Lower bounds for text indexing with mismatches and differences

Abstract: In this paper we study lower bounds for the fundamental problem of text indexing with mismatches and differences. In this problem we are given a long string of length $n$, the "text", and the task is to preprocess it into a data structure such that given a query string $Q$, one can quickly identify substrings that are within Hamming or edit distance at most $k$ from $Q$. This problem is at the core of various problems arising in biology and text processing. While exact text indexing allows linear-size data structures with linear query time, text indexing with $k$ mismatches (or $k$ differences) seems to be much harder: All known data structures have exponential dependency on $k$ either in the space, or in the time bound. We provide conditional and pointer-machine lower bounds that make a step toward explaining this phenomenon. We start by demonstrating lower bounds for $k = \Theta(\log n)$. We show that assuming the Strong Exponential Time Hypothesis, any data structure for text indexing that can be constructed in polynomial time cannot have $\mathcal{O}(n^{1-\delta})$ query time, for any $\delta>0$. This bound also extends to the setting where we only ask for $(1+\varepsilon)$-approximate solutions for text indexing. However, in many applications the value of $k$ is rather small, and one might hope that for small~$k$ we can develop more efficient solutions. We show that this would require a radically new approach as using the current methods one cannot avoid exponential dependency on $k$ either in the space, or in the time bound for all even $\frac{8}{\sqrt{3}} \sqrt{\log n} \le k = o(\log n)$. Our lower bounds also apply to the dictionary look-up problem, where instead of a text one is given a set of strings.