Efficient Construction of Probabilistic Tree Embeddings

In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and ${\bf{E}}_{T}[d_T(u,v)]\leq O(\log n)\cdot d_G(u,v)$. Such embeddings have proved highly useful in designing fast approximation algorithms, as many hard problems on graphs are easy to solve on tree instances. For a graph with $n$ vertices and $m$ edges, our algorithm runs in $O(m\log n)$ time with high probability, which improves the previous upper bound of $O(m\log^3 n)$ shown by Mendel et al.\,in 2009. The key component of our algorithm is a new approximate single-source shortest-path algorithm, which implements the priority queue with a new data structure, the "bucket-tree structure". The algorithm has three properties: it only requires linear time in the number of edges in the input graph; the computed distances have a distance preserving property; and when computing the shortest-paths to the $k$-nearest vertices from the source, it only requires to visit these vertices and their edge lists. These properties are essential to guarantee the correctness and the stated time bound. Using this shortest-path algorithm, we show how to generate an intermediate structure, the approximate dominance sequences of the input graph, in $O(m \log n)$ time, and further propose a simple yet efficient algorithm to converted this sequence to a tree embedding in $O(n\log n)$ time, both with high probability. Combining the three subroutines gives the stated time bound of the algorithm. Then we show that this efficient construction can facilitate some applications. We proved that FRT trees (the generated tree embedding) are Ramsey partitions with asymptotically tight bound, so the construction of a series of distance oracles can be accelerated

An Analysis of the Influence of CEO Characteristics on Research and Development Expenditures in Large American Corporations (2005 Data)

This study analyzes the influence exerted by CEO characteristics (specifically, CEO stockholding percentages and CEO age) on research and development (R&D) expenditures in large American corporations over the twenty year period from 1986 to 2005. Using the Herfindahl-Hirschman Index (HHI) to measure market share concentrations, and making specific reference to two Schumpeterian hypotheses on the correlation between R&D and increases in firm size, this study establishes a positive linear relationship between the dependent variable, R&D expenditure, and the independent variables of CEO stockholding, CEO age, firm size, and market share. This study next describes the corporate and market conditions which promote the development of a positive linear relationship between CEO characteristics and R&D and concludes by identifying the point of high market concentration at which the prominence of R&D activity is superseded by expenditures for advertising