48,647 research outputs found
Constructions of Generalized Sidon Sets
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
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 -ary Trees, Prefix Sums and Multisets
We consider the {\it indexable dictionary} problem, which consists of storing
a set for some integer , while supporting the
operations of \Rank(x), which returns the number of elements in that are
less than if , and -1 otherwise; and \Select(i) which returns
the -th smallest element in . We give a data structure that supports both
operations in O(1) time on the RAM model and requires bits to store a set of size , where {\cal B}(n,m) = \ceil{\lg
{m \choose n}} is the minimum number of bits required to store any -element
subset from a universe of size . Previous dictionaries taking this space
only supported (yes/no) membership queries in O(1) time. In the cell probe
model we can remove the 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 -ary cardinal
tree that supports standard operations in constant time,
A representation of a multiset of size from in bits that supports (appropriate generalizations of) \Rank
and \Select operations in constant time, and
A representation of a sequence of non-negative integers summing up to
in 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
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
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-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
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 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 factor. Finally, we use our weighted
sampling algorithm to improve the message complexity of distributed
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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