Algorithms. Algorithms 3.3 BALANCED SEARCH TREES. 2 3 search trees red black BSTs ROBERT SEDGEWICK KEVIN WAYNE.

Transcription

1 lgorithms OBT DGWICK KVIN WYN 3.3 BLNCD C T 2 3 search trees red black BTs lgorithms F O U T D I T I O N OBT DGWICK KVIN WYN Last updated on 10/11/16 9:22 M

2 BT: ordered symbol table operations summary sequential search binary search BT search N log N h insert N N h min / max N 1 h h = height of BT (proportional to log N if keys inserted in random order) floor / ceiling N log N h rank N log N h select N 1 h ordered iteration N log N N N order of growth of running time of ordered symbol table operations 2

3 Many BTs correspond to same set of keys. Number of compares for search/insert = 1 + depth of node. Bottom line. Tree shape depends on order of insertion. 3 Tree shape X C best case C X typical case X C worst case

4 ymbol table review implementation guarantee search insert delete search hit average case insert delete ordered ops? key interface sequential search (unordered list) binary search (ordered array) N N N N N N equals() log N N N log N N N compareto() BT N N N log N log N N compareto() goal log N log N log N log N log N log N compareto() Challenge. Guarantee performance. This lecture. 2 3 trees and left-leaning red black BTs. Textbook: B-trees. 4

5 3.3 BLNCD C T 2 3 search trees red black BTs lgorithms OBT DGWICK KVIN WYN

6 2 3 tree llow 1 or 2 keys per node. 2-node: one key, two children. 3-node: two keys, three children. ymmetric order. In order traversal yields keys in ascending order. Perfect balance. very path from root to null link has same length. M how to maintain? 3-node 2-node smaller than J larger than J C L P X between and J null link 6

7 2 3 tree demo earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for M J C L P X 7

8 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for is less than M (go left) M J C L P X

9 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for is between and J (go middle) M J C L P X

10 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for M J C L P X found (search hit)

11 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for B B is less than M (go left) B M J C L P X

12 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for B B is less than (go left) M B J C L P X

13 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for B M J B is between and C (go middle) B C L P X

14 2 3 tree demo: search earch. Compare search key against keys in node. Find interval containing search key. Follow associated link (recursively). search for B M J C L P X B link is null (search miss)

15 2 3 tree demo: insertion Insert into a 2-node at bottom. earch for key, as usual. eplace 2-node with 3-node. insert K K is less than M (go left) K M J C L P X

16 2 3 tree demo: insertion Insert into a 2-node at bottom. earch for key, as usual. eplace 2-node with 3-node. insert K M K is greater than J (go right) K J C L P X

17 2 3 tree demo: insertion Insert into a 2-node at bottom. earch for key, as usual. eplace 2-node with 3-node. insert K M J C K L P X search ends here

18 2 3 tree demo: insertion Insert into a 2-node at bottom. earch for key, as usual. eplace 2-node with 3-node. insert K M J C K L P X replace 2-node with 3-node containing K

19 2 3 tree demo: insertion Insert into a 2-node at bottom. earch for key, as usual. eplace 2-node with 3-node. insert K M J C KLL P X

20 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z Z is greater than M (go right) M Z J C K L P X

21 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M Z is greater than (go right) J Z C K L P X

22 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J C K L P X Z search ends here

23 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J C K L P X Z replace 3-node with temporary 4-node containing Z

24 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J C K L P X Z

25 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J C K L P X Z split 4-node into two 2-nodes (pass middle key to parent)

26 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J X C K L P Z

27 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. insert Z M J X C K L P Z

28 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L C P L X convert 3-node into 4-node

29 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L C L P X

30 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L C L P X split 4-node (move L to parent)

31 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L L C P X

32 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L split 4-node (move L to parent) L C P X

33 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. height of tree increases by 1 insert L L C P X

34 2 3 tree demo: insertion Insert into a 3-node at bottom. dd new key to 3-node to create temporary 4-node. Move middle key in 4-node into parent. epeat up the tree, as necessary. If you reach the root and it's a 4-node, split it into three 2-nodes. insert L L C P X

35 2 3 tree construction demo insert 35

36 2 3 tree demo: construction 2 3 tree

37 2 3 tree demo: construction insert convert 2-node into 3-node

38 2 3 tree demo: construction insert

39 2 3 tree demo: construction 2 3 tree

40 2 3 tree demo: construction insert convert 3-node into 4-node

41 2 3 tree demo: construction insert

42 2 3 tree demo: construction insert split 4-node (move to parent)

