Reconstructing semi-directed level-1 networks using few quarnets

Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary \(n\)-leaf semi-directed level-1 networks in \(O( n^2)\) time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of \(O(n \log n)\) quarnets. Under group-based models of evolution with the Jukes-Cantor or Kimura 2-parameter constraints, it has been shown that only four-cycle quarnets and the splits of the other quarnets can practically be inferred with high accuracy from nucleotide sequence data. Our algorithm uses only this information, assuming the network contains no triangles. Additionally, we provide an \(O(n^3)\) time algorithm that reconstructs the blobtree (or tree-of-blobs) of any binary \(n\)-leaf semi-directed network with unbounded level from \(O(n^3)\) splits of its quarnets.