A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics

A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics J. Fakcharoenphol, S. Rao and K. Talwar Presented By Noam Arkind and Liah Kor

Paper Result • Any n-point metric space can be probabilistically embedded into a distribution of Tree metrics with distortion

Outline • Introduction • Probabilistic Embeddings • The Algorithm • Analysis of the algorithm • Derandomization • Applications

Introduction • Goal: approximating a given metric by a “simpler” metric • Tree metrics are favored from algorithmic point of view • There are simple graphs for which the distortion is • Probabilistic embeddings may give better bounds on the distortion

Probabilistic Embeddings • Probability distribution over a family of metrics • Distance between points is the expected value over the distribution • Distortion is computed with respect to the expected distances between points

The Algorithm – Assumptions and Definitions • Let ∆ be the diameter of the metric (V,d) • We assume w.l.o.g that the smallest distance is more than 1 and that for some • A metric (V’,d’) is said to dominate (V,d) if for all

The Algorithm – Assumptions and Definitions cont. • Let S be a family of metrics over V and D a probability distribution over S. (S,D) - probabilistically approximates (V,d) if : • Every metric in S dominates d • For every pair of vertices

The Algorithm – Assumptions and Definitions cont. • r-cut decompositionof (V, d) is a partitioning of V into clusters, each centered around a vertex and having radius at most r (diameter at most 2r). • hierarchical–cut-decomposition of (V, d) is sequence of clusters s.t : • is –cut decomposition of s.t each cluster in is contained in some cluster in (Each cluster in has radius 1 thus must be a singleton vertex)

The Algorithm – Assumptions and Defintions cont. • Hierarchical decomposition defines a laminar family and corresponds to a rooted tree:

The Algorithm – Assumptions and Definitions • We define a distance function on the resulting tree as follows: • A link between a node in and each of it’s child nodes has length • is the length of the shortest path in T between node {u} and node {v} So dominates • if (u,v) is first cut at level :

The Algorithm –Partition(V,