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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 266 (14 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
Representing trees of higher degree
 ALGORITHMICA
, 2005
"... This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or kary tries), where each node has k ..."
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Cited by 96 (16 self)
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This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or kary tries), where each node has k slots, labelled {1,..., k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time. For ordinal trees we support the operations of finding the degree, parent, ith child and subtree size. For cardinal trees the structure also supports finding the child labeled i of a given node apart from the ordinal tree operations. These representations also provide a mapping from the n nodes of the tree onto the integers {1,..., n}, giving unique labels to the nodes of the tree. This labelling can be used to store satellite information with the nodes efficiently.
Membership in constant time and almostminimum space
 SIAM Journal on Computing
, 1999
"... Abstract. This paper deals with the problem of storing a subset of elements from the bounded universeM = {0,...,M−1} so that membership queries can be performed efficiently. In particular, we introduce a data structure to represent a subset of N elements of M in a number of bits close to the informa ..."
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Cited by 73 (3 self)
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Abstract. This paper deals with the problem of storing a subset of elements from the bounded universeM = {0,...,M−1} so that membership queries can be performed efficiently. In particular, we introduce a data structure to represent a subset of N elements of M in a number of bits close to the informationtheoretic minimum, B = lg
Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions
 Algorithmica
, 1996
"... Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, so ..."
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Cited by 65 (6 self)
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Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given nsubset of {1,..., m}.
Splitters and nearoptimal derandomization
"... We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of lengt ..."
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Cited by 64 (1 self)
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We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2k configurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixedsubgraph finding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits.
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 63 (10 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.
Monotone Minimal Perfect Hashing: Searching a Sorted Table with O(1) Accesses
"... A minimal perfect hash function maps a set S of n keys into the set { 0, 1,..., n − 1} bijectively. Classical results state that minimal perfect hashing is possible in constant time using a structure occupying space close to the lower bound of log e bits per element. Here we consider the problem of ..."
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Cited by 35 (11 self)
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A minimal perfect hash function maps a set S of n keys into the set { 0, 1,..., n − 1} bijectively. Classical results state that minimal perfect hashing is possible in constant time using a structure occupying space close to the lower bound of log e bits per element. Here we consider the problem of monotone minimal perfect hashing, in which the bijection is required to preserve the lexicographical ordering of the keys. A monotone minimal perfect hash function can be seen as a very weak form of index that provides ranking just on the set S (and answers randomly outside of S). Our goal is to minimise the description size of the hash function: we show that, for a set S of n elements out of a universe of 2 w elements, O(n log log w) bits are sufficient to hash monotonically with evaluation time O(log w). Alternatively, we can get space O(n log w) bits with O(1) query time. Both of these data structures improve a straightforward construction with O(n log w) space and O(log w) query time. As a consequence, it is possible to search a sorted table with O(1) accesses to the table (using additional O(n log log w) bits). Our results are based on a structure (of independent interest) that represents a trie in a very compact way, but admits errors. As a further application of the same structure, we show how to compute the predecessor (in the sorted order of S) of an arbitrary element, using O(1) accesses in expectation and an index of O(n log w) bits, improving the trivial result of O(nw) bits. This implies an efficient index for searching a blocked memory.
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 29 (6 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.
Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension
, 1993
"... Given d, m and epsilon, we deterministically produce a sequence of points S that hits every combinatorial rectangle in [m]^d of volume at least epsilon. Both the running time of the algorithm and S are polynomial in m log(d)/epsilon. This algorithm has applications to deterministic constructions o ..."
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Cited by 27 (8 self)
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Given d, m and epsilon, we deterministically produce a sequence of points S that hits every combinatorial rectangle in [m]^d of volume at least epsilon. Both the running time of the algorithm and S are polynomial in m log(d)/epsilon. This algorithm has applications to deterministic constructions of small sample spaces for general multivalued random variables. 1 Introduction The problem we consider in this paper is motivated by the following basic witness finding problem: design an efficient algorithm that on Hebrew University, Computer Science Department, Jerusalem Israel. Research partially done while visiting the International Computer Science Institute. y International Computer Science Institute and UC Berkeley. Research partially supported by NSF Grant CCR9016468 and grant No. 8900312 from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel. z Department of Mathematics, Rutgers University and Department of CSE, UCSD. Supported in part by NSF under grant C...