### Article

## Characterizing (quasi-)ultrametric finite spaces in terms of (directed) graphs

We consider the hub label optimization problem, which arises in designing fast preprocessing-based shortest- path algorithms. We give O(log n)-approximation algorithms for the objectives of minimizing the maximum label size (l∞-norm) and simultaneously minimizing a constant number of lp-norms. Prior to this, an O(log n)- approximation algorithm was known [Cohen et al. 2003] only for minimizing the total label size (l1-norm).

We consider leader election and spanning tree construction problems in a synchronous network. Protocols are designed under the assumption that nodes in the network have identifiers but the size of an identifier is unlimited. We present fast protocols with runtime *O*(*D*log *L*+*L*), where *L* is the size of the minimal identifier and *D* is the network diameter.

Ranking is an important part of several areas of contemporary research, including social sciences, decision theory, data analysis and information retrieval. The goal of this project is to align developments in quantitative social sciences and decision theory with the current thought in computer science, including a few novel results. Specifically, we consider binary preference relations, the so-called weak orders that are in one-to-one correspondence with rankings. We show that the conventional symmetric difference distance between weak orders, considered as sets of ordered pairs, coincides with the celebrated Kemeny distance between the corresponding rankings, despite the seemingly much simpler structure of the former. Based on this, we review several properties of the geometric space of weak orders involving the ternary relation “between”, and contingency tables for cross-partitions. Next we reformulate the consensus ranking problem as a variant of finding an optimal linear ordering, given a correspondingly defined consensus matrix. The difference is in a subtracted term, the partition concentration, that depends only on the distribution of the objects in the individual parts. We apply our results to the conventional Likert scale to show that the Kemeny consensus rule is rather insensitive to the data under consideration and, therefore, should be supplemented with more sensitive consensus schemes.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.