Trees and graph packing

In this thesis we investigate two main topics, namely, suffix trees and graph packing problems. In Chapter 2, we present the suffix trees. The main result of this chapter is a lower bound on the size of simple suffix trees. In the rest of the thesis we deal with packing problems. In Chapter 3 we give almost tight conditions on a bipartite packing problem. In Chapter 4 we consider an embedding problem regarding degree sequences. In Chapter 5 we show the existence of bounded degree bipartite graphs with a small separator and large bandwidth and we prove that under certain conditions these graphs can be embedded into graphs with minimum degree slightly over n/2

A friendly introduction to Fourier analysis on polytopes

This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field. We assume a familiarity with Linear Algebra, and some Calculus. Of the many applications, we have chosen to focus on: (a) formulations for the Fourier transform of a polytope, (b) Minkowski and Siegel's theorems in the geometry of numbers, (c) tilings and multi-tilings of Euclidean space by translations of a polytope, (d) Computing discrete volumes of polytopes, which are combinatorial approximations to the continuous volume, (e) Optimizing sphere packings and their densities, and (f) use iterations of the divergence theorem to give new formulations for the Fourier transform of a polytope, with an application. Throughout, we give many examples and exercises, so that this book is also appropriate for a course, or for self-study.Comment: 204 pages, 46 figure

Network Characteristics and Dynamics: Reciprocity, Competition and Information Dissemination

Networks are commonly used to study complex systems. This often requires a good understanding of the structural characteristics and evolution dynamics of networks, and also their impacts on a variety of dynamic processes taking place on top of them. In this thesis, we study various aspects of networks characteristics and dynamics, with a focus on reciprocity, competition and information dissemination. We first formulate the maximum reciprocity problem and study its use in the interpretation of reciprocity in real networks. We propose to interpret reciprocity based on its comparison with the maximum possible reciprocity for a network exhibiting the same degrees. We show that the maximum reciprocity problem is NP-hard, and use an upper bound instead of the maximum. We find that this bound is surprisingly close to the empirical reciprocity in a wide range of real networks, and that there is a surprisingly strong linear relationship between the two. We also show that certain small suboptimal motifs called 3-paths are the major cause for suboptimality in real networks. Secondly, we analyze competition dynamics under cumulative advantage, where accumulated resource promotes gathering even more resource. We characterize the tail distributions of duration and intensity for pairwise competition. We show that duration always has a power-law tail irrespective of competitors\u27 fitness, while intensity has either a power-law tail or an exponential tail depending on whether the competitors are equally fit. We observe a struggle-of-the-fitness phenomenon, where a slight different in fitness results in an extremely heavy tail of duration distribution. Lastly, we study the efficiency of information dissemination in social networks with limited budget of attention. We quantify the efficiency of information dissemination for both cooperative and selfish user behaviors in various network topologies. We identify topologies where cooperation plays a critical role in efficient information propagation. We propose an incentive mechanism called plus-one to coax users into cooperation in such cases

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

Quantum Contextuality

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here, compatible measurements are those that can be performed simultaneously or in any order without disturbance. This conflict is generically called quantum contextuality. In this article, we present an introduction to this subject and its current status. We review several proofs of the Kochen-Specker theorem and different notions of contextuality. We explain how to experimentally test some of these notions and discuss connections between contextuality and nonlocality or graph theory. Finally, we review some applications of contextuality in quantum information processing.Comment: 63 pages, 20 figures. Updated version. Comments still welcome

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum