Bounded ordered dictionaries in O(log log N) time and O(n) space

Mehlhorn, Kurt; Näher, Stefan

doi:10.1016/0020-0190(90)90022-p

“…The van Emde Boas tree structure [van Emde Boas et al 1976] supports findNext, insert and delete in O(log log m) time. By using dynamic perfect hashing [Dietzfelbinger et al 1994] to store the elements of the tree, the structure can be implemented with O(n log m) bits, but the time for updates becomes O(log log m) expected amortized time [Mehlhorn and Naher 1990]. To reduce the space, the set can be partitioned into groups of Θ(log n) contiguous elements each (except perhaps the first and last groups which can be smaller).…”

Section: Ordered Integer Setsmentioning

confidence: 99%

Compact dictionaries for variable-length keys and data with applications

Blandford

¹

,

Blelloch

²

2008

ACM Trans. Algorithms

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We consider the problem of maintaining a dynamic dictionary T of keys and associated data for which both the keys and data are bit strings that can vary in length from zero up to the length w of a machine word. We present a data structure for this variable-bit-length dictionary problem that supports constant time lookup and expected amortized constant time insertion and deletion. It uses O(m + 3n − n log 2 n) bits, where n is the number of elements in T , and m is the total number of bits across all strings in T (keys and data). Our dictionary uses an array A[1 . . . n] in which locations store variable-bit-length strings. We present a data structure for this variable-bit-length array problem that supports worst-case constant-time lookups and updates and uses O(m + n) bits, where m is the total number of bits across all strings stored in A.The motivation for these structures is to support applications for which it is helpful to efficiently store short varying length bit strings. We present several applications, including representations for semi-dynamic graphs, order queries on integers sets, cardinal trees with varying cardinality, and simplicial meshes of d dimensions. These results either generalize or simplify previous results.

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“…We assume the keys come from a totally ordered universe U with comparison < and Predecessor(key k) returns the entry with the greatest key less or equal to k. Our solutions are straightforward given the SHI hash structure described in the previous section. If the keys come from a universe U = [m] we can use the variant of the Van Emde Boas structure [23] described by Mehlhorn and Naher [12]. The following theorem follows directly from the use of our SHI hash table in the algorithm described by Mehlhorn and Naher.…”

Section: Implementing Other Shi Data Structuresmentioning

confidence: 99%

Strongly History-Independent Hashing with Applications

Blelloch

¹

,

Golovin

²

2007

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)

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We present a strongly history independent (SHI) hash table that supports search in O(1) worst-case time, and insert and delete in O(1) expected time using O(n) data space. This matches the bounds for dynamic perfect hashing, and improves on the best previous results by Naor and Teague on history independent hashing, which were either weakly history independent, or only supported insertion and search (no delete) each in O(1) expected time.The results can be used to construct many other SHI data structures. We show straightforward constructions for SHI ordered dictionaries: for n keys from {1, . . . , n k } searches take O(log log n) worst-case time and updates (insertions and deletions) O(log log n) expected time, and for keys in the comparison model searches take O(log n) worst-case time and updates O(log n) expected time. We also describe a SHI data structure for the order-maintenance problem. It supports comparisons in O(1) worst-case time, and updates in O(1) expected time. All structures use O(n) data space.

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“…We assume the keys come from a totally ordered universe U with comparison < and Predecessor(key k) returns the entry with the greatest key less or equal to k. Our solutions are straightforward given the SHI hash structure described in the previous section. If the keys come from a universe U = [m] we can use the variant of the Van Emde Boas structure [24] described by Mehlhorn and Naher [12]. The following theorem follows directly from the use of our SHI hash table in the algorithm described by Mehlhorn and Naher.…”

Section: Implementing Other Shi Data Structuresmentioning

confidence: 99%

Strongly History-Independent Hashing with Applications

Blelloch¹,

Golovin²

2007

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)

View full text Add to dashboard Cite

We present a strongly history independent (SHI) hash table that supports search in O(1) worst-case time, and insert and delete in O(1) expected time using O(n) data space. This matches the bounds for dynamic perfect hashing, and improves on the best previous results by Naor and Teague on history independent hashing, which were either weakly history independent, or only supported insertion and search (no delete) each in O(1) expected time.The results can be used to construct many other SHI data structures. We show straightforward constructions for SHI ordered dictionaries: for n keys from {1, . . . , n k } searches take O(log log n) worst-case time and updates (insertions and deletions) O(log log n) expected time, and for keys in the comparison model searches take O(log n) worst-case time and updates O(log n) expected time. We also describe a SHI data structure for the order-maintenance problem. It supports comparisons in O(1) worst-case time, and updates in O(1) expected time. All structures use O(n) data space.

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Bounded ordered dictionaries in O(log log N) time and O(n) space

Cited by 61 publications

References 6 publications

Compact dictionaries for variable-length keys and data with applications

Compact dictionaries for variable-length keys and data with applications

Strongly History-Independent Hashing with Applications

Strongly History-Independent Hashing with Applications

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