Indexing a dictionary for subset matching queries

We consider a subset matching variant of the Dictionary Query problem. Consider a dictionary D of n strings, where each string location contains a set of characters drawn from some alphabet E. Our goal is to preprocess D so when given a query pattern p, where each location in p contains a single character from Σ, we answer if p matches to D. p is said to match to D if there is some s ∈ D where |p| = |s| and p[i] ∈ s[i] for every 1 ≤ i ≤ |p|. To achieve a query time of O(|p|), we construct a compressed trie of all possible patterns that appear in D. Assuming that for every a ∈ D there are at most k locations where |s[i]| > 1, we present two constructions of the trie that yield a preprocessing time of O(nm + |Σ|^k n lg(min{n, m})), where n is the number of strings in D and m is the maximum length of a string in D. The first construction is based on divide and conquer and the second construction uses ideas introduced in [2] for text fingerprinting. Furthermore, we show how to obtain O(nm + |Σ|^k n + |Σ|^k/2 n lg(min{n, m})) preprocessing time and O(|p|lg lg|Σ| + min{|p|lg(|Σ|^kn)}lg lg(|Σ|^kn)) query time by cutting the dictionary strings and constructing two compressed tries. Our problem is motivated by haplotype inference from a library of genotypes [14,17]. There, D is a known library of genotypes (|Σ| = 2), and p is a haplotype. Indexing all possible haplotypes that can be inferred from D as well as gathering statistical information about them can be used to accelerate various haplotype inference algorithms. In particular, algorithms based on the "pure parsimony criteria" [13,16], greedy heuristics such as "Clarks rule" [6,18], EM based algorithms [1,11,12,20,26,30], and algorithms for inferring haplotypes from a set of Trios [4,27].