Trace Reconstruction: Generalized and Parameterized

Abstract

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 p < 1. 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 exp(O(n 1/4 √ log n)) 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(n 1/3)). 2. An optimal result for random matrix reconstruction: we show that Θ(log n) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a superlogarithmic lower bound and the best known upper bound is exp(O(log 1/3 n)). 3. We show that exp(O(k 1/3 log 2/3 n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/ log 2 n). 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 log n).

Trace Reconstruction with Bounded Edit Distance

“…Without loss of generality, assume that w = (w , 0) ∈ {0, 1} 10k is non-periodic. Then, similar to the arguments in [8], [12], [13], [16], [19], the core part of the proof is to show that the difference of multi-bit statistics EX [1X k i =wi,∀i∈ [10k] ] and EỸ [1Ỹ k i =wi,∀i∈ [10k] ], is at least 1 n O(k) for some integers 1 ≤ i 1 < . .…”

“…As indicated in [12], [13], [15], [16], [19], the problem of worst case trace reconstruction is essentially equivalent to a hypothesis testing problem of distinguishing any two strings using noisy samples. More specifically, the sample complexity needed for trace reconstruction is at most poly(n) times the sample complexity needed to distinguish arbitrary two strings.…”

“…Hence, for convenience, we consider the problem in the form of distinguishing any two strings x ∈ {0, 1} n and y ∈ {0, 1} n when x is within edit distance k to y. One special case of the problem is to distinguish two stings within Hamming distance k, which was addressed in [16] and n O(k) sample complexity was achieved. Recently, an independent work [13] studied the limitations of mean-based algorithms (see [12] and [19]) in distinguishing two strings with bounded edit distance.…”

Approximate trace reconstruction of random strings from a constant number of traces

“…There have also been generalizations to objects other than strings. For matrices, see [23]. For trees, see [15], [25], and [6].…”

Tree trace reconstruction using subtraces

“…Given the difficulty of the trace reconstruction problem, several variants have been introduced, in part to study the strengths and weaknesses of various techniques. These include generalizing trace reconstruction from strings to trees [9] and matrices [18], coded trace reconstruction [8,4], population recovery [1,2,21], and more [22,7,10].…”