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Sec. 13.2 Balanced Trees 437that is found to be unbalanced. Deletion is similar; however, consideration forunbalanced nodes must begin at the level of the deletemin operation.Example 13.3 In Figure 13.4 (b), the bottom-most unbalanced node hasvalue 7. The excess node (with value 5) is in the right subtree of the leftchild of 7, so we have an example of Case 2. This requires a double rotationto fix. After the rotation, 5 becomes the left child of 24, 2 becomes the leftchild of 5, and 7 becomes the right child of 5.13.2.2 The Splay TreeLike the AVL tree, the splay tree is not actually a distinct data structure, but ratherreimplements the BST insert, delete, and search methods to improve the performanceof a BST. The goal of these revised methods is to provide guarantees on thetime required by a series of operations, thereby avoiding the worst-case linear timebehavior of standard BST operations. No single operation in the splay tree is guaranteedto be efficient. Instead, the splay tree access rules guarantee that a seriesof m operations will take O(m log n) time for a tree of n nodes whenever m ≥ n.Thus, a single insert or search operation could take O(n) time. However, m suchoperations are guaranteed to require a total of O(m log n) time, for an average costof O(log n) per access operation. This is a desirable performance guarantee for anysearch-tree structure.Unlike the AVL tree, the splay tree is not guaranteed to be height balanced.What is guaranteed is that the total cost of the entire series of accesses will becheap. Ultimately, it is the cost of the series of operations that matters, not whetherthe tree is balanced. Maintaining balance is really done only for the sake of reachingthis time efficiency goal.The splay tree access functions operate in a manner reminiscent of the moveto-frontrule for self-organizing lists from Section 9.2, and of the path compressiontechnique for managing parent-pointer trees from Section 6.2. These accessfunctions tend to make the tree more balanced, but an individual access will notnecessarily result in a more balanced tree.Whenever a node S is accessed (e.g., when S is inserted, deleted, or is the goalof a search), the splay tree performs a process called splaying. Splaying moves Sto the root of the BST. When S is being deleted, splaying moves the parent of S tothe root. As in the AVL tree, a splay of node S consists of a series of rotations.A rotation moves S higher in the tree by adjusting its position with respect to itsparent and grandparent. A side effect of the rotations is a tendency to balance thetree. There are three types of rotation.A single rotation is performed only if S is a child of the root node. The singlerotation is illustrated by Figure 13.7. It basically switches S with its parent in a
438 Chap. 13 Advanced Tree StructuresPSSCAPA B B C(a)(b)Figure 13.7 Splay tree single rotation. This rotation takes place only whenthe node being splayed is a child of the root. Here, node S is promoted to theroot, rotating with node P. Because the value of S is less than the value of P,P must become S’s right child. The positions of subtrees A, B, and C are alteredas appropriate to maintain the BST property, but the contents of these subtreesremains unchanged. (a) The original tree with P as the parent. (b) The tree aftera rotation takes place. Performing a single rotation a second time will return thetree to its original shape. Equivalently, if (b) is the initial configuration of the tree(i.e., S is at the root and P is its right child), then (a) shows the result of a singlerotation to splay P to the root.way that retains the BST property. While Figure 13.7 is slightly different fromFigure 13.5, in fact the splay tree single rotation is identical to the AVL tree singlerotation.Unlike the AVL tree, the splay tree requires two types of double rotation. Doublerotations involve S, its parent (call it P), and S’s grandparent (call it G). Theeffect of a double rotation is to move S up two levels in the tree.The first double rotation is called a zigzag rotation. It takes place when eitherof the following two conditions are met:1. S is the left child of P, and P is the right child of G.2. S is the right child of P, and P is the left child of G.In other words, a zigzag rotation is used when G, P, and S form a zigzag. Thezigzag rotation is illustrated by Figure 13.8.The other double rotation is known as a zigzig rotation. A zigzig rotation takesplace when either of the following two conditions are met:1. S is the left child of P, which is in turn the left child of G.2. S is the right child of P, which is in turn the right child of G.Thus, a zigzig rotation takes place in those situations where a zigzag rotation is notappropriate. The zigzig rotation is illustrated by Figure 13.9. While Figure 13.9appears somewhat different from Figure 13.6, in fact the zigzig rotation is identicalto the AVL tree double rotation.
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Data Structures and AlgorithmAnalys
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viContents6.1 General Tree Definiti
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viiiContents10.5.2 B-Tree Analysis
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PrefaceWe study data structures so
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PrefacexvA sophomore-level class wh
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Prefacexviithe form ofcalls to meth
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PrefacexixFor the second edition, I
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1Data Structures and AlgorithmsHow
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.2 Abstract Data Types and Da
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Sec. 1.3 Design Patterns 13that tra
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Sec. 1.4 Problems, Algorithms, and
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Sec. 1.5 Further Reading 19vides po
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Sec. 1.6 Exercises 21than 10,000 of
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2Mathematical PreliminariesThis cha
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Sec. 2.1 Sets and Relations 25A seq
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Sec. 2.3 Logarithms 29Unfortunately
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3Algorithm AnalysisHow long will it
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Sec. 3.1 Introduction 55efficient.
