File Maintenance: When in Doubt, Change the Layout!

Bender, Michael A.; Fineman, Jeremy T.; Gilbert, Seth; Kopelowitz, Tsvi; Montes, Pablo

doi:10.1137/1.9781611974782.98

Cited by 8 publications

(8 citation statements)

References 21 publications

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“…These assumptions can easily be removed by rounding all of the quantities to powers of four, and performing the analysis using the rounded quantities. 17 The grid G is parameterized so that the expected number of blue dots (resp. red dots) in each cell is exactly 1.…”

Section: Bounds On Insertion Surplusmentioning

confidence: 99%

“…Relationship with other data structures. One of the interesting features of our results in that it reveals an unexpected connection between the linear probing and several other problems in data structures (e.g., list labeling [26], file maintenance [17,64,65,[148][149][150], cache-oblivious and locality-preserving B-trees [12,13,23], and even sorting [16]). Solutions to these problems all share a commonality, which is that they strategically leave extra space between elements of a data structure in order to enable fast modifications.…”

Section: Related Workmentioning

confidence: 99%

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Linear Probing Revisited: Tombstones Mark the Death of Primary Clustering

Bender,

Kuszmaul,

Kuszmaul

2021

Preprint

Self Cite

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First introduced in 1954, the linear-probing hash table is among the oldest data structures in computer science, and thanks to its unrivaled data locality, linear probing continues to be one of the fastest hash tables in practice. It is widely believed and taught, however, that linear probing should never be used at high load factors; this is because of an effect known as primary clustering which causes insertions at a load factor of 1 − 1/x to take expected time Θ(x 2 ) (rather than the intuitive running time of Θ(x)). The dangers of primary clustering, first discovered by Knuth in 1963, have now been taught to generations of computer scientists, and have influenced the design of some of the most widely used hash tables in production.We show that primary clustering is not the foregone conclusion that it is reputed to be. We demonstrate that seemingly small design decisions in how deletions are implemented have dramatic effects on the asymptotic performance of insertions: if these design decisions are made correctly, then even if a hash table operates continuously at a load factor of 1 − Θ(1/x), the expected amortized cost per insertion/deletion is Õ(x). This is because the tombstones left behind by deletions can actually cause an anti-clustering effect that combats primary clustering. Interestingly, these design decisions, despite their remarkable effects, have historically been viewed as simply implementation-level engineering choices.We also present a new variant of linear probing (which we call graveyard hashing) that completely eliminates primary clustering on any sequence of operations: if, when an operation is performed, the current load factor is 1 − 1/x for some x, then the expected cost of the operation is O(x). Thus we can achieve the data locality of traditional linear probing without any of the disadvantages of primary clustering. One corollary is that, in the external-memory model with a data blocks of size B, graveyard hashing offers the following remarkably strong guarantee: at any load factor 1 − 1/x satisfying x = o(B), graveyard hashing achieves 1 + o(1) expected block transfers per operation. In contrast, past external-memory hash tables have only been able to offer a 1 + o(1) guarantee when the block size B is at least Ω(x 2 ).Our results come with actionable lessons for both theoreticians and practitioners, in particular, that welldesigned use of tombstones can completely change the asymptotic landscape of how the linear probing behaves (and even in workloads without deletions).

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Section: Bounds On Insertion Surplusmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Linear Probing Revisited: Tombstones Mark the Death of Primary Clustering

Bender,

Kuszmaul,

Kuszmaul

2021

Preprint

Self Cite

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show abstract

“…Reference [17] remarks that the calibrator tree for rebalancing is actually unnecessary, leading to a form of circular array. The recent paper [8] deals with the problem of dearmortisation, i.e. reducing the (non amortised) worst case complexity of updates, and surveys some of the earlier schemes proposed.…”

Section: Related Workmentioning

confidence: 99%

Packed Memory Arrays - Rewired

Leo

Boncz

2019

2019 IEEE 35th International Conference on Data Engineering (ICDE)

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The physical memory layout of a tree-based index structure deteriorates over time as it sustains more updates; such that sequential scans on the physical level become nonsequential, and therefore slower. Packed Memory Arrays (PMAs) prevent this by managing all data in a sequential sparse array. PMAs have been studied mostly theoretically but suffer from practical problems, as we show in this paper. We study and fix these problems, resulting in an improved data structure: the Rewired Memory Array (RMA). We compare RMA with the main previous PMA implementations as well as state-of-the-art tree index structures and show on a wide variety of data and query distributions that RMA can reach competitive update and point lookup performance, while always providing superior scan performance -close to dense column scans.

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“…Our solution is based on the same approach as our dynamic tree structure, but we employ a different method to maintain the underlying tree. This method is based on the list maintenance problem [8,9,27]. The full details of this result are omitted from this extended abstract but are presented in the Appendix, in Section 8.1.…”

Section: Previous and Related Workmentioning

confidence: 99%

“…When a new pseudo-leaf is inserted, positions of some other pseudo-leaves in B may change; that is, some pseudo-leaves are shifted to the right (or to the left) in B, but their relative order remains unchanged. We can maintain elements in B in such a way that at most log 2 n pseudo-leaves are moved to different positions in B after every insertion [9,27]. For every a i , there is a node ε i such that all pseudo-leaves below ε i are associated to a i .…”

Section: Appendix 81 Alphabetic Codesmentioning

confidence: 99%

Dynamic Trees with Almost-Optimal Access Cost

Golin¹,

Iacono²,

Langerman³

et al. 2018

Preprint

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An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost.In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element e i does not exceed min(log(W/w i ), log n)+O(1) where w i is the number of times e i was accessed and W is the total length of the access sequence.Our techniques can also be used to encode a sequence of m symbols with a dynamic alphabetic code in O(m) time so that the encoding length is bounded by m(H + O(1)), where H is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol.

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File Maintenance: When in Doubt, Change the Layout!

Cited by 8 publications

References 21 publications

Linear Probing Revisited: Tombstones Mark the Death of Primary Clustering

Linear Probing Revisited: Tombstones Mark the Death of Primary Clustering

Packed Memory Arrays - Rewired

Dynamic Trees with Almost-Optimal Access Cost

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