Approximating Graphs by Trees Marcin Bieńkowski University of Wrocław

the distances between any pair of vertices: The problem Given an arbitrary graph of nodes… … construct a tree with … … that preserves (approximately) the distances between any pair of vertices: Distortion

Applications Basically, most of optimization problems on graphs, whose objective function is linear in distances. In particular: metrical task systems group Steiner tree buy-at-bulk network design distributed paging k-server k-median …

Is small distortion possible? (1) The distortion is n-1. Actually, this is more complicated as T may contain additional nodes, but the distortion is [Rabinovich, Raz 2005] No good deterministic choice of T, what is about random one?

Is small distortion possible? (2) Cut a random edge from the circle (each with probability 1/n) obtaining a random tree. Fix any two nodes and , let Then, distortion < 2

The (modified) problem Given an arbitrary graph of nodes… … construct a random tree with … … that preserves (approximately) the distances between any pair of vertices: and

Asymptotically optimal Historical Notes Bounding distortion: [Alon, Karp, Peleg, West, SIAM Jcomp ’91] [Bartal, FOCS ’96] [Bartal, STOC ‘98] [Fakcharoenphol, Rao, Talwar, STOC ’03] The last three results use hierarchical partitioning of the graph. Asymptotically optimal

Hierarchical decomposition Iteratively partition set Leaves of = singleton sets Nodes of correspond to other sets in partitioning How to choose partitioning of G? How to choose distances in T?

Choosing tree distances W.l.o.g. = the smallest power of 2 greater than FRT decomp. guarantee: for a set S on level , there exists a ball of radius < (centered at some node) containing nodes of S. Observation 1. For nodes s.t. is on level it holds that Lemma 1.

Partitioning of a single set At the beginning randomly choose and a random permu-tation of all nodes. Partitioning of S on level i+1: for do if then is the new set on level

Bounding distances in tree (1) Random variables permutation Lemma 2. with the following events: = exactly one from belongs to = for all s.t. it holds that

Bounding distances in tree (2) = exactly one from belongs to = for all s.t. it holds that Lem 2. Lemma 3. for any : Lemma 4. for any :

Thank you for you attention!