1. Approximating Graphs by Trees Marcin Bieńkowski University of Wrocław

2. 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

3. 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 …

4. 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?

5. 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

6. 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

7. 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

8. 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?

9. 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.

10. 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

11. 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

12. 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 :

13. Thank you for you attention!