74 research outputs found
Set-Codes with Small Intersections and Small Discrepancies
We are concerned with the problem of designing large families of subsets over
a common labeled ground set that have small pairwise intersections and the
property that the maximum discrepancy of the label values within each of the
sets is less than or equal to one. Our results, based on transversal designs,
factorizations of packings and Latin rectangles, show that by jointly
constructing the sets and labeling scheme, one can achieve optimal family sizes
for many parameter choices. Probabilistic arguments akin to those used for
pseudorandom generators lead to significantly suboptimal results when compared
to the proposed combinatorial methods. The design problem considered is
motivated by applications in molecular data storage and theoretical computer
science
Better Pseudorandom Generators from Milder Pseudorandom Restrictions
We present an iterative approach to constructing pseudorandom generators,
based on the repeated application of mild pseudorandom restrictions. We use
this template to construct pseudorandom generators for combinatorial rectangles
and read-once CNFs and a hitting set generator for width-3 branching programs,
all of which achieve near-optimal seed-length even in the low-error regime: We
get seed-length O(log (n/epsilon)) for error epsilon. Previously, only
constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for
these classes with polynomially small error.
The (pseudo)random restrictions we use are milder than those typically used
for proving circuit lower bounds in that we only set a constant fraction of the
bits at a time. While such restrictions do not simplify the functions
drastically, we show that they can be derandomized using small-bias spaces.Comment: To appear in FOCS 201
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Pseudorandomness for Regular Branching Programs via Fourier Analysis
We present an explicit pseudorandom generator for oblivious, read-once,
permutation branching programs of constant width that can read their input bits
in any order. The seed length is , where is the length of the
branching program. The previous best seed length known for this model was
, which follows as a special case of a generator due to
Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of
for arbitrary branching programs of size ). Our techniques
also give seed length for general oblivious, read-once branching
programs of width , which is incomparable to the results of
Impagliazzo et al.Our pseudorandom generator is similar to the one used by
Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite
different; ours is based on Fourier analysis of branching programs. In
particular, we show that an oblivious, read-once, regular branching program of
width has Fourier mass at most at level , independent of the
length of the program.Comment: RANDOM 201
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Pseudorandomness for Regular Branching Programs via Fourier Analysis
We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is , where n is the length of the branching program. The previous best seed length known for this model was , which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of for arbitrary branching programs of size s). Our techniques also give seed length for general oblivious, read-once branching programs of width , which is incomparable to the results of Impagliazzo et al.
Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width w has Fourier mass at most at level k, independent of the length of the program.Engineering and Applied Science
Bounded Independence Plus Noise Fools Products
Let D be a b-wise independent distribution over {0,1}^m. Let E be the "noise" distribution over {0,1}^m where the bits are independent and each bit is 1 with probability eta/2. We study which tests f: {0,1}^m -> [-1,1] are epsilon-fooled by D+E, i.e., |E[f(D+E)] - E[f(U)]| <= epsilon where U is the uniform distribution.
We show that D+E epsilon-fools product tests f: ({0,1}^n)^k -> [-1,1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1-eta)^{Omega(b^2/m)}, where m = nk and b >= n. This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m^{2/3} log m and any constant eta the distribution D+E also 0.1-fools log-space algorithms.
We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f: ({0,1}^n)^k -> [-1,1] under any permutation of the bits with seed lengths 2n + O~(k^2 log(1/epsilon)) and O(n) + O~(sqrt{nk log 1/epsilon}). Previous generators have seed lengths >= nk/2 or >= n sqrt{n k}. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n+log k)log(1/epsilon)
Fourier Bounds and Pseudorandom Generators for Product Tests
We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest.
As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths.
We also extend our results to functions f_i whose range is [-1,1]
Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers
This is basically a review of the field of Quasi-Monte Carlo intended for
computational physicists and other potential users of quasi-random numbers. As
such, much of the material is not new, but is presented here in a style
hopefully more accessible to physicists than the specialized mathematical
literature. There are also some new results: On the practical side we give
important empirical properties of large quasi-random point sets, especially the
exact quadratic discrepancies; on the theoretical side, there is the exact
distribution of quadratic discrepancy for random point sets.Comment: 51 pages. Full paper, including all figures also available at:
ftp://ftp.nikhef.nl/pub/preprints/96-017.ps.gz Accepted for publication in
Comp.Phys.Comm. Fixed some typos, corrected formula 108,figure 11 and table
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
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