Trace Reconstruction: Generalized and Parameterized
Akshay Krishnamurthy, Arya Mazumdar, Andrew McGregor, Soumyabrata Pal · IEEE Transactions on Information Theory · 2021
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random “traces” of x where each trace is generated by deleting each coordinate of x independently with probability p1/4√{logn})) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown √n×√n matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n1/3)). 2) An optimal result for random matrix reconstruction: we show that Θ(logn) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp(O(log1/3n)). 3) We show that exp(O(k1/3log2/3n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log2n). 4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a “separation” promise, specifically that the indices of 1's in x all differ by Ω(k logn).