5 research outputs found
Edit Distance: Sketching, Streaming and Document Exchange
We show that in the document exchange problem, where Alice holds and Bob holds , Alice can send Bob a message of
size bits such that Bob can recover using the
message and his input if the edit distance between and is no more
than , and output "error" otherwise. Both the encoding and decoding can be
done in time . This result significantly
improves the previous communication bounds under polynomial encoding/decoding
time. We also show that in the referee model, where Alice and Bob hold and
respectively, they can compute sketches of and of sizes
bits (the encoding), and send to the referee, who can
then compute the edit distance between and together with all the edit
operations if the edit distance is no more than , and output "error"
otherwise (the decoding). To the best of our knowledge, this is the first
result for sketching edit distance using bits.
Moreover, the encoding phase of our sketching algorithm can be performed by
scanning the input string in one pass. Thus our sketching algorithm also
implies the first streaming algorithm for computing edit distance and all the
edits exactly using bits of space.Comment: Full version of an article to be presented at the 57th Annual IEEE
Symposium on Foundations of Computer Science (FOCS 2016
Locally Decodable Codes with Randomized Encoding
We initiate a study of locally decodable codes with randomized encoding.
Standard locally decodable codes are error correcting codes with a
deterministic encoding function and a randomized decoding function, such that
any desired message bit can be recovered with good probability by querying only
a small number of positions in the corrupted codeword. This allows one to
recover any message bit very efficiently in sub-linear or even logarithmic
time. Besides this straightforward application, locally decodable codes have
also found many other applications such as private information retrieval,
secure multiparty computation, and average-case complexity.
However, despite extensive research, the tradeoff between the rate of the
code and the number of queries is somewhat disappointing. For example, the best
known constructions still need super-polynomially long codeword length even
with a logarithmic number of queries, and need a polynomial number of queries
to achieve a constant rate. In this paper, we show that by using a randomized
encoding, in several models we can achieve significantly better rate-query
tradeoff. In addition, our codes work for both the standard Hamming errors, and
the more general and harder edit errors.Comment: 23 page