Retrieving random media

Benjamini asked whether the scenery reconstruction methods of Matzinger (see e.g. [21], [22], [20]) can be done in polynomial time. In this article, we give the following answer for a 2-color scenery and simple random walk with holding: We prove that a piece of the scenery of length of the order 3 ⁿ around the origin can be reconstructed - up to a reflection and a small translation - with high probability from the first 2 • 3^{10 αn} observations with a constant α > 0 independent of n. Thus, the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct. The probability that the reconstruction fails tends to 0 as n→∞. In contrast to [21], [22], and [20], the proofs in this article are all constructive. Our reconstruction algorithm is an algorithm in the sense of computer science. This is the first article which shows that the scenery reconstruction is also possible in the 2-color case with holding. The case with holding is much more difficult than [22] and requires completely different methods.