Algorithms and Complexity on Indexing Founder Graphs

TL;DR

This paper investigates the complexity of indexing founder graphs derived from multiple sequence alignments, introduces subclasses that are efficiently indexable, and proposes algorithms for their construction, bridging the gap between complex graph classes and efficient indexing.

Contribution

It introduces elastic founder graphs and subclasses that are efficiently indexable, along with algorithms for their construction from MSAs, addressing the bottleneck in Wheeler graph recognition.

Findings

Repeat-free elastic founder graphs can be constructed in near-linear time.

Semi-repeat-free graphs can be built efficiently from general MSAs.

Repeat-free elastic founder graphs can be reduced to Wheeler graphs in polynomial time.

Abstract

We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: E.g. Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (MSAs). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free non-elastic founder graph from a gapless MSA, and (parameterized) near-linear time algorithms to construct semi-repeat-free (repeat-free, respectively) elastic founder graphs from general MSAs. Finally, we show that repeat-free elastic founder graphs admit a reduction to Wheeler graphs in polynomial time.