The trace reconstruction problem for spider graphs

Abstract

We study the trace reconstruction problem for spider graphs. Let n be the number of nodes of a spider and d be the length of each leg, and suppose that we are given independent traces of the spider from a deletion channel in which each non-root node is deleted with probability q. This is a natural generalization of the string trace reconstruction problem in theoretical computer science, which corresponds to the special case where the spider has one leg. In the regime where d≥log1/q⁡(n), the problem can be reduced to the vanilla string trace reconstruction problem. We thus study the more interesting regime d≤log1/q⁡(n), in which entire legs of the spider are deleted with non-negligible probability. We describe an algorithm that reconstructs spiders with high probability using exp⁡(O((nqd)1/3d1/3(log⁡n)2/3)) traces. Our algorithm works for all deletion probabilities q∈(0,1).