Related papers: A Tale of Two Trees: New Analysis for AVL Tree and…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
Large tree structures are ubiquitous and real-world relational datasets often have information associated with nodes (e.g., labels or other attributes) and edges (e.g., weights or distances) that need to be communicated to the viewers. Yet,…
In this work, we study two types of constraints on two-dimensional binary arrays. In particular, given $p,\epsilon>0$, we study (i) The $p$-bounded constraint: a binary vector of size $m$ is said to be $p$-bounded if its weight is at most…
The entities in directed networks arising from real-world interactions are often naturally organized under some hierarchical structure. Given a directed, weighted, graph with edges and node labels, we introduce ranking problem where the…
This is a short exposition--mostly by way of the toy models ``double logarithm'' and ``triple logarithm''--which should serve as an introduction to a forthcoming article in which we establish a connection between multiple polylogarithms,…
Phylogenetic trees summarize evolutionary relationships between organisms, and tools to analyze collections of phylogenetic trees enable contrasts between different genes' ancestry. The BHV metric space has enabled the analysis of…
Null models of binary phylogenetic trees are useful for testing hypotheses on real world phylogenies. In this paper we consider phylogenies as binary trees without edge lengths together with a sampling measure and encode them as algebraic…
We investigate hierarchical structure in various complex systems according to Minimum Spanning Tree methods. Firstly, we investigate stock markets where the graphis obtained from the matrix of correlations coefficient computed between all…
Real-world applications such as the internet of things, wireless sensor networks, smart grids, transportation networks, communication networks, social networks, and computer grid systems are typically modeled as network structures. Network…
In reliability engineering, we need to understand system dependencies, cause-effect relations, identify critical components, and analyze how they trigger failures. Three prominent graph models commonly used for these purposes are fault…
Decision trees are well-known due to their ease of interpretability. To improve accuracy, we need to grow deep trees or ensembles of trees. These are hard to interpret, offsetting their original benefits. Shapley values have recently become…
We study the problem of learning a latent tree graphical model where samples are available only from a subset of variables. We propose two consistent and computationally efficient algorithms for learning minimal latent trees, that is, trees…
We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples…
This paper presents a new kind of self-balancing ternary search trie that uses a randomized balancing strategy adapted from Aragon and Seidel's randomized binary search trees ("treaps"). After any sequence of insertions and deletions of…
When analysing binary data, the ease at which one can interpret results is very important. Many existing methods, however, discover either models that are difficult to read, or return so many results interpretation becomes impossible. Here,…
An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to…
It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with $n$ given labeled leaves is of order between $n^{0.366}$ and $n^{1/2}$. We improve the lower bound to…
Describing the geometry of the dual amplituhedron without reference to a particular triangulation is an open problem. In this note we introduce a new way of determining the volume of the tree-level NMHV dual amplituhedron. We show that…
Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure…
This article briefly describes four algorithmic problems where the notion of treewidth is very useful. Even though the problems themselves have nothing to do with treewidth, it turns out that combining known results on treewidth allows us…