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19
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 62 (5 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}.
Are bitvectors optimal?
"... ... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must u ..."
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Cited by 54 (7 self)
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... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must use \Omega ( n2ffl2 log(n=ffl) log m) bits of storage. We also
Extracting randomness using few independent sources
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
"... In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over ..."
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Cited by 49 (6 self)
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In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, 1} n, each having minentropy δn. These extractors output n bits, which are 2 −n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with minentropy Ω(log n) and outputs every possible mbit string with positive probability. The main tool we use is a variant of the “steppingup lemma ” used in establishing lower bound
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 49 (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.
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 19 (8 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.
Membership in Constant Time and Minimum Space
 Lecture Notes in Computer Science
, 1994
"... . We investigate the problem of storing a subset of the elements of a boundeduniverse so that searches canbe performed in constant time and the space used is within a constant factor of the minimum required. Initially we focus on the static version of this problem and conclude with an enhancement th ..."
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Cited by 18 (5 self)
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. We investigate the problem of storing a subset of the elements of a boundeduniverse so that searches canbe performed in constant time and the space used is within a constant factor of the minimum required. Initially we focus on the static version of this problem and conclude with an enhancement that permits insertions and deletions. 1 Introduction Given a universal set M = f0; : : : ; M \Gamma 1g and any subset N = fe 1 ; : : : ; e N g the membership problem is to determine whether given query element in M is an element of N . There are two standard approaches to solve this problem: to list all elements of N (e.g. in a hash table) or to list all the answers (e.g. a bit map of size M ). When N is small the former approach comes close to the information theoretic lower bound on the number of bits needed to represent an arbitrary subset of the given size (i.e. a function of both N and M , l lg \Gamma M N \Delta m ). Similarly, when N is large (say ffM ) the later approach is near...
Tables Should Be Sorted (on Random Access Machines)
, 1995
"... We consider the problem of storing an n element subset S of a universe of size m, so that membership queries (is x 2 S?) can be answered efficiently. The model of computation is a random access machine with the standard instruction set (direct and indirect adressing, conditional branching, addit ..."
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Cited by 14 (4 self)
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We consider the problem of storing an n element subset S of a universe of size m, so that membership queries (is x 2 S?) can be answered efficiently. The model of computation is a random access machine with the standard instruction set (direct and indirect adressing, conditional branching, addition, subtraction, and multiplication). We show that if s memory registers are used to store S, where n s m=n , then query time \Omega\Gammame/ n) is necessary in the worst case. That is, under these conditions, the solution consisting of storing S as a sorted table and doing binary search is optimal. The condition s m=n is essentially optimal; we show that if n + m=n o(1) registers may be used, query time o(log n) is possible.
Explicit ordispersers with polylogarithmic degree
 J. ACM
, 1998
"... An (N,M,T)ORdisperser is a bipartite multigraph G = (V,W,E) withV  = N, and W  = M, having the following expansion property: any subset of V having at least T vertices has a neighbor set of size at least M/2. For any pair of constants ξ,λ,1 ≥ ξ>λ ≥ 0, any sufficiently large N, andforany ( ..."
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Cited by 13 (1 self)
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An (N,M,T)ORdisperser is a bipartite multigraph G = (V,W,E) withV  = N, and W  = M, having the following expansion property: any subset of V having at least T vertices has a neighbor set of size at least M/2. For any pair of constants ξ,λ,1 ≥ ξ>λ ≥ 0, any sufficiently large N, andforany (log N)ξ (log N)λ T ≥ 2, M ≤ 2, we give an explicit elementary construction of an (N,M,T)ORdisperser such that the outdegree of any vertex in V is at most polylogarithmic in N. Using this with known applications of ORdispersers yields several results. First, our construction implies that the complexity class StrongRP defined by Sipser, equals RP. Second, for any fixed η>0, we give the first polynomialtime simulation of RP algorithms using the output of any “ηminimally random ” source. For any integral R>0, such a source accepts a single request for an Rbit string and generates the string according to a distribution that assigns probability at most 2−Rη to any string. It is minimally random in the sense that any weaker source is
Increasing the Output Length of ZeroError Dispersers
, 2008
"... Let C be a class of probability distributions over a finite set Ω. A function D: Ω ↦ → {0, 1} m is a disperser for C with entropy threshold k and error ɛ if for any distribution X in C such that X gives positive probability to at least 2k elements we have that the distribution D(X) gives positive pr ..."
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Cited by 6 (5 self)
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Let C be a class of probability distributions over a finite set Ω. A function D: Ω ↦ → {0, 1} m is a disperser for C with entropy threshold k and error ɛ if for any distribution X in C such that X gives positive probability to at least 2k elements we have that the distribution D(X) gives positive probability to at least (1 − ɛ)2m elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called “extractors”) for various classes of distributions while trying to maximize m as a function of k. In this paper we are interested in explicitly constructing zeroerror dispersers (that is dispersers with error ɛ = 0). For several interesting classes of distributions there are explicit constructions in the literature of zeroerror dispersers with “small ” output length m and we give improved constructions that achieve “large ” output length, namely m = Ω(k). We achieve this by developing a general technique to improve the output length of zeroerror dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given in [31] building on earlier work
Efficient searching for multidimensional data made simple
 7th Annual European Symposium on Algorithms (ESA’99), Prague, Czech Republic, Jul.99, Lecture Notes in Computer Science vol.1643
, 1999
"... Abstract. We introduce an innovative decomposition technique which reduces a multi–dimensional searching problem to a sequence of one–dimensional problems, each one easily manageable in optimal time×space complexity using traditional searching strategies. The reduction has no additional storage requ ..."
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Cited by 5 (2 self)
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Abstract. We introduce an innovative decomposition technique which reduces a multi–dimensional searching problem to a sequence of one–dimensional problems, each one easily manageable in optimal time×space complexity using traditional searching strategies. The reduction has no additional storage requirement and the time complexity to reconstruct the result of the original multi–dimensional query is linear in the dimension. More precisely, we show how to preprocess a set of S ⊆ IN d of multi–dimensional objects into a data structure requiring O(m log n) space, where m = S  and n is the maximum number of different values for each coordinate. The obtained data structure is implicit, i.e. does not use pointers, and is able to answer the exact match query in 7(d − 1) steps. Additionally, the model of computation required for querying the data structure is very simple; the only arithmetic operation needed is the addition and no shift operation is used. The technique introduced, overcoming the multi–dimensional bottleneck, can be also applied to non traditional models of computation as external memory, distributed, and hierarchical environments. Additionally, we will show how the proposed technique permits the effective realizability of the well known perfect hashing techniques on real data. The algorithms for building the data structure are easy to implement and run in polynomial time. 1