3,917 research outputs found
On taking advantage of multiple requests in error correcting codes
In most notions of locality in error correcting codes -- notably locally
recoverable codes (LRCs) and locally decodable codes (LDCs) -- a decoder seeks
to learn a single symbol of a message while looking at only a few symbols of
the corresponding codeword. However, suppose that one wants to recover r > 1
symbols of the message. The two extremes are repeating the single-query
algorithm r times (this is the intuition behind LRCs with availability,
primitive multiset batch codes, and PIR codes) or simply running a global
decoding algorithm to recover the whole thing. In this paper, we investigate
what can happen in between these two extremes: at what value of r does
repetition stop being a good idea? In order to begin to study this question we
introduce robust batch codes, which seek to find r symbols of the message using
m queries to the codeword, in the presence of erasures. We focus on the case
where r = m, which can be seen as a generalization of the MDS property.
Surprisingly, we show that for this notion of locality, repetition is optimal
even up to very large values of
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
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