Algorithms and Data Structures for External Memory

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11 Multiway Tree Data Structures In this chapter, we explore some important search-tree data structures in external memory. An advantage of search trees over hashing methods is that the data items in a tree are sorted, and thus the tree can be used readily for one-dimensional range search. The items in a range [x,y] can be found by searching for x in the tree, and then performing an inorder traversal in the tree from x to y. 11.1 B-trees and Variants Tree-based data structures arise naturally in the online setting, in which the data can be updated and queries must be processed immediately. Binary trees have a host of applications in the (internal memory) RAM model. One primary application is search. Some binary search trees, for example, support lookup queries for a dictionary of N items in O(logN) time, which is optimal in the comparison model of computation. The generalization to external memory, in which data are transferred in blocks of B items and B comparisons can be done per I/O, provides an Ω(log B N) I/O lower bound. These lower bounds depend heavily 89
90 Multiway Tree Data Structures upon the model; we saw in the last chapter that hashing can support dictionary lookup in a constant number of I/Os. In order to exploit block transfer, trees in external memory generally represent each node by a block that can store Θ(B) pointers and data values and can thus achieve Θ(B)-way branching. The well-known balanced multiway B-tree due to Bayer and McCreight [74, 114, 220], is the most widely used nontrivial EM data structure. The degree of each node in the B-tree (with the exception of the root) is required to be Θ(B), which guarantees that the height of a B-tree storing N items is roughly log B N. B-trees support dynamic dictionary operations and one-dimensional range search optimally in linear space using O(log B N) I/Os per insert or delete and O(log B N + z) I/Os per query, where Z = zB is the number of items output. When a node overflows during an insertion, it splits into two half-full nodes, and if the splitting causes the parent node to have too many children and overflow, the parent node splits, and so on. Splittings can thus propagate up to the root, which is how the tree grows in height. Deletions are handled in a symmetric way by merging nodes. Franceschini et al. [167] show how to achieve the same I/O bounds without space for pointers. In the B + -tree variant, pictured in Figure 11.1, all the items are stored in the leaves, and the leaves are linked together in symmetric order to facilitate range queries and sequential access. The internal nodes store only key values and pointers and thus can have a higher branching factor. In the most popular variant of B + -trees, called B*trees, splitting can usually be postponed when a node overflows, by Level 2 Level 1 Leaves Fig. 11.1 B + -tree multiway search tree. Each internal and leaf node corresponds to a disk block. All the items are stored in the leaves; the darker portion of each leaf block indicates its relative fullness. The internal nodes store only key values and pointers, Θ(B) of them per node. Although not indicated here, the leaf blocks are linked together sequentially.
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Algorithms and Data Structures for
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Foundations and Trends R○ in Theo
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Foundations and Trends R○ in Theo
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Preface I first became fascinated a
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Preface xi I would like to thank my
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xiv Contents 6 Lower Bounds on I/O
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1 Introduction The world is drownin
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However, not all memory references
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1.1 Overview 5 Our general goal is
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Table 1.1 Paradigms for I/O efficie
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10 Parallel Disk Model (PDM) spindl
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12 Parallel Disk Model (PDM) D = nu
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14 Parallel Disk Model (PDM) The pr
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16 Parallel Disk Model (PDM) 2.3 Re
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18 Parallel Disk Model (PDM) archit
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3 Fundamental I/O Operations and Bo
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As Table 3.1 indicates, the multipl
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26 Exploiting Locality and Load Bal
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28 Exploiting Locality and Load Bal
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30 External Sorting and Related Pro
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140 Dynamic Memory Allocation For s
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142 External Memory Programming Env
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144 External Memory Programming Env
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146 Conclusions intriguing connecti
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148 Notations and Acronyms lowercas
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150 Notations and Acronyms � �
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152 References Proceedings of the A
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154 References [41] L. Arge, K. H.
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156 References Computer Science, pp
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158 References ACM Conference on In
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160 References [132] F. K. H. A. De
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162 References [165] R. W. Floyd,
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164 References [195] M. R. Henzinge
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166 References [228] K. Küspert,
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168 References [261] D. R. Morrison
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170 References [293] J. T. Robinson
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172 References [325] “TerraServer
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174 References [356] O. Wolfson, P.
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