Comparing Degenerate Strings

Abstract

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

Quantum Linear Algorithm for Edit Distance Using the Word QRAM Model

“…Although we studied level DAGs here just to give an example of breaking the quadratic barrier, such DAGs have connections to earlier literature. Namely, degenerate strings [1] are a special case of level DAGs, so the algorithms developed here apply directly on them. We leave it as open question if similar solutions can be derived for arbitrary DAGs, or e.g.…”

“…Since these two register are in a superposition, we are actually checking every pair of nodes at the same time. Thus, what the whole algorithm is doing is accessing every possible |P |-tuple of nodes (v (1) , v (2) , . .…”

“…The superposition is indicated with the tensor product notation, otherwise it would take too much space, but it is indeed representing every possible tuple (v (1) , v (2) , v (3) ) ∈ V × V × V . Assume that we can check in constant time whether an edge between two nodes exists.…”

“…For instance, we could assume to have adjacency matrix A G of G stored in our QuRAM, that we can access to check the existence of an edge. For nodes (v (1) , v (2) , v (3) ), our algorithm computes A G [v (1) ][v (2) ] ∧ A G [v (2) ][v (3) ] ∧ e and stores the result in |e , which is an operation that we can perform Then, we check whether the pattern matches these paths, that is we compute (v (1)…”

“…Operation 1 sets the least significant bit of D to 1, which is needed to test P [1] against T [i]. Operation 2 computes a bit-wise and between D and the column of B corresponding to character…”