Trace Reconstruction: Generalized and Parameterized

In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> given random “traces” of <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> where each trace is generated by deleting each coordinate of <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> independently with probability <inline-formula> <tex-math notation="LaTeX">$p < 1$ </tex-math></inline-formula>. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. Perhaps our most surprising results are: 1) We prove that <inline-formula> <tex-math notation="LaTeX">$\exp (O(n^{1/4} \sqrt {\log n}))$ </tex-math></inline-formula> traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown <inline-formula> <tex-math notation="LaTeX">$\sqrt {n}\times \sqrt {n}$ </tex-math></inline-formula> matrix is deleted independently with probability <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is <inline-formula> <tex-math notation="LaTeX">$\exp (O(n^{1/3}))$ </tex-math></inline-formula>. 2) An optimal result for random matrix reconstruction: we show that <inline-formula> <tex-math notation="LaTeX">$\Theta (\log n)$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\exp ({O}(\log ^{1/3} n))$ </tex-math></inline-formula>. 3) We show that <inline-formula> <tex-math notation="LaTeX">$\exp (O(k^{1/3}\log ^{2/3} n))$ </tex-math></inline-formula> traces suffice to reconstruct <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-sparse strings, providing an improvement over the best known sequence reconstruction results when <inline-formula> <tex-math notation="LaTeX">$k = o(n/\log ^{2}\,\,n)$ </tex-math></inline-formula>. 4) We show that <inline-formula> <tex-math notation="LaTeX">$\textrm {poly}(n)$ </tex-math></inline-formula> traces suffice if <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-sparse and we additionally have a “separation” promise, specifically that the indices of <monospace>1</monospace>’s in <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> all differ by <inline-formula> <tex-math notation="LaTeX">$\Omega (k \log n)$ </tex-math></inline-formula>.