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 $O\left(n^2\right)$ 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. When the network is assumed to contain no triangles, our method instead relies only on four-cycle quarnets and the splits of the other quarnets. A variant of our algorithm works with quartets rather than quarnets and we show that it reconstructs most of a semi-directed level-1 network from an asymptotically optimal $O(n\log n)$ of the quartets it displays. Additionally, we provide an $O\left(n^3\right)$ time algorithm that reconstructs the tree-of-blobs of any binary n-leaf semi-directed network with unbounded level from $O\left(n^3\right)$ splits of its quarnets.