Algorithms and Data Structures for External Memory

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11.3 Parent Pointers and Level-Balanced B-trees 93 preferable to BB[α]-trees for purposes of augmenting one-dimensional data structures with range restriction capabilities [354]. 11.3 Parent Pointers and Level-Balanced B-trees It is sometimes useful to augment B-trees with parent pointers. For example, if we represent a total order via the leaves in a B-tree, we can answer order queries such as “Is x
94 Multiway Tree Data Structures purposes, nodes are packed together into disk blocks. Each node in the forest is stored as a node record (which points to the parent’s node record) and a doubly linked list of child records (which point to the node records of the children). There are also pointers between the node record and the list of child records. Every disk block stores only node records or only child records, but all the child records for a given node must be stored in the same block (possibly with child records for other nodes). The advantage of this extra level of indirection is that cuts and concatenates can usually be done in only O(1) I/Os per level of the forest. For example, during a cut, a node record gets split into two, and its list of child nodes is chopped into two separate lists. The parent node must therefore get a new child record to point to the new node. These updates require O(1) I/Os except when there is not enough space in the disk block of the parent’s child records, in which case the block must be split into two, and extra I/Os are needed to update the pointers to the moved child records. The amortized I/O cost, however, is only O(1) per level, since each update creates at most one node record and child record at each level. The other dynamic update operations can be handled similarly. All that remains is to reestablish the global level invariant when a level gets too many nodes as a result of an update. If level h is the lowest such level out of balance, then level h and all the levels above it are reconstructed via a postorder traversal in O(Nh) I/Os so that the new nodes get degree Θ(b) and the invariant is restored. The final trick is to construct the new parent pointers that point from the Θ(Nh−1) = Θ(bNh) node records on level h − 1 to the Θ(Nh) level-h nodes. The parent pointers can be accessed in a blocked manner with respect to the new ordering of the nodes on level h. By sorting, the pointers can be rearranged to correspond to the ordering of the nodes on level h − 1, after which the parent pointer values can be written via a linear scan. The resulting I/O cost is O � Nh + Sort(bNh) +Scan(bNh) � , which can be amortized against the Θ(Nh) updates that have occurred since the last time the level-h invariant was violated, yielding an amortized update cost of O � 1+(b/B)log m n � I/Os per level. Order queries such as “Does leaf x precede leaf y in the total order represented by the tree?” can be answered using O(log B N)
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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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Page 59 and 60: 42 External Sorting and Related Pro
Page 74 and 75: 6 Lower Bounds on I/O In this chapt
Page 76 and 77: 6.1 Permuting 59 in backwards order
Page 78 and 79: 6.2 Lower Bounds for Sorting and Ot
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Page 86 and 87: 8 Batched Problems in Computational
Page 88 and 89: 8.1 Distribution Sweep 71 Goodrich
Page 90 and 91: 8.1 Distribution Sweep 73 the curre
Page 92 and 93: Time (seconds) Time (seconds) 7000
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Page 98 and 99: 9.2 Special Cases 81 that contains
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Page 108 and 109: 11.1 B-trees and Variants 91 “sha
Page 112 and 113: 11.4 Buffer Trees 95 I/Os by follow
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Page 117 and 118: 100 Spatial Data Structures and Ran
Page 136 and 137: 13 Dynamic and Kinetic Data Structu
Page 138 and 139: 13.2 Continuously Moving Items 121
Page 140 and 141: 14 String Processing In this chapte
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Page 144 and 145: 14.3 Suffix Trees and Suffix Arrays
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Page 148 and 149: 15.1 Data Representations and Compr
Page 150 and 151: 15.2 External Memory Compressed Dat
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Page 157 and 158: 140 Dynamic Memory Allocation For s
Page 159 and 160: 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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Algorithms and Data Structures for External Memory

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