43 2 3 tree demo: construction insert

44 2 3 tree demo: construction 2 3 tree

45 2 3 tree demo: construction insert convert 2-node into 3-node

46 2 3 tree demo: construction insert

47 2 3 tree demo: construction 2 3 tree

48 2 3 tree demo: construction insert C C convert 2-node into 3-node

49 2 3 tree demo: construction insert C C

50 2 3 tree demo: construction 2 3 tree C

51 2 3 tree demo: construction insert C convert 3-node into 4-node

52 2 3 tree demo: construction insert C

53 2 3 tree demo: construction insert C split 4-node (move to parent)

54 2 3 tree demo: construction insert C

55 2 3 tree demo: construction 2 3 tree C

56 2 3 tree demo: construction insert X C X convert 2-node into 3-node

57 2 3 tree demo: construction insert X C X

58 2 3 tree demo: construction 2 3 tree C X

59 2 3 tree demo: construction insert M C M X convert 2-node into 3-node

60 2 3 tree demo: construction insert M C M X

61 2 3 tree demo: construction 2 3 tree C M X

62 2 3 tree demo: construction insert P C M P X convert 3-node into 4-node

63 2 3 tree demo: construction insert P C M P X

64 2 3 tree demo: construction insert P C M P X split 4-node (move L to parent)

65 2 3 tree demo: construction insert P M C P X

66 2 3 tree demo: construction insert P split 4-node (move M to parent) M C P X

67 2 3 tree demo: construction insert P M C P X

68 2 3 tree demo: construction 2 3 tree M C P X

69 2 3 tree demo: construction insert L M C L P X convert 2-node into 3-node

70 2 3 tree demo: construction insert L M C L P X convert 2-node into 3-node

71 2 3 tree demo: construction 2-3 tree M C L P X

72 2 3 tree: global properties Invariants. Maintains symmetric order and perfect balance. Pf. ach transformation maintains symmetric order and perfect balance. root a b c a b c parent is a 3-node left d e b d e a b c a c parent is a 2-node left a b c d a b d c middle a e b c d a c e b d right a a c right a b a b d b c d b d c d e c e 72

73 2 3 tree: performance plitting a 4-node is a local transformation: constant number of operations. a e b c d less than a between a and b between b and c between c and d between d and e greater than e a c e b d less than a between a and b between b and c between c and d between d and e greater than e 73

74 2 3 tree: performance Perfect balance. very path from root to null link has same length. Tree height. Worst case: lg N. Best case: log [all 2-nodes] 3 N.631 lg N. [all 3-nodes] The tree will have a height of between 12 and 20 for a million nodes. The tree will have a height of between 18 and 30 for a billion nodes. Bottom line. Guaranteed logarithmic performance for search and insert. 74

75 T implementations: summary implementation guarantee search insert delete search hit average case insert delete ordered ops? key interface sequential search (unordered list) N N N N N N equals() binary search (ordered array) log N N N log N N N compareto() BT N N N log N log N N compareto() 2 3 tree log N log N log N log N log N log N compareto() but hidden constant c is large (depends upon implementation) 75

76 2 3 tree: implementation? Direct implementation is complicated, because: Maintaining multiple node types is cumbersome. Need multiple compares to move down tree. Need to move back up the tree to split 4-nodes. Large number of cases for splitting. fantasy code public void put(key key, Value val) { Node x = root; Beautiful while (x.getthecorrectchild(key) algorithms are not always!= null) the most useful. } { x = x.getthecorrectchildkey(); Donald Knuth if (x.is4node()) x.split(); } if (x.is2node()) x.make3node(key, val); else if (x.is3node()) x.make4node(key, val); Bottom line. Could do it, but there's a better way. 76

77 3.3 BLNCD C T 2 3 search trees red black BTs lgorithms OBT DGWICK KVIN WYN

78 ow to implement 2 3 trees with binary trees? Challenge. ow to represent a 3 node? pproach 1. egular BT. No way to tell a 3-node from a 2-node. Cannot map from BT back to 2 3 tree. pproach 2. egular BT with red "glue" nodes. Wastes space, wasted link. Code probably messy. pproach 3. egular BT with red "glue" links. Widely used in practice. rbitrary restriction: red links lean left. 78

79 Left-leaning red black BTs (Guibas-edgewick 1979 and edgewick 2007) 3-node lack tree 2. Use "internal"mleft-leaning links as "glue" 3 nodes. greater less for between J C than a 3-node L a b less than a P b between a and b a M J C less than a between a and b a less than a M X C L P between a and b black links connect 2-nodes and 3-nodes red links "glue" nodes within a 3-node red black tree M J J greater than b ncoding a 3-node with two 2-nodes connected by a left-leaning red link greater than b ncoding a 3-node with two 2-nodes connected by a left-leaning red link ee larger key is root greater than b P L than b a and b X b ontal red links a b 1. epresent 2 3 tree as a BT. C X P L X 2 3 tree corresponding red black BT correspondence between red-black and 2-3 trees horizontal red links M 79

