# EADS Talk by Arnold Filtser – Københavns Universitet

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# EADS Talk by Arnold Filtser

## Title

On Notions of Distortion and an Almost Minimum Spanning Tree with constant Average Distortion

## Abstract

Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, Abraham et al showed that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a {\em spanning tree} whose weight is at most [image: (1+\rho)] times that of the MST, providing {\em constant average distortion} [image: O(1/\rho)].

The constant average distortion bound is implied by a stronger property of {\em scaling distortion}, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight {\em spanner} with small {\em prioritized distortion}, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.

## Biografy

I am a PhD student at the Computer Science Department at Ben Gurion University of the Negev. My advisors are Ofer Neiman and Robert Krauthgamer .

My research interest is theoretical computer science. More specifically: Metric Spaces, Low-Distortion Embeddings, Randomized Algorithms and Approximation.

### Co-authors

Yair Bartal and Ofer Neiman