Subset Wavelet Trees

Abstract

Given an alphabet Σ of σ = | Σ | symbols, a degenerate (or indeterminate) string X is a sequence X = X [0] , X [1] . . . , X [ n − 1] of n subsets of Σ. Since their introduction in the mid 70s, degenerate strings have been widely studied, with applications driven by their being a natural model for sequences in which there is a degree of uncertainty about the precise symbol at a given position, such as those arising in genomics and proteomics. In this paper we introduce a new data structural tool for degenerate strings, called the subset wavelet tree (SubsetWT). A SubsetWT supports two basic operations on degenerate strings: subset-rank( i, c ), which returns the number of subsets up to the i -th subset in the degenerate string that contain the symbol c ; and subset-select( i, c ), which returns the index in the degenerate string of the i -th subset that contains symbol c . These queries are analogs of rank and select queries that have been widely studied for ordinary strings. Via experiments in a real genomics application in which degenerate strings are fundamental, we show that subset wavelet trees are practical data structures, and in particular offer an attractive space-time tradeoff. Along the way we investigate data structures for supporting (normal) rank queries on base-4 and base-3 sequences, which may be of independent interest. Our C++ implementations of the data structures are available at https://github.com/jnalanko/SubsetWT .