CS880: Approximations Algorithms
Scribe: Matt Darnall Lecturer: Shuchi Chawla
Topic: Tree Embeddings Date: 4/10/07
23.1 Problem Statement and Results
REcall that in the previous lecture, we discussed the use of probabilistic tree embeddings for approx-
imations algorithms. We now present an algorithm for obtaining low distortion tree embeddings.
Let Gbe a graph and let Tbe a collection of trees on the same vertices as G. We say the T
is a γdistortion if, for any vertices xand yand any T∈ T :
dG(x, y)≤dT(x, y)
and
E[dT(x, y)] ≤γdG(x, y )
The expected value is taken over a distribution given to the elements in T. It has been shown in
[2] that for an arbitrary graph we can have γ=O(log nlog log n). This result has been improved
to a O(log n) distortion in [1], which is essentially optimal.
23.2 Construction
Let Gbe a graph with nnodes and diameter ∆ = 2δ. The constructions for good tree embeddings
have the set Tmade of trees of the following form. They are calle Hierarchically well seperated
trees, which means the distance from each parent to its children is the same, and at each level the
distance decrease by a constant factor. A tree will have a root that corresponds to the entire graph.
Then, considering each node in the tree as a subset of the graph, the children nodes of each node,
v, will be a partition of the vertices of the graph in the subset corresponding to v. We shall make
it so that a node at a depth of ishall correspond to a subset of the graph with diameter less than
2δ−i. Notice that at depth δwe are left with the vertices of the original graph.
We set the cost of traveling on an edge of depth ito be 2δ−i. Thus, the cost of getting from
a vertex uto a vertex vin one of our trees is at least the cost of getting from uto vin the graph
since in a tree you have to travel up to a subset that contains both uand vand then back down.
This cost will be large enough from the cost put on edges of depth i. Thus, we have that a set of
trees with this property satisfies the first requirement to be a γdistotion embedding of G. The key
to the argument, is that if the graph is partitioned well then the expected distance between two
points won’t change much. As seen by Bartal, if a metric graph can be probabilistically partitioned
into pieces where the diameter has decreased by a constant factor and the chance an edge is cut
is about its length times λover the diameter, then, by recursively using these partitions for our
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