Inferring a Tree from Lowest Common Ancestors with an Application to the Optimization of Relational Expressions

Abstract

We present an algorithm for constructing a tree to satisfy a set of lineage constraints on common ancestors. We then apply this algorithm to synthesize a relational algebra expression from a simple tableau, a problem arising in the theory of relational databases.

Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets

“…We end this section by giving a short description of the Build algorithm [1,23], which decides if there exists a rooted tree (i.e. a network without reticulations) displaying a given set of triplets L. The Build algorithm constructs a graph R L (L) with a vertex for each taxon and an edge {x, y} precisely if there exists a triplet…”

“…The first step of the algorithm is to construct the graph R b (B) 1 , which has a vertex for each taxon and an edge {x, y} if (at least) one of N (x; y) and N (y; x) is contained in B. The algorithm searches for a nonempty strict subset U of the vertices of Ω b (B) such that there is no arc (π 1 , π 2 ) with π 1 / ∈ U and π 2 ∈ U .…”

“…4, we present an exponential-time algorithm for an arbitrary set consisting of binets and trinets (see Theorem 2). This algorithm uses a top-down approach that is somewhat similar in nature to the Build algorithm [1,23] but it is considerably more intricate. The algorithm can be generalised to instances containing level-1 networks with arbitrarily many leaves since trinets encode level-1 networks [11].…”

“…We consider binets as well as trinets since they can provide important information to help piece together sets of trinets. Our approach can be regarded as a generalisation of the wellknown supertree algorithm called Build [1,23] for checking whether or not a set of triplets is displayed by a phylogenetic tree and constructing such a tree if this is the case. In particular, the algorithm we present in Sect.…”

Efficient Algorithms for Descendent Subtrees Comparison of Phylogenetic Trees with Applications to Co-evolutionary Classifications in Bacterial Genome

“…Numerous phylogenetic inference methods, e.g. maximum parsimony, maximum likelihood, distance matrix fitting, subtrees consistency, and quartet based methods have been proposed over the years [15,1,14,26,17,27,4]; furthermore, it is rather common to compare the same set of species w.r.t. different biological sequences or different genes, hence obtaining various trees.…”