80 Left-leaning red black BTs: 1 1 correspondence with 2 3 trees Key property. 1 1 correspondence between 2 3 and LLB. red black tree J M C L P X horizontal red links C J L M P X 2-3 tree M J C L P X 80

81 n equivalent definition BT such that: No node has two red links connected to it. very path from root to null link has the same number of black links. ed links lean left. "perfect black balance" ck tree M J C L P X 81

82 earch implementation for red black BTs Observation. earch is the same as for elementary BT (ignore color). public Value get(key key) { Node x = root; while (x!= null) { int cmp = key.compareto(x.key); if (cmp < 0) x = x.left; else if (cmp > 0) x = x.right; else if (cmp == 0) return x.val; } return null; } but runs faster because of better balance ck tree emark. Most other ops (e.g., floor, iteration, selection) are also identical. C J L M P X 82

83 ed black BT representation ach node is pointed to by precisely one link (from its parent) can encode color of links in nodes. private static final boolean D = true; private static final boolean BLCK = false; private class Node { Key key; Value val; Node left, right; boolean color; // color of parent link } h.left.color is D C D h G J h.right.color is BLCK private boolean ised(node x) { if (x == null) return false; return x.color == D; } null links are black 83

84 Insertion into a LLB tree: overview Basic strategy. Maintain 1 1 correspondence with 2 3 trees. During internal operations, maintain: ymmetric order. Perfect black balance. [ but not necessarily color invariants ] right-leaning red link two red children (a temporary 4-node) ow? pply elementary red black BT operations: rotation and color flip. left-left red (a temporary 4-node) left-right red (a temporary 4-node) 84

85 lementary red black BT operations Left rotation. Orient a (temporarily) right-leaning red link to lean left. rotate left (before) less than h between and x greater than private Node rotateleft(node h) { assert ised(h.right); Node x = h.right; h.right = x.left; x.left = h; x.color = h.color; h.color = D; return x; } Invariants. Maintains symmetric order and perfect black balance. 85

86 lementary red black BT operations Left rotation. Orient a (temporarily) right-leaning red link to lean left. rotate left (after) less than h between and x greater than private Node rotateleft(node h) { assert ised(h.right); Node x = h.right; h.right = x.left; x.left = h; x.color = h.color; h.color = D; return x; } Invariants. Maintains symmetric order and perfect black balance. 86

87 lementary red black BT operations ight rotation. Orient a left-leaning red link to (temporarily) lean right. rotate right (before) x less than between and h greater than private Node rotateight(node h) { assert ised(h.left); Node x = h.left; h.left = x.right; x.right = h; x.color = h.color; h.color = D; return x; } Invariants. Maintains symmetric order and perfect black balance. 87

88 lementary red black BT operations ight rotation. Orient a left-leaning red link to (temporarily) lean right. rotate right (after) less than x between and h greater than private Node rotateight(node h) { assert ised(h.left); Node x = h.left; h.left = x.right; x.right = h; x.color = h.color; h.color = D; return x; } Invariants. Maintains symmetric order and perfect black balance. 88

89 lementary red black BT operations Color flip. ecolor to split a (temporary) 4-node. flip colors less (before) between h between greater private void flipcolors(node h) { assert!ised(h); assert ised(h.left); assert ised(h.right); h.color = D; h.left.color = BLCK; h.right.color = BLCK; } than and and than Invariants. Maintains symmetric order and perfect black balance. 89

90 lementary red black BT operations Color flip. ecolor to split a (temporary) 4-node. flip colors (after) less between h between greater private void flipcolors(node h) { assert!ised(h); assert ised(h.left); assert ised(h.right); h.color = D; h.left.color = BLCK; h.right.color = BLCK; } than and and than Invariants. Maintains symmetric order and perfect black balance. 90

91 Insertion into a LLB tree Warmup 1. Insert into a tree with exactly 1 node. left a b b root search ends at this null link root red link to new node containing a converts 2-node to 3-node right a a a search ends at this null link attached new node with red link b b root root rotated left to make a legal 3-node 91

92 Insertion into a LLB tree Case 1. Insert into a 2-node at the bottom. Do standard BT insert; color new link red. If new red link is a right link, rotate left. to maintain symmetric order and perfect black balance to fix color invariants insert C add new node here right link red so rotate left C C C Insert into a 2-node 92

