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Succinct indexable dictionaries with applications to encoding kary trees and multisets
 In Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
"... We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the ith smallest ele ..."
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Cited by 191 (7 self)
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We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n,m)+ o(n)+O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈ lg ( m) ⌉ n is the minimum number of bits required to store any nelement subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: • an informationtheoretically optimal representation of a kary cardinal tree that supports standard operations in constant time, • a representation of a multiset of size n from {0,...,m − 1} in B(n,m+n) + o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time, and • a representation of a sequence of n nonnegative integers summing up to m in B(n,m + n) + o(n) bits that supports prefix sum queries in constant time. 1
Optimal Bounds for the Predecessor Problem
 In Proceedings of the ThirtyFirst Annual ACM Symposium on Theory of Computing
"... We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed efficiently stored set. Our algorithms are for the unitcost wordlevel RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds ar ..."
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Cited by 63 (0 self)
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We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed efficiently stored set. Our algorithms are for the unitcost wordlevel RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds are proved in a much stronger communication game model, but they apply to the cell probe and RAM models and to both static and dynamic predecessor problems.
LOW REDUNDANCY IN STATIC DICTIONARIES WITH CONSTANT QUERY TIME
 SIAM J. COMPUT.
, 2001
"... A static dictionary is a data structure storing subsets of a finite universe U, answering membership queries. We show that on a unit cost RAM with word size Θ(log U), a static dictionary for nelement sets with constant worst case query time can be obtained using B +O(log log U)+o(n) (U) bits ..."
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Cited by 50 (7 self)
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A static dictionary is a data structure storing subsets of a finite universe U, answering membership queries. We show that on a unit cost RAM with word size Θ(log U), a static dictionary for nelement sets with constant worst case query time can be obtained using B +O(log log U)+o(n) (U) bits of storage, where B = ⌈log2 ⌉ is the minimum number of bits needed to represent all nn element subsets of U.
Low Redundancy in Static Dictionaries with O(1) Worst Case Lookup Time
 IN PROCEEDINGS OF THE 26TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP '99
, 1999
"... A static dictionary is a data structure for storing subsets of a nite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size (log jU j), and show that for nelement subsets, constant worst case query time can be obtained us ..."
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Cited by 21 (5 self)
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A static dictionary is a data structure for storing subsets of a nite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size (log jU j), and show that for nelement subsets, constant worst case query time can be obtained using B +O(log log jU j) + o(n) bits of storage, where B = dlog 2 jUj n e is the minimum number of bits needed to represent all such subsets. For jU j = n log O(1) n the dictionary supports constant time rank queries.
Static Dictionaries on AC^0 RAMs: Query time Θ(,/log n / log log n) is necessary and sufficient
, 1996
"... In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©���������������� ..."
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Cited by 19 (5 self)
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In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©��������������������� of on the time for answering membership queries in a set of � size when reasonable space is used for the data structure storing the set; the upper bound can be obtained using space ������ � �� � ���� �. Several variations of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unitcost assumption: any RAM instruction set which permits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth �������©���������������©���� � , where � is the word size of the machine (and ���© � the size of the universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by Fredman, Komlós and Szemerédi.
Low Redundancy in Dictionaries with O(1) Worst Case Lookup Time
 IN PROC. 26TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP
, 1998
"... A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size ze jU j), and show that for nelement subsets, constant worst case query time can be obtain ..."
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Cited by 18 (0 self)
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A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size ze jU j), and show that for nelement subsets, constant worst case query time can be obtained using B +O(log log jU j) + o(n) bits of storage, where B = dlog jU j e is the minimum number of bits needed to represent all such subsets. The solution for dense subsets uses B + O( jU j log log jU j log jU j ) bits of storage, and supports constant time rank queries. In a dynamic setting, allowing insertions and deletions, our techniques give an O(B) bit space usage.
Lower bounds for static dictionaries on RAMs with bit operations but no multiplication
 In Automata, languages and programming (Paderborn
, 1996
"... . We consider solving the static dictionary problem with n keys from the universe f0; : : : ; m \Gamma 1g on a RAM with direct and indirect addressing, conditional jump, addition, bitwise Boolean operations, and arbitrary shifts (a Practical RAM). For any ffl ? 0, tries yield constant query time us ..."
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Cited by 7 (2 self)
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. We consider solving the static dictionary problem with n keys from the universe f0; : : : ; m \Gamma 1g on a RAM with direct and indirect addressing, conditional jump, addition, bitwise Boolean operations, and arbitrary shifts (a Practical RAM). For any ffl ? 0, tries yield constant query time using space m ffl , provided that n = m o(1) . We show that this is essentially optimal: Any scheme with constant query time requires space m ffl for some ffl ? 0, even if n (log m) 2 . 1 Introduction The static dictionary problem is the following: Given a subset S of size n of the universe U = f0; : : : ; m \Gamma 1g, store it as a data structure OE S in the memory of a unit cost random access machine, using few memory registers, each containing O(logm) bits, so that membership queries "Is x 2 S?" can be answered efficiently for any value of x. The set S can be stored as a sorted table using n memory registers. Then queries can be answered using binary search in O(logn) time. Yao...
Optimal spacetime dictionaries over an unbounded universe with flat implicit trees
, 2003
"... In the classical dictionary problem, a set of n distinct keys over an unbounded and ordered universe is maintained under insertions and deletions of individual keys while supporting search operations. An implicit dictionary has the additional constraint of occupying the space merely required by stor ..."
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Cited by 2 (1 self)
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In the classical dictionary problem, a set of n distinct keys over an unbounded and ordered universe is maintained under insertions and deletions of individual keys while supporting search operations. An implicit dictionary has the additional constraint of occupying the space merely required by storing the n keys, that is, exactly n contiguous words of space in total. All what is known is the starting position of the memory segment hosting the keys, as the rest of the information is implicitly encoded by a suitable permutation of the keys. This paper describes the
at implicit tree, which is the rst implicit dictionary requiring O(log n) time per search and update operation.
Static Dictionaries on ACO RAMs: Query time 0 (,/log n / log log n) is necessary and sufficient*
"... In this paper we consider solutions to the static dictionary problem on ACo RAMs, i.e. random access machines where the only restriction on theJinite instruction set is that all computationcil instructions are in ACO. Our main result is a tight upper and lower bound of 0 (Jlog n / log log n) on the ..."
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In this paper we consider solutions to the static dictionary problem on ACo RAMs, i.e. random access machines where the only restriction on theJinite instruction set is that all computationcil instructions are in ACO. Our main result is a tight upper and lower bound of 0 (Jlog n / log log n) on the time for (answering membership queries in a set of size n when reasonable space is used for the data structure storing the set; the upper bound can be obtained using O(n) space, and the lower bound holds even if we allow space 2 PolYlog *. Several varitztions of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unitcost assumption: any RAM instruction set which purrtits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth R(1og w / log log w), where w is the word size of the machine (and log the size ofthe universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by Fredman, Komlds and Szemerkdi. 1