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Sec. 10.5 B-Trees 3634845 4748 50 52(a)231833101215 18 19 20 2122 23 30 31 4547485052(b)Figure 10.24 Deleting the record with key value 33 from the B + -tree of Figure10.18 via collapsing siblings. (a) The two leftmost leaf nodes merge togetherto form a single leaf. Unfortunately, the parent node now has only one child.(b) Because the left subtree has a spare leaf node, that node is passed to the rightsubtree. The placeholder values of the root and the right internal node are updatedto reflect the changes. Value 23 moves to the root, and old root value 33 moves tothe rightmost internal node./** Delete a record with the given key value, andreturn true if the root underflows */private boolean removehelp(BPNode rt, Key k) {int currec = binaryle(rt.keys(), rt.numrecs(), k);if (rt.isLeaf())if (((BPLeaf)rt).keys()[currec] == k)return ((BPLeaf)rt).delete(currec);else return false;else // Process internal nodeif (removehelp(((BPInternal)rt).pointers(currec),k))// Child will merge if necessaryreturn ((BPInternal)rt).underflow(currec);else return false;}Figure 10.25 Java-like pseudocode for the B + -tree delete algorithm.
364 Chap. 10 Indexingneighboring sibling, if possible. If the sibling is also full, then these two nodes splitinto three. Similarly, when a node underflows, it is combined with its two siblings,and the total reduced to two nodes. Thus, the nodes are always at least two thirdsfull. 210.5.2 B-Tree AnalysisThe asymptotic cost of search, insertion, and deletion of records from B-trees,B + -trees, and B ∗ -trees is Θ(log n) where n is the total number of records in thetree. However, the base of the log is the (average) branching factor of the tree.Typical database applications use extremely high branching factors, perhaps 100 ormore. Thus, in practice the B-tree and its variants are extremely shallow.As an illustration, consider a B + -tree of order 100 and leaf nodes that containup to 100 records. A B + -tree with height one (that is, just a single leaf node) canhave at most 100 records. A B + -tree with height two (a root internal node whosechildren are leaves) must have at least 100 records (2 leaves with 50 records each).It has at most 10,000 records (100 leaves with 100 records each). A B + -tree withheight three must have at least 5000 records (two second-level nodes with 50 childrencontaining 50 records each) and at most one million records (100 second-levelnodes with 100 full children each). A B + -tree with height four must have at least250,000 records and at most 100 million records. Thus, it would require an extremelylarge database to generate a B + -tree of more than height four.The B + -tree split and insert rules guarantee that every node (except perhaps theroot) is at least half full. So they are on average about 3/4 full. But the internalnodes are purely overhead, since the keys stored there are used only by the tree todirect search, rather than store actual data. Does this overhead amount to a significantuse of space? No, because once again the high fan-out rate of the tree structuremeans that the vast majority of nodes are leaf nodes. Recall (from Section 6.4) thata full K-ary tree has approximately 1/K of its nodes as internal nodes. This meansthat while half of a full binary tree’s nodes are internal nodes, in a B + -tree of order100 probably only about 1/75 of its nodes are internal nodes. This means that theoverhead associated with internal nodes is very low.We can reduce the number of disk fetches required for the B-tree even moreby using the following methods. First, the upper levels of the tree can be stored inmain memory at all times. Because the tree branches so quickly, the top two levels(levels 0 and 1) require relatively little space. If the B-tree is only height four, then2 This concept can be extended further if higher space utilization is required. However, the updateroutines become much more complicated. I once worked on a project where we implemented 3-for-4node split and merge routines. This gave better performance than the 2-for-3 node split and mergeroutines of the B ∗ -tree. However, the spitting and merging routines were so complicated that eventheir author could no longer understand them once they were completed!
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Data Structures and AlgorithmAnalys
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ivContents2.6.1 Direct Proof 372.6.
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viContents6.1 General Tree Definiti
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viiiContents10.5.2 B-Tree Analysis
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xContents15.7 Optimal Sorting 50115
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PrefaceWe study data structures so
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PrefacexvA sophomore-level class wh
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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. 1.5 Further Reading 19vides po
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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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4Lists, Stacks, and QueuesIf your p
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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. 4.1 Lists 115/** Insert "it" a
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Sec. 4.2 Stacks 121top1top2Figure 4
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Sec. 4.3 Queues 125/** Queue ADT */
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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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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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212 Chap. 6 Non-Binary Trees(a)(b)F
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7Internal SortingWe sort many thing
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Sec. 12.3 Memory Management 413/**
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13Advanced Tree StructuresThis chap
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14Analysis TechniquesOften it is ea
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486 Chap. 15 Lower Boundsthe concep
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500 Chap. 15 Lower BoundsFigure 15.
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16Patterns of AlgorithmsThis chapte
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536 Chap. 17 Limits to ComputationT
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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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560 Chap. 17 Limits to Computationt
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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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