93 Insertion into a LLB tree Warmup 2. Insert into a tree with exactly 2 nodes. larger a a a b b b c search ends at this null link attached new node with red link colors flipped to black c smaller a a a b b search ends at this null link b b c c attached new node with red link c rotated right b colors flipped to black c ed between a a a a a b b b b c search ends at this null link c c attached new node with red link rotated left c rotated right colors flipped to black c 93

94 Insertion into a LLB tree Case 2. Insert into a 3-node at the bottom. Do standard BT insert; color new link red. otate to balance the 4-node (if needed). Flip colors to pass red link up one level. otate to make lean left (if needed). to maintain symmetric order and perfect black balance to fix color invariants inserting C add new node here two lefts C both children red so flip colors node here C two lefts in a row so rotate right right link red so rotate left C C Insert into a 3-node right link red 94

95 Insertion into a LLB tree: passing red links up the tree Case 2. Insert into a 3-node at the bottom. inserting P Do standard BT insert; color new link red. otate to balance the 4-node (if needed). Flip colors to pass red link up one Clevel. otate to make lean left (if needed). inserting P epeat case 1 or case 2 up the tree (if needed). C M add new node here C right link red so rotate left C M M M P two lefts in a row so rotate right C M P inserting P C right link red both so rotate children left red so flip colors C C M M M M P add new node here P two lefts in a row so rotate right M P both children red so flip colors both children red so flip colors C M add new node here both children red so flip colors right link red C M both children so rotate left P red so to maintain symmetric order flip colors and perfect black balance C Mright link red so P rotate left two lefts in a rowto so rotate right C fix color M invariants P C C C both children red so flip colors C M M two lefts in a row so rotate right M C P Passing a red link up C the tree P P P M Passing a red link up the tree P P both children red so flip colors M M P 95

96 ed-black BT construction demo insert

97 ed-black BT construction demo insert

98 ed-black BT construction demo insert

99 ed-black BT construction demo insert two left reds in a row (rotate right)

100 ed-black BT construction demo both children red (flip colors)

101 ed-black BT construction demo both children red (flip colors)

102 ed-black BT construction demo red-black BT

103 ed-black BT construction demo red-black BT

104 ed-black BT construction demo insert

105 ed-black BT construction demo red-black BT

106 ed-black BT construction demo red-black BT

107 ed-black BT construction demo insert C C

108 ed-black BT construction demo right link red (rotate left) C

109 ed-black BT construction demo red-black BT C

110 ed-black BT construction demo red-black BT C

111 ed-black BT construction demo red-black BT C

112 ed-black BT construction demo insert C

113 ed-black BT construction demo two left reds in a row (rotate right) C

114 ed-black BT construction demo both children red (flip colors) C

115 ed-black BT construction demo right link red (rotate left) C

116 ed-black BT construction demo red-black BT C

117 ed-black BT construction demo red-black BT C

118 ed-black BT construction demo red-black BT C

119 ed-black BT construction demo insert X C X

120 ed-black BT construction demo insert X right link red (rotate left) C X

121 ed-black BT construction demo red-black BT X C

122 ed-black BT construction demo red-black BT X C

123 ed-black BT construction demo red-black BT X C

124 ed-black BT construction demo insert M X C M

125 ed-black BT construction demo insert M right link red (rotate left) X C M

126 ed-black BT construction demo red-black BT X C M

127 ed-black BT construction demo insert P two red children (flip colors) X C M P

128 ed-black BT construction demo right link red (rotate left) X C M P

129 ed-black BT construction demo two left reds in a row (rotate right) M X P C

130 ed-black BT construction demo two red children (flip colors) M C P X

131 ed-black BT construction demo two red children (flip colors) M C P X

132 ed-black BT construction demo red-black BT M C P X

133 ed-black BT construction demo red-black BT M C P X

134 ed-black BT construction demo red-black BT M C P X

135 Insertion into a LLB tree: Java implementation ame code for all cases. ight child red, left child black: rotate left. Left child, left-left grandchild red: rotate right. Both children red: flip colors. h h left rotate h right rotate flip colors private Node put(node h, Key key, Value val) { if (h == null) return new Node(key, val, D); int cmp = key.compareto(h.key); if (cmp < 0) h.left = put(h.left, key, val); else if (cmp > 0) h.right = put(h.right, key, val); else if (cmp == 0) h.val = val; Passing a red link up a red-black tree insert at bottom (and color it red) if (ised(h.right) &&!ised(h.left)) h = rotateleft(h); if (ised(h.left) && ised(h.left.left)) h = rotateight(h); if (ised(h.left) && ised(h.right)) flipcolors(h); lean left balance 4-node split 4-node } return h; only a few extra lines of code provides near-perfect balance 135

