Reconstructing a random scenery in polynomial time

Benjamini asked whether the scenery reconstruction problem can be solved in polynomial time In this article we answer his question in the a rmative for an i i d uniformly colored scenery on Z observed along a random walk path with bounded jumps We assume the random walk is recurrent can reach every integer with positive probability and the number of possible single steps for the random walk exceeds the number of colors We prove that a nite piece of scenery of length l around the origin can be reconstructed up to re ection and a small translation from the rst p l observations with high probability here p is a polynomial and the probability that the reconstruction succeeds converges to as l Introduction and Result We call a coloring of Z with colors from the nite set C f Cg a scenery The scenery reconstruction problem can be described as follows Let be a scenery and let S Sk k N be a recurrent random walk on Z If we are given the color record Sk k N observed along the random walk path can we almost surely reconstruct from these observations of course without knowing S It is not hard to see that in general the reconstruction works only up to a re ection and translation Recently Lindenstrauss proved the existence of uncountably many sceneries which cannot be reconstructed Nevertheless it turns out that in many situations typical sceneries can be reconstructed if the scenery is randomly colored and the random walk S is independent of Work on the scenery reconstruction problem started with the question how much information can be extracted from the color record This question was Eurandom PO Box MB Eindhoven The Netherlands Matzinger eurandom tue nl Eurandom PO Box MB Eindhoven The Netherlands Rolles eurandom tue nl