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Sec. 3.1 Introduction 57n! 2 n2n 25
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Sec. 3.10 Speeding Up Your Programs
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Sec. 3.13 Exercises 853.13 Exercise
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Sec. 3.13 Exercises 87(f) sum = 0;f
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4Lists, Stacks, and QueuesIf your p
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Sec. 4.1 Lists 95list ADT. Interfac
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Sec. 4.1 Lists 97for (L.moveToStart
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Sec. 4.1 Lists 99public void moveTo
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Sec. 4.1 Lists 101/** Singly linked
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Sec. 4.1 Lists 105/** Set curr at l
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Sec. 4.3 Queues 125/** Queue ADT */
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Sec. 4.3 Queues 127frontfront20 512
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Sec. 4.3 Queues 129/** Array-based
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Sec. 4.4 Dictionaries 135/** Contai
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Sec. 4.4 Dictionaries 137}/** @retu
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146 Chap. 5 Binary TreesABCDEFGHIFi
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148 Chap. 5 Binary TreesAny number
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150 Chap. 5 Binary Trees/** ADT for
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152 Chap. 5 Binary Treestends to be
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154 Chap. 5 Binary Trees5.3 Binary
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156 Chap. 5 Binary TreesABCDEFGHIFi
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158 Chap. 5 Binary Treesto its call
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160 Chap. 5 Binary Trees5.3.2 Space
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162 Chap. 5 Binary Trees012345 678
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164 Chap. 5 Binary Trees12037422442
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166 Chap. 5 Binary Trees37244273240
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168 Chap. 5 Binary Treessubroot105
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170 Chap. 5 Binary Treesin the case
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172 Chap. 5 Binary Treessynonymous
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174 Chap. 5 Binary Trees/** Heapify
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176 Chap. 5 Binary TreesRH1H2Figure
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178 Chap. 5 Binary Trees5.6 Huffman
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180 Chap. 5 Binary TreesLetter C D
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182 Chap. 5 Binary Trees/** Huffman
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184 Chap. 5 Binary Trees/** Build a
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186 Chap. 5 Binary Treesa reverse p
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188 Chap. 5 Binary Trees18 bits, mo
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190 Chap. 5 Binary Trees5.12 Write
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192 Chap. 5 Binary Trees(c) What is
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194 Chap. 5 Binary Trees5.7 Complet
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196 Chap. 6 Non-Binary TreesRootRAn
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198 Chap. 6 Non-Binary TreesRABCD E
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200 Chap. 6 Non-Binary Trees/** Gen
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204 Chap. 6 Non-Binary Treesditiona
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206 Chap. 6 Non-Binary Treeslence o
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208 Chap. 6 Non-Binary TreesR’RA
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210 Chap. 6 Non-Binary TreesRRABABC
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212 Chap. 6 Non-Binary Trees(a)(b)F
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7Internal SortingWe sort many thing
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Sec. 7.10 Further Reading 257• Eq
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Sec. 7.11 Exercises 259algorithm to
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308 Chap. 9 Searchingrecords by exp
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324 Chap. 9 Searching/** Insert rec
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328 Chap. 9 SearchingExample 9.9 Co
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332 Chap. 9 SearchingThe cost for a
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10IndexingMany large-scale computin
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Sec. 10.4 2-3 Trees 351/** 2-3 tree
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Sec. 10.5 B-Trees 355private TTNode
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Sec. 10.6 Further Reading 365at mos
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PART IVAdvanced Data Structures369
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372 Chap. 11 Graphs04123147123(a)(b
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374 Chap. 11 Graphs0 1 2 3 40201 11
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376 Chap. 11 Graphstime. This is a
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378 Chap. 11 GraphsIt is reasonably
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380 Chap. 11 Graphs}/** @return an
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382 Chap. 11 Graphs}/** Set the wei
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384 Chap. 11 GraphsABABCCDDFFEE(a)(
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488 Chap. 15 Lower Boundsat least o
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490 Chap. 15 Lower BoundsABCGDEFFig
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492 Chap. 15 Lower BoundsProof 1: T
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494 Chap. 15 Lower BoundsFigure 15.
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496 Chap. 15 Lower BoundsTo minimiz
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498 Chap. 15 Lower BoundsThis recur
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500 Chap. 15 Lower BoundsFigure 15.
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502 Chap. 15 Lower Boundsnot only t
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504 Chap. 15 Lower Bounds1234Figure
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506 Chap. 15 Lower Bounds(a) Assume
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16Patterns of AlgorithmsThis chapte
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Sec. 16.2 Randomized Algorithms 515
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Sec. 16.3 Numerical Algorithms 523
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Sec. 16.3 Numerical Algorithms 529o
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Sec. 16.6 Projects 533value depth5
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536 Chap. 17 Limits to ComputationT
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540 Chap. 17 Limits to ComputationS
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542 Chap. 17 Limits to Computationa
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544 Chap. 17 Limits to Computation3
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548 Chap. 17 Limits to Computationp
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550 Chap. 17 Limits to ComputationE
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552 Chap. 17 Limits to Computation1
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554 Chap. 17 Limits to ComputationM
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556 Chap. 17 Limits to Computationw
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558 Chap. 17 Limits to Computationx
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560 Chap. 17 Limits to Computationt
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562 Chap. 17 Limits to Computationm
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564 Chap. 17 Limits to Computation(
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Bibliography[AG06] Ken Arnold and J
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BIBLIOGRAPHY 569[GI91] Zvi Galil an
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BIBLIOGRAPHY 571[SJH93] Clifford A.
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Index80/20 rule, 309, 333abstract d
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INDEX 575image space, 430key space,
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INDEX 577building, 172-177for memor
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INDEX 579octree, 451Ω notation, 65
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INDEX 581comparing algorithms, 224-
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