136 Insertion into a LLB tree: visualization 255 insertions in ascending order 136

137 Insertion into a LLB tree: visualization 255 insertions in descending order 137

138 Insertion into a LLB tree: visualization 255 random insertions 138

139 Balance in LLB trees Proposition. eight of tree is 2 lg N in the worst case. Pf. Black height = height of corresponding 2 3 tree lg N. Never two red links in-a-row. Property. eight of tree is ~ 1.0 lg N in typical applications. 139

140 T implementations: summary implementation guarantee search insert delete search hit average case insert delete ordered ops? key interface sequential search (unordered list) N N N N N N equals() binary search (ordered array) log N N N log N N N compareto() BT N N N log N log N N compareto() 2 3 tree log N log N log N log N log N log N compareto() red black BT log N log N log N log N log N log N compareto() hidden constant c is small (at most 2 lg N compares) 140

141 ed black BTs in the wild Common sense. ixth sense. Together they're the FBI's newest team. 141

142 ed black BTs in the wild 142

143 War story: why red black? Xerox PC innovations. [1970s] lto. GUI. thernet. malltalk. InterPress. Laser printing. Bitmapped display. WYIWYG text editor.... Xerox lto DIClIOlV1TIC FUl\l\V()K Fon BLNCD T Leo J. Guibas.Xerox Palo lto esearch Center, Palo lto, California, and Carnegie-fellon University and obert edgewick* Program in Computer cience Brown University Providence,. I. BTUCT I() this paper we present a uniform framework for the implementation and study of halanced tree algorithms. \Ve show how to imhcd in this the way down towards a leaf. s we will see, this has a number of significant advantages ovcr the older methods. We shall cxamine a numhcr of variations on a common theme and exhibit full implementations which are notable for their brcvity. One imp1cn1entation is exatnined carefully, and some properties about its 143

144 War story: red black BTs Telephone company contracted with database provider to build real-time database to store customer information. Database implementation. ed Black BT. xceeding height limit of 80 triggered error-recovery process. xtended telephone service outage. show allow for for up to 2 40 keys Main cause = height bound exceeded! Telephone company sues database provider. Legal testimony: did not rebalance BT during delete If implemented properly, the height of a red black BT with N keys is at most 2 lg N. expert witness 144

145 File system model Page. Contiguous block of data (e.g., a 4,096-byte chunk). Probe. First access to a page (e.g., from disk to memory). slow fast Property. Time required for a probe is much larger than time to access data within a page. Cost model. Number of probes. Goal. ccess data using minimum number of probes. 145

146 B-trees (Bayer-McCreight, 1972) B-tree. Generalize 2 3 trees by allowing up to M keys per node. t least M / 2 keys in all nodes (except root). very path from root to leaf has same number of links. choose M as large as possible so that M keys fit in a page (M = 1,024 is typical) G P U C D F I J K L O Q T V W X Y Z a B-tree (M = 6) 146

147 earch in a B-tree tart at root. Check if node contains key. Otherwise, find interval for search key and take corresponding link. could use binary search (but all ops are considered free) G P U C D F I J K L O Q T V W X Y Z a B-tree (M = 6) 147

148 Insertion in a B-tree earch for new key. Insert at bottom. plit nodes with M + 1 keys on the way back up the B-tree (moving middle key to parent). G P U C D F I J K L O Q T V W X Y Z a B-tree (M = 6) 148

149 Balance in B-tree Proposition. search or an insertion in a B-tree of order M with N keys requires between ~ log M N and ~ log M/2 N probes. Pf. ll nodes (except possibly root) have between M / 2 and M keys. In practice. Number of probes is at most 4. M = 1024; N = 62 billion log M/2 N 4 149

150 Balanced search trees: quiz 3 What of the following does the B in B-tree not mean?. Bayer B. Balanced C. Binary D. Boeing. I don't know. the more you think about what the B in B-trees could mean, the more you learn about B-trees and that is good. udolph Bayer 150

151 Balanced trees in the wild ed Black trees are widely used as system symbol tables. Java: java.util.treemap, java.util.treeet. C++ TL: map, multimap, multiset. Linux kernel: completely fair scheduler, linux/rbtree.h. macs: conservative stack scanning. B-tree cousins. B+ tree, B*tree, B# tree, B-trees (and cousins) are widely used for file systems and databases. Windows: NTF. Mac: F, F+. Linux: eiserf, XF, xt3f, JF, BTF. Databases: OCL, DB2, ING, QL, PostgreQL. 151