Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let $D$ be a finite domain and let $D^n$ be the set of $n$-length vectors (tuples) of $D$. Let $f : D \times D \to D$ be a function and let $\oplus_f$ be a coordinate-wise application of $f$. The $f$-Convolution of two functions $g,h : D^n \to \{-M,\ldots,M\}$ is \begin{displaymath} (g \circledast_f h)(v) := \sum_{\substack{v_g,v_h \in D^n\\ \text{s.t. } v = v_g \oplus_f v_h}} g(v_g) \cdot h(v_h) \end{displaymath} for every $v \in D^n$. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function $f$ and domain $D$ we can compute $f$-Convolution via brute-force enumeration in $\tilde O{|D|^{2n}}$ time. Our main result is an improvement over this naive algorithm. We show that $f$-Convolution can be computed exactly in $\tilde O{ (c \cdot |D|^2)^{n}}$ for constant $c := 5/6$ when $D$ has even cardinality. Our main observation is that a \emph{cyclic partition} of a function $f : D \times D \to D$ can be used to speed up the computation of $f$-Convolution, and we show that an appropriate cyclic partition exists for every $f$. Furthermore, we demonstrate that a single entry of the $f$-Convolution can be computed more efficiently. In this variant, we are given two functions $g,h : D^n \to \{-M,\ldots,M\}$ alongside with a vector $v \in D^n$ and the task of the $f$-Query problem is to compute integer $(g \circledast_f h)(v)$. This is a generalization of the well-known Orthogonal Vectors problem. We show that $f$-Query can be computed in $\tilde O{|D|^{\frac{\omega}{2} n}}$ time, where $\omega \in [2,2.373)$ is the exponent of currently fastest matrix multiplication algorithm

Mediating archaeology : the relationship between archaeology, the media and the public in Britain (1996-2002)

This thesis is an inter-disciplinary study of how archaeologists, the mass media and the public interrelate in the process of archaeological communication.  The study takes a critical look at the relationship between these three communities in Britain, and at the representation of archaeology in British national newspapers and television documentaries, during a dynamic seven-year period (1996-2002). Understanding the nature of this current relationship requires foundational work on the historical, contemporary and theoretical contexts that shape archaeological mediation.  This thesis begins with an historical account of archaeological communication and a discussion of recent trends that together explain the current timeliness of the subject.  It then goes on to explore the relationship from each partner’s perspective in order to establish the defining features of the relationship and how the perceptions that they have of each other and the values they place on the relationship affect their approaches to archaeological communication. Through a critique of existing models of how communication works and an introduction of new models, an analysis is put forward that demonstrates how the public communication of archaeology functions as a process of mediation and how mediators play different roles in the process.  The results of this analysis suggest that the sources and especially the means involved in releasing archaeological information have a significant impact on archaeology’s public communication, a proposition that is then tested in detailed press and television case studies. This thesis argues for, and makes a contribution to, a fundamental reconceptualisation of the relationship between archaeologists, the media and the public and the establishing of more evidence-based research in the field.</p

The accessibility of convex bodies and derandomization of the hit and run algorithm

We introduce the concept of accessibility and prove that any convex body $X$ in $\mathbb R^d$ is accessible with relevant constants depending on $d$ only. This property leads to a new algorithm which may be considered as a natural derandomization of the hit and run algorithm applied to generate a sequence of random points covering $X$ uniformly. We prove stability of the Markov chain generated by the proposed algorithm and provide its rate of convergence.Comment: revised version, new title and abstract; 13 pages, 2 figure

Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let \Dd be a finite domain and let \Dd^n be the set of $n$-length vectors (tuples) of \Dd. Let f \from \Dd \times \Dd \to \Dd be a function and let $\oplus_f$ be a coordinate-wise application of $f$. The \fConv of two functions g,h \from \Dd^n \to \{-M,\ldots,M\} is \begin{displaymath} (g \fconv h)(\bv) \deff \sum_{\substack{\bv_g,\bv_h \in \Dd^n\\ \text{s.t. } \bv = \bv_g \oplus_f \bv_h}} g(\bv_g) \cdot h(\bv_h) \end{displaymath} for every \bv \in \Dd^n. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function $f$ and domain \Dd we can compute \fConv via brute-force enumeration in \Oc{|\Dd|^{2n}} time. Our main result is an improvement over this naive algorithm. We show that \fConv can be computed exactly in \Oc{ (c \cdot |\Dd|^2)^{n}} for constant c \deff 5/6 when \Dd has even cardinality. Our main observation is that a \emph{cyclic partition} of a function f \from \Dd \times \Dd \to \Dd can be used to speed up the computation of \fConv, and we show that an appropriate cyclic partition exists for every $f$. Furthermore, we demonstrate that a single entry of the \fConv can be computed more efficiently. In this variant, we are given two functions g,h \from \Dd^n \to \{-M,\ldots,M\} alongside with a vector \bv \in \Dd^n and the task of the \fquery problem is to compute integer (g \fconv h)(\bv). This is a generalization of the well-known Orthogonal Vectors problem. We show that \fquery can be computed in \Oc{|\Dd|^{\frac{\omega}{2} n}} time, where $\omega \in [2,2.373)$ is the exponent of currently fastest matrix multiplication algorithm

Late Effects of Ionizing Radiation on the Ultrastructure of Hepatocytes and Activity of Lysosomal Enzymes in Mouse Liver Irradiated In Vivo

The study aimed to investigate late radiation-induced changes in the histology, ultrastructure, and activity of lysosomal enzymes in mouse liver exposed to ionizing radiation. The experiment was conducted on C57BL/6J male mice whose distal part of the liver was exposed occasionally to single doses of radiation (6 MV photons) during targeted heart irradiation; estimated doses delivered to analyzed tissue were 0.025 Gy, 0.25 Gy, 1 Gy, and 2 Gy. Tissues were collected 40 weeks after irradiation. We have observed that late effects of radiation have an adaptive nature and their intensity was dose-dependent. Morphological changes in hepatocytes included an increased number of primary lysosomes and autophagic vacuoles, which were visible in tissues irradiated with 0.25 Gy and higher doses. On the other hand, a significant increase in the activity of lysosomal hydrolases was observed only in tissues exposed to 2 Gy. The etiology of these changes may be multifactorial and result, among others, from unintentional irradiation of the distal part of the liver and/or functional interaction of the liver with an irradiated heart. In conclusion, we confirmed the presence of late dose-dependent ultrastructural and biochemical changes in mouse hepatocytes after liver irradiation in vivo