEY32, nil) is returned.
   470  func (t *T) Max() (k int32, d interface{}) {
   471  	return t.root.max().nilOrKeyAndData()
   472  }
   473  
   474  // Glb returns the greatest-lower-bound-exclusive of x and the associated
   475  // data.  If x has no glb in the tree, then (NOT_KEY32, nil) is returned.
   476  func (t *T) Glb(x int32) (k int32, d interface{}) {
   477  	return t.root.glb(x, false).nilOrKeyAndData()
   478  }
   479  
   480  // GlbEq returns the greatest-lower-bound-inclusive of x and the associated
   481  // data.  If x has no glbEQ in the tree, then (NOT_KEY32, nil) is returned.
   482  func (t *T) GlbEq(x int32) (k int32, d interface{}) {
   483  	return t.root.glb(x, true).nilOrKeyAndData()
   484  }
   485  
   486  // Lub returns the least-upper-bound-exclusive of x and the associated
   487  // data.  If x has no lub in the tree, then (NOT_KEY32, nil) is returned.
   488  func (t *T) Lub(x int32) (k int32, d interface{}) {
   489  	return t.root.lub(x, false).nilOrKeyAndData()
   490  }
   491  
   492  // LubEq returns the least-upper-bound-inclusive of x and the associated
   493  // data.  If x has no lubEq in the tree, then (NOT_KEY32, nil) is returned.
   494  func (t *T) LubEq(x int32) (k int32, d interface{}) {
   495  	return t.root.lub(x, true).nilOrKeyAndData()
   496  }
   497  
   498  func (t *node32) isLeaf() bool {
   499  	return t.left == nil && t.right == nil && t.height_ == LEAF_HEIGHT
   500  }
   501  
   502  func (t *node32) visitInOrder(f func(int32, interface{})) {
   503  	if t.left != nil {
   504  		t.left.visitInOrder(f)
   505  	}
   506  	f(t.key, t.data)
   507  	if t.right != nil {
   508  		t.right.visitInOrder(f)
   509  	}
   510  }
   511  
   512  func (t *node32) find(key int32) *node32 {
   513  	for t != nil {
   514  		if key < t.key {
   515  			t = t.left
   516  		} else if key > t.key {
   517  			t = t.right
   518  		} else {
   519  			return t
   520  		}
   521  	}
   522  	return nil
   523  }
   524  
   525  func (t *node32) min() *node32 {
   526  	if t == nil {
   527  		return t
   528  	}
   529  	for t.left != nil {
   530  		t = t.left
   531  	}
   532  	return t
   533  }
   534  
   535  func (t *node32) max() *node32 {
   536  	if t == nil {
   537  		return t
   538  	}
   539  	for t.right != nil {
   540  		t = t.right
   541  	}
   542  	return t
   543  }
   544  
   545  func (t *node32) glb(key int32, allow_eq bool) *node32 {
   546  	var best *node32 = nil
   547  	for t != nil {
   548  		if key <= t.key {
   549  			if allow_eq && key == t.key {
   550  				return t
   551  			}
   552  			// t is too big, glb is to left.
   553  			t = t.left
   554  		} else {
   555  			// t is a lower bound, record it and seek a better one.
   556  			best = t
   557  			t = t.right
   558  		}
   559  	}
   560  	return best
   561  }
   562  
   563  func (t *node32) lub(key int32, allow_eq bool) *node32 {
   564  	var best *node32 = nil
   565  	for t != nil {
   566  		if key >= t.key {
   567  			if allow_eq && key == t.key {
   568  				return t
   569  			}
   570  			// t is too small, lub is to right.
   571  			t = t.right
   572  		} else {
   573  			// t is an upper bound, record it and seek a better one.
   574  			best = t
   575  			t = t.left
   576  		}
   577  	}
   578  	return best
   579  }
   580  
   581  func (t *node32) aInsert(x int32) (newroot, newnode, oldnode *node32) {
   582  	// oldnode default of nil is good, others should be assigned.
   583  	if x == t.key {
   584  		oldnode = t
   585  		newt := *t
   586  		newnode = &newt
   587  		newroot = newnode
   588  		return
   589  	}
   590  	if x < t.key {
   591  		if t.left == nil {
   592  			t = t.copy()
   593  			n := makeNode(x)
   594  			t.left = n
   595  			newnode = n
   596  			newroot = t
   597  			t.height_ = 2 // was balanced w/ 0, sibling is height 0 or 1
   598  			return
   599  		}
   600  		var new_l *node32
   601  		new_l, newnode, oldnode = t.left.aInsert(x)
   602  		t = t.copy()
   603  		t.left = new_l
   604  		if new_l.height() > 1+t.right.height() {
   605  			newroot = t.aLeftIsHigh(newnode)
   606  		} else {
   607  			t.height_ = 1 + max(t.left.height(), t.right.height())
   608  			newroot = t
   609  		}
   610  	} else { // x > t.key
   611  		if t.right == nil {
   612  			t = t.copy()
   613  			n := makeNode(x)
   614  			t.right = n
   615  			newnode = n
   616  			newroot = t
   617  			t.height_ = 2 // was balanced w/ 0, sibling is height 0 or 1
   618  			return
   619  		}
   620  		var new_r *node32
   621  		new_r, newnode, oldnode = t.right.aInsert(x)
   622  		t = t.copy()
   623  		t.right = new_r
   624  		if new_r.height() > 1+t.left.height() {
   625  			newroot = t.aRightIsHigh(newnode)
   626  		} else {
   627  			t.height_ = 1 + max(t.left.height(), t.right.height())
   628  			newroot = t
   629  		}
   630  	}
   631  	return
   632  }
   633  
   634  func (t *node32) aDelete(key int32) (deleted, newSubTree *node32) {
   635  	if t == nil {
   636  		return nil, nil
   637  	}
   638  
   639  	if key < t.key {
   640  		oh := t.left.height()
   641  		d, tleft := t.left.aDelete(key)
   642  		if tleft == t.left {
   643  			return d, t
   644  		}
   645  		return d, t.copy().aRebalanceAfterLeftDeletion(oh, tleft)
   646  	} else if key > t.key {
   647  		oh := t.right.height()
   648  		d, tright := t.right.aDelete(key)
   649  		if tright == t.right {
   650  			return d, t
   651  		}
   652  		return d, t.copy().aRebalanceAfterRightDeletion(oh, tright)
   653  	}
   654  
   655  	if t.height() == LEAF_HEIGHT {
   656  		return t, nil
   657  	}
   658  
   659  	// Interior delete by removing left.Max or right.Min,
   660  	// then swapping contents
   661  	if t.left.height() > t.right.height() {
   662  		oh := t.left.height()
   663  		d, tleft := t.left.aDeleteMax()
   664  		r := t
   665  		t = t.copy()
   666  		t.data, t.key = d.data, d.key
   667  		return r, t.aRebalanceAfterLeftDeletion(oh, tleft)
   668  	}
   669  
   670  	oh := t.right.height()
   671  	d, tright := t.right.aDeleteMin()
   672  	r := t
   673  	t = t.copy()
   674  	t.data, t.key = d.data, d.key
   675  	return r, t.aRebalanceAfterRightDeletion(oh, tright)
   676  }
   677  
   678  func (t *node32) aDeleteMin() (deleted, newSubTree *node32) {
   679  	if t == nil {
   680  		return nil, nil
   681  	}
   682  	if t.left == nil { // leaf or left-most
   683  		return t, t.right
   684  	}
   685  	oh := t.left.height()
   686  	d, tleft := t.left.aDeleteMin()
   687  	if tleft == t.left {
   688  		return d, t
   689  	}
   690  	return d, t.copy().aRebalanceAfterLeftDeletion(oh, tleft)
   691  }
   692  
   693  func (t *node32) aDeleteMax() (deleted, newSubTree *node32) {
   694  	if t == nil {
   695  		return nil, nil
   696  	}
   697  
   698  	if t.right == nil { // leaf or right-most
   699  		return t, t.left
   700  	}
   701  
   702  	oh := t.right.height()
   703  	d, tright := t.right.aDeleteMax()
   704  	if tright == t.right {
   705  		return d, t
   706  	}
   707  	return d, t.copy().aRebalanceAfterRightDeletion(oh, tright)
   708  }
   709  
   710  func (t *node32) aRebalanceAfterLeftDeletion(oldLeftHeight int8, tleft *node32) *node32 {
   711  	t.left = tleft
   712  
   713  	if oldLeftHeight == tleft.height() || oldLeftHeight == t.right.height() {
   714  		// this node is still balanced and its height is unchanged
   715  		return t
   716  	}
   717  
   718  	if oldLeftHeight > t.right.height() {
   719  		// left was larger
   720  		t.height_--
   721  		return t
   722  	}
   723  
   724  	// left height fell by 1 and it was already less than right height
   725  	t.right = t.right.copy()
   726  	return t.aRightIsHigh(nil)
   727  }
   728  
   729  func (t *node32) aRebalanceAfterRightDeletion(oldRightHeight int8, tright *node32) *node32 {
   730  	t.right = tright
   731  
   732  	if oldRightHeight == tright.height() || oldRightHeight == t.left.height() {
   733  		// this node is still balanced and its height is unchanged
   734  		return t
   735  	}
   736  
   737  	if oldRightHeight > t.left.height() {
   738  		// left was larger
   739  		t.height_--
   740  		return t
   741  	}
   742  
   743  	// right height fell by 1 and it was already less than left height
   744  	t.left = t.left.copy()
   745  	return t.aLeftIsHigh(nil)
   746  }
   747  
   748  // aRightIsHigh does rotations necessary to fix a high right child
   749  // assume that t and t.right are already fresh copies.
   750  func (t *node32) aRightIsHigh(newnode *node32) *node32 {
   751  	right := t.right
   752  	if right.right.height() < right.left.height() {
   753  		// double rotation
   754  		if newnode != right.left {
   755  			right.left = right.left.copy()
   756  		}
   757  		t.right = right.leftToRoot()
   758  	}
   759  	t = t.rightToRoot()
   760  	return t
   761  }
   762  
   763  // aLeftIsHigh does rotations necessary to fix a high left child
   764  // assume that t and t.left are already fresh copies.
   765  func (t *node32) aLeftIsHigh(newnode *node32) *node32 {
   766  	left := t.left
   767  	if left.left.height() < left.right.height() {
   768  		// double rotation
   769  		if newnode != left.right {
   770  			left.right = left.right.copy()
   771  		}
   772  		t.left = left.rightToRoot()
   773  	}
   774  	t = t.leftToRoot()
   775  	return t
   776  }
   777  
   778  // rightToRoot does that rotation, modifying t and t.right in the process.
   779  func (t *node32) rightToRoot() *node32 {
   780  	//    this
   781  	// left  right
   782  	//      rl   rr
   783  	//
   784  	// becomes
   785  	//
   786  	//       right
   787  	//    this   rr
   788  	// left  rl
   789  	//
   790  	right := t.right
   791  	rl := right.left
   792  	right.left = t
   793  	// parent's child ptr fixed in caller
   794  	t.right = rl
   795  	t.height_ = 1 + max(rl.height(), t.left.height())
   796  	right.height_ = 1 + max(t.height(), right.right.height())
   797  	return right
   798  }
   799  
   800  // leftToRoot does that rotation, modifying t and t.left in the process.
   801  func (t *node32) leftToRoot() *node32 {
   802  	//     this
   803  	//  left  right
   804  	// ll  lr
   805  	//
   806  	// becomes
   807  	//
   808  	//    left
   809  	//   ll  this
   810  	//      lr  right
   811  	//
   812  	left := t.left
   813  	lr := left.right
   814  	left.right = t
   815  	// parent's child ptr fixed in caller
   816  	t.left = lr
   817  	t.height_ = 1 + max(lr.height(), t.right.height())
   818  	left.height_ = 1 + max(t.height(), left.left.height())
   819  	return left
   820  }
   821  
   822  func max(a, b int8) int8 {
   823  	if a > b {
   824  		return a
   825  	}
   826  	return b
   827  }
   828  
   829  func (t *node32) copy() *node32 {
   830  	u := *t
   831  	return &u
   832  }
   833
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