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# Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions

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## Cite As

Frank Ban, Xi Chen, Rocco A. Servedio, and Sandip Sinha. Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.44

## Abstract

A number of recent works have considered the trace reconstruction problem, in which an unknown source string x in {0,1}^n is transmitted through a probabilistic channel which may randomly delete coordinates or insert random bits, resulting in a trace of x. The goal is to reconstruct the original string x from independent traces of x. While the asymptotically best algorithms known for worst-case strings use exp(O(n^{1/3})) traces [De et al., 2017; Fedor Nazarov and Yuval Peres, 2017], several highly efficient algorithms are known [Yuval Peres and Alex Zhai, 2017; Nina Holden et al., 2018] for the average-case version of the problem, in which the source string x is chosen uniformly at random from {0,1}^n. In this paper we consider a generalization of the above-described average-case trace reconstruction problem, which we call average-case population recovery in the presence of insertions and deletions. In this problem, rather than a single unknown source string there is an unknown distribution over s unknown source strings x^1,...,x^s in {0,1}^n, and each sample given to the algorithm is independently generated by drawing some x^i from this distribution and returning an independent trace of x^i. Building on the results of [Yuval Peres and Alex Zhai, 2017] and [Nina Holden et al., 2018], we give an efficient algorithm for the average-case population recovery problem in the presence of insertions and deletions. For any support size 1 <= s <= exp(Theta(n^{1/3})), for a 1-o(1) fraction of all s-element support sets {x^1,...,x^s} subset {0,1}^n, for every distribution D supported on {x^1,...,x^s}, our algorithm can efficiently recover D up to total variation distance at most epsilon with high probability, given access to independent traces of independent draws from D. The running time of our algorithm is poly(n,s,1/epsilon) and its sample complexity is poly (s,1/epsilon,exp(log^{1/3} n)). This polynomial dependence on the support size s is in sharp contrast with the worst-case version of the problem (when x^1,...,x^s may be any strings in {0,1}^n), in which the sample complexity of the most efficient known algorithm [Frank Ban et al., 2019] is doubly exponential in s.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Information theory
• Theory of computation → Machine learning theory
##### Keywords
• population recovery
• deletion channel
• trace reconstruction

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## References

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