Abstract
We investigate the problem of local reconstruction, as deflned by Saks and Seshadhri (2008), in the context of error correcting codes. The flrst problem we address is that of message reconstruction, where given oracle access to a corrupted encoding w 2 f0;1g n of some message x 2 f0;1g k our goal is to probabilistically recover x (or some portion of it). This should be done by a procedure (reconstructor) that given an index i as input, probes w at few locations and outputs the value of xi. The reconstructor can (and indeed must) be randomized, with all its randomness specifled in advance by a single random seed, and the main requirement is that for most random seeds, all values are reconstructed correctly (notice that swapping the order of \for most random seeds and \for all x1;:::;xk makes the deflnition equivalent to standard Local Decoding). A message reconstructor can serve as a \fllter that allows evaluating certain classes of algorithms on x safely and e-ciently. For instance, to run a parallel algorithm, one can initialize several copies of the reconstructor with the same random seed, and then they can autonomously handle decoding requests while producing outputs that are consistent with the original message x. Another motivation for studying message reconstruction arises from the theory of Locally Decodable Codes. The second problem that we address is codeword reconstruction, which is similarly deflned, but instead of reconstructing the message the goal is to reconstruct the codeword itself, given an oracle access to its corrupted version. Error correcting codes that admit message and codeword reconstruction can be obtained from Locally Decodable Codes (LDC) and Self Correctible Codes (SCC) respectively. The main contribution of this paper is a proof that in terms of query complexity, these are close to be the best possible constructions, even when we disregard the length of the encoding.
Highlights
Consider the following problem: a large data set x ∈ {0, 1}k is stored on a storage device in encoded form, but a small fraction of the encoding may be corrupted
The second problem that we address is codeword reconstruction, which is defined, but instead of reconstructing the message the goal is to reconstruct the codeword itself, given oracle access to its corrupted version
We want to execute an algorithm M on x, but most likely M will only need a small fraction of x for its execution. This can be the case if M is a single process in a large parallelized system, or if M is a querying algorithm with limited memory, that can even be adaptive, not knowing which bits of the input it will need in advance. In all these cases M should have the ability to efficiently decode any bit of x only when the need arises, and to ensure correctness it is necessary that M succeeds in correctly decoding all bits that are required for its execution
Summary
Consider the following problem: a large data set x ∈ {0, 1}k is stored on a storage device in encoded form, but a small fraction of the encoding may be corrupted. The first theorem of this paper (see Theorem 3.2) states that for any encoding of any length, a non-adaptive message reconstructor must make Ω(log k) queries per decoding request Another family of error correcting codes related to LDCs are self-correctable codes (SCC) [5]. The second problem that we study here is of codeword reconstruction, which is related to SCCs in the same manner that message reconstruction is related to LDCs. Concretely, a codeword reconstructor is an algorithm that can recover a codeword y ∈ {0, 1}n from its corrupted version w ∈ {0, 1}n with two conditions: (locality) for every i ∈ [n] reconstructing yi requires reading w only at very few locations; (consistency) with high probability, all indices i ∈ [n] should be reconstructed correctly.
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