Reconstructing semi-directed level-1 networks using few quarnets

TL;DR

This paper introduces efficient algorithms for reconstructing semi-directed level-1 networks from minimal sets of quarnets or quartets, advancing phylogenetic network reconstruction methods.

Contribution

It presents the first polynomial-time algorithms for reconstructing semi-directed level-1 networks using only an asymptotically optimal number of quarnets or quartets.

Findings

Reconstruction algorithm runs in O(n^2) time from all quarnets.

Method works with a minimal O(n log n) quarnets.

Alternative approach reconstructs the tree-of-blobs in O(n^3) time.

Abstract

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. 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 $O(n^3)$ time algorithm that reconstructs the tree-of-blobs of any binary $n$-leaf semi-directed network with unbounded level from $O(n^3)$ splits of its quarnets.