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Register a new userConstructions from Dots and Lines
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Added by
Marko A. Rodriguez
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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A graph is a data structure composed of dots (i.e. vertices) and lines (i.e. edges). The dots and lines of a graph can be organized into intricate arrangements. The ability for a graph to denote objects and their relationships to one another allow for a surprisingly large number of things to be modeled as a graph. From the dependencies that link software packages to the wood beams that provide the framing to a house, most anything has a corresponding graph representation. However, just because it is possible to represent something as a graph does not necessarily mean that its graph representation will be useful. If a modeler can leverage the plethora of tools and algorithms that store and process graphs, then such a mapping is worthwhile. This article explores the world of graphs in computing and exposes situations in which graphical models are beneficial.
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In this note we illustrate how common matrix approximation methods, such as random projection and random sampling, yield projection-cost-preserving sketches, as introduced in [FSS13, CEM+15]. A projection-cost-preserving sketch is a matrix approximation which, for a given parameter $k$, approximately preserves the distance of the target matrix to all $k$-dimensional subspaces. Such sketches have applications to scalable algorithms for linear algebra, data science, and machine learning. Our goal is to simplify the presentation of proof techniques introduced in [CEM+15] and [CMM17] so that they can serve as a guide for future work. We also refer the reader to [CYD19], which gives a similar simplified exposition of the proof covered in Section 2.
Let $P$ be a set (called points), $Q$ be a set (called queries) and a function $ f:Ptimes Qto [0,infty)$ (called cost). For an error parameter $epsilon>0$, a set $Ssubseteq P$ with a emph{weight function} $w:P rightarrow [0,infty)$ is an $epsilon$-coreset if $sum_{sin S}w(s) f(s,q)$ approximates $sum_{pin P} f(p,q)$ up to a multiplicative factor of $1pmepsilon$ for every given query $qin Q$. We construct coresets for the $k$-means clustering of $n$ input points, both in an arbitrary metric space and $d$-dimensional Euclidean space. For Euclidean space, we present the first coreset whose size is simultaneously independent of both $d$ and $n$. In particular, this is the first coreset of size $o(n)$ for a stream of $n$ sparse points in a $d ge n$ dimensional space (e.g. adjacency matrices of graphs). We also provide the first generalizations of such coresets for handling outliers. For arbitrary metric spaces, we improve the dependence on $k$ to $k log k$ and present a matching lower bound. For $M$-estimator clustering (special cases include the well-known $k$-median and $k$-means clustering), we introduce a new technique for converting an offline coreset construction to the streaming setting. Our method yields streaming coreset algorithms requiring the storage of $O(S + k log n)$ points, where $S$ is the size of the offline coreset. In comparison, the previous state-of-the-art was the merge-and-reduce technique that required $O(S log^{2a+1} n)$ points, where $a$ is the exponent in the offline constructions dependence on $epsilon^{-1}$. For example, combining our offline and streaming results, we produce a streaming metric $k$-means coreset algorithm using $O(epsilon^{-2} k log k log n)$ points of storage. The previous state-of-the-art required $O(epsilon^{-4} k log k log^{6} n)$ points.
We study the problem of finding a mapping $f$ from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points $(u,v,w)$ asserts that $|f(u)-f(v)|<|f(u)-f(w)|$. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies $(1-varepsilon)$-fraction of all constraints, our algorithm computes a solution that satisfies $(1-O(varepsilon^{1/8}))$-fraction of all constraints, in time $O(n^7) + (1/varepsilon)^{O(1/varepsilon^{1/8})} n$.
Cathodoluminescence measurements on single InGaN/GaN quantum dots (QDs) are reported. Complex spectra with up to five emission lines per QD are observed. The lines are polarized along the orthogonal crystal directions [1 1 -2 0] and [-1 1 0 0]. Realistic eight-band k.p electronic structure calculations show that the polarization of the lines can be explained by excitonic recombinations involving hole states which are either formed by the A or the B valence band.
Systematic constructions of MDS self-dual codes is widely concerned. In this paper, we consider the constructions of MDS Euclidean self-dual codes from short length. Indeed, the exact constructions of MDS Euclidean self-dual codes from short length ($n=3,4,5,6$) are given. In general, we construct more new of $q$-ary MDS Euclidean self-dual codes from MDS self-dual codes of known length via generalized Reed-Solomon (GRS for short) codes and extended GRS codes.
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