48,647 research outputs found

    Constructions of Generalized Sidon Sets

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    We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis' idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia, and Habsieger & Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.Comment: 15 pages, 1 figure (revision fixes typos, adds a few details, and adjusts notation

    Multivariate sparse interpolation using randomized Kronecker substitutions

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    We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial, that is, its sparsity. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201

    Succinct Indexable Dictionaries with Applications to Encoding kk-ary Trees, Prefix Sums and Multisets

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    We consider the {\it indexable dictionary} problem, which consists of storing a set S{0,...,m1}S \subseteq \{0,...,m-1\} for some integer mm, while supporting the operations of \Rank(x), which returns the number of elements in SS that are less than xx if xSx \in S, and -1 otherwise; and \Select(i) which returns the ii-th smallest element in SS. 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(lglgm){\cal B}(n,m) + o(n) + O(\lg \lg m) bits to store a set of size nn, where {\cal B}(n,m) = \ceil{\lg {m \choose n}} is the minimum number of bits required to store any nn-element subset from a universe of size mm. 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(lglgm)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 information-theoretically optimal representation of a kk-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size nn from {0,...,m1}\{0,...,m-1\} in B(n,m+n)+o(n){\cal 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 nn non-negative integers summing up to mm in B(n,m+n)+o(n){\cal B}(n,m+n) + o(n) bits that supports prefix sum queries in constant time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report 2002/1

    The structure of maximal zero-sum free Sequences

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    Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.Comment: 27 pages, 3 figure

    Tight Size-Degree Bounds for Sums-of-Squares Proofs

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    We exhibit families of 44-CNF formulas over nn variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) dd but require SOS proofs of size nΩ(d)n^{\Omega(d)} for values of d=d(n)d = d(n) from constant all the way up to nδn^{\delta} for some universal constantδ\delta. This shows that the nO(d)n^{O(d)} running time obtained by using the Lasserre semidefinite programming relaxations to find degree-dd SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP\mathsf{NP}-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively

    Weighted Reservoir Sampling from Distributed Streams

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    We consider message-efficient continuous random sampling from a distributed stream, where the probability of inclusion of an item in the sample is proportional to a weight associated with the item. The unweighted version, where all weights are equal, is well studied, and admits tight upper and lower bounds on message complexity. For weighted sampling with replacement, there is a simple reduction to unweighted sampling with replacement. However, in many applications the stream has only a few heavy items which may dominate a random sample when chosen with replacement. Weighted sampling \textit{without replacement} (weighted SWOR) eludes this issue, since such heavy items can be sampled at most once. In this work, we present the first message-optimal algorithm for weighted SWOR from a distributed stream. Our algorithm also has optimal space and time complexity. As an application of our algorithm for weighted SWOR, we derive the first distributed streaming algorithms for tracking \textit{heavy hitters with residual error}. Here the goal is to identify stream items that contribute significantly to the residual stream, once the heaviest items are removed. Residual heavy hitters generalize the notion of 1\ell_1 heavy hitters and are important in streams that have a skewed distribution of weights. In addition to the upper bound, we also provide a lower bound on the message complexity that is nearly tight up to a log(1/ϵ)\log(1/\epsilon) factor. Finally, we use our weighted sampling algorithm to improve the message complexity of distributed L1L_1 tracking, also known as count tracking, which is a widely studied problem in distributed streaming. We also derive a tight message lower bound, which closes the message complexity of this fundamental problem.Comment: To appear in PODS 201
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