Algorithms and Complexity on Indexing Founder Graphs

We introduce a compact pangenome representation based on an optimal segmentation concept that aims to reconstruct founder sequences from a multiple sequence alignment (MSA). Such founder sequences have the feature that each row of the MSA is a recombination of the founders. Several linear time dynamic programming algorithms have been previously devised to optimize segmentations that induce founder blocks that then can be concatenated into a set of founder sequences. All possible concatenation orders can be expressed as a founder graph. We observe a key property of such graphs: if the node labels (founder segments) do not repeat in the paths of the graph, such graphs can be indexed for efficient string matching. We call such graphs repeat-free founder graphs when constructed from a gapless MSA and repeat-free elastic founder graphs when constructed from a general MSA with gaps. We give a linear time algorithm and a parameterized near linear time algorithm to construct a repeat-free founder graph and a repeat-free elastic founder graph, respectively. We derive a tailored succinct index structure to support queries of arbitrary length in the paths of a repeat-free (elastic) founder graph. In addition, we show how to turn a repeat-free (elastic) founder graph into a Wheeler graph in polynomial time. Furthermore, we show that a property such as repeat-freeness is essential for indexability. In particular, we show that unless the Strong Exponential Time Hypothesis (SETH) fails, one cannot build an index on an elastic founder graph in polynomial time to support fast queries.