
Approximate Trace Reconstruction
In the usual trace reconstruction problem, the goal is to exactly recons...
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Approximate Trace Reconstruction via Median String (in AverageCase)
We consider an approximate version of the trace reconstruction problem, ...
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Lower bounds for trace reconstruction
In the trace reconstruction problem, an unknown bit string xโ{0,1}^n is...
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Polynomialtime trace reconstruction in the smoothed complexity model
In the trace reconstruction problem, an unknown source string x โ{0,1}^n...
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Reconstructing Trees from Traces
We study the problem of learning a nodelabeled tree given independent t...
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Limitations of MeanBased Algorithms for Trace Reconstruction at Small Distance
Trace reconstruction considers the task of recovering an unknown string ...
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Trace Reconstruction: Generalized and Parameterized
In the beautifully simpletostate problem of trace reconstruction, the ...
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NearOptimal AverageCase Approximate Trace Reconstruction from Few Traces
In the standard trace reconstruction problem, the goal is to exactly reconstruct an unknown source string ๐โ{0,1}^n from independent "traces", which are copies of ๐ that have been corrupted by a ฮดdeletion channel which independently deletes each bit of ๐ with probability ฮด and concatenates the surviving bits. We study the approximate trace reconstruction problem, in which the goal is only to obtain a highaccuracy approximation of ๐ rather than an exact reconstruction. We give an efficient algorithm, and a nearmatching lower bound, for approximate reconstruction of a random source string ๐โ{0,1}^n from few traces. Our main algorithmic result is a polynomialtime algorithm with the following property: for any deletion rate 0 < ฮด < 1 (which may depend on n), for almost every source string ๐โ{0,1}^n, given any number M โคฮ(1/ฮด) of traces from Del_ฮด(๐), the algorithm constructs a hypothesis string ๐ that has edit distance at most n ยท (ฮด M)^ฮฉ(M) from ๐. We also prove a nearmatching informationtheoretic lower bound showing that given M โคฮ(1/ฮด) traces from Del_ฮด(๐) for a random nbit string ๐, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is n ยท (ฮด M)^O(M).
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