Edit Distance between Unrooted Trees in Cubic Time

Edit distance between trees is a natural generalization of the classical edit distance between strings, in which the allowed elementary operations are contraction, uncontraction and relabeling of an edge. Demaine et al. [ACM Trans. on Algorithms, 6(1), 2009] showed how to compute the edit distance between rooted trees on n nodes in O(n^3) time. However, generalizing their method to unrooted trees seems quite problematic, and the most efficient known solution remains to be the previous O(n^3 log n) time algorithm by Klein [ESA 1998]. Given the lack of progress on improving this complexity, it might appear that unrooted trees are simply more difficult than rooted trees. We show that this is, in fact, not the case, and edit distance between unrooted trees on n nodes can be computed in O(n^3) time. A significantly faster solution is unlikely to exist, as Bringmann et al. [SODA 2018] proved that the complexity of computing the edit distance between rooted trees cannot be decreased to O(n^{3-epsilon}) unless some popular conjecture fails, and the lower bound easily extends to unrooted trees. We also show that for two unrooted trees of size m and n, where m <=n, our algorithm can be modified to run in O(nm^2(1+log(n/m))). This, again, matches the complexity achieved by Demaine et al. for rooted trees, who also showed that this is optimal if we restrict ourselves to the so-called decomposition algorithms

Cactus group and monodromy of Bethe vectors

Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra $\mathfrak{g}$. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$ (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope.Comment: 23 pages, sections 2 and 3 revise

Koszul Operads Governing Props and Wheeled Props

In this paper, we construct groupoid coloured operads governing props and wheeled props, and show they are Koszul. This is accomplished by new biased definitions for (wheeled) props, and an extension of the theory of Groebner bases for operads to apply to groupoid coloured operads. Using the Koszul machine, we define homotopy (wheeled) props, and show they are not formed by polytope based models. Finally, using homotopy transfer theory, we construct Massey products for (wheeled) props, show these products characterise the formality of these structures, and re-obtain a theorem of Mac Lane on the existence of higher homotopies of (co)commutative Hopf algebras.Comment: Questions or comments are most welcome

Barking up the wrong tree : some obstacles to phylogenetic reconstruction

Phylogenetics is the study of evolutionary relationships between entities, usually biological in nature. The primary aim of such study is to elucidate the structure of these evolutionary histories. Unfortunately, such study can run into a variety of obstacles, both practical and theoretical. In this thesis we explore theoretical obstacles to phylogenetic reconstruction, by examining several scenarios in which distinguishing between similar structures can become quite difficult. In Chapter 2, we consider when metrics on trees and metrics on networks can become indistinguishable, and present several novel results in this area, showing that it is possible for any tree metric to be represented on a non-trivial network, and provide early results on the possible structures of these networks. In Chapter 3, we consider tree-based networks - a phenomenon in which networks have a strong tree-like signal. We present the first findings on these networks in the context of unrooted non-binary networks. We characterise the circumstances under which such networks can become `saturated' by these signals, and provide some graph theoretical results in this area as well. In Chapter 4 we consider the scenario in which two trees can appear similar due to their hierarchical structure. We present a new metric to quantify this similarity, and use simulations to show several promising properties of the metric and the relative accuracy of a function that gives an upper bound to the metric