The Olympic medals ranks, lexicographic ordering and numerical infinities

Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. How can we quantify what do these words, more precious, mean? Can we introduce a counter that for any possible number of medals would allow us to compute a numerical rank of a country using the number of gold, silver, and bronze medals in such a way that the higher resulting number would put the country in the higher position in the rank? Here we show that it is impossible to solve this problem using the positional numeral system with any finite base. Then we demonstrate that this problem can be easily solved by applying numerical computations with recently developed actual infinite numbers. These computations can be done on a new kind of a computer – the recently patented Infinity Computer. Its working software prototype is described briefly and examples of computations are given. It is shown that the new way of counting can be used in all situations where the lexicographic ordering is required

Probabilistic Metric Embedding via Metric Labeling

We consider probabilistic embedding of metric spaces into ultra-metrics (or equivalently to a constant factor, into hierarchically separated trees) to minimize the expected distortion of any pairwise distance. Such embeddings have been widely used in network design and online algorithms. Our main result is a polynomial time algorithm that approximates the optimal distortion on any instance to within a constant factor. We achieve this via a novel LP formulation that reduces this problem to a probabilistic version of uniform metric labeling