The skip list is an elegant dictionary data structure that is com- monly deployed in RAM. A skip list with N elements supports searches, inserts, and deletes in O(logN) operations with high probability (w.h.p.) and range queries returning K elements in O(log N + K) operations w.h.p.
A seemingly natural way to generalize the skip list to external memory with block size B is to “promote” with probability 1/B, rather than 1/2. However, there are practical and theoretical obsta- cles to getting the skip list to retain its efficient performance, space bounds, and high-probability guarantees.
We give an external-memory skip list that achieves write- optimized bounds. That is, for 0 < ε < 1, range queries take O(logBε N + K/B) I/Os w.h.p. and insertions and deletions take O((logBε N)/B1−ε) amortized I/Os w.h.p.
Our write-optimized skip list inherits the virtue of simplicity from RAM skip lists. Moreover, it matches or beats the asymptotic bounds of prior write-optimized data structures such as Bε trees or LSM trees. These data structures are deployed in high-performance databases and file systems.
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Cross-Referenced Dictionaries and the Limits of Write Optimization
Dictionaries remain the most well studied class of data structures. A dictionary supports insertions, deletions, membership queries, and usually successor, predecessor, and extract-min. In a RAM, all such operations take O(log n) time on n elements.
Dictionaries are often cross-referenced as follows. Consider a set of tuples {〈ai,bi,ci…〉}. A database might include more than one dictionary on such a set, for example, one indexed on the a ‘s, another on the b‘s, and so on. Once again, in a RAM, inserting into a set of L cross-referenced dictionaries takes O(L log n) time, as does deleting.
The situation is more interesting in external memory. On a Disk Access Machine (DAM), B-trees achieve O(logB N) I/Os for insertions and deletions on a single dictionary and K-element range queries take optimal O(logB N + K/B) I/Os. These bounds are also achievable by a B-tree on cross-referenced dictionaries, with a slowdown of an L factor on insertion and deletions.
In recent years, both the theory and practice of external- memory dictionaries has been revolutionized by write- optimization techniques. A dictionary is write optimized if it is close to a B-tree for query time while beating B-trees on insertions. The best (and optimal) dictionaries achieve a substantially improved insertion and deletion cost of amortized I/Os on a single dictionary while maintaining optimal O(log1+B∊ N + K/B)- I/O range queries.
Although write optimization still helps for insertions into cross-referenced dictionaries, its value for deletions would seem to be greatly reduced. A deletion into a cross- referenced dictionary only specifies a key a. It seems to be necessary to look up the associated values b, c … in order to delete them from the other dictionaries. This takes Ω(logB N) I/Os, well above the per-dictionary write-optimization budget of So the total deletion cost is
In short, for deletions, write optimization offers an advantage over B-trees in that L multiplies a lower order term, but when L = 2, write optimization seems to offer no asymptotic advantage over B-trees. That is, no known query- optimal solution for pairs of cross-referenced dictionaries seem to beat B-trees for deletions.
In this paper, we show a lower bound establishing that a pair of cross-referenced dictionaries that are optimal for range queries and that supports deletions cannot match the write optimization bound available to insert-only dictionaries.
This result thus establishes a limit to the applicability of write-optimization techniques on which many new databases and file systems are based.
Read More: http://epubs.siam.org/doi/10.1137/1.9781611974782.99
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- Award ID(s):
- 1637458
- PAR ID:
- 10027801
- Date Published:
- Journal Name:
- Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611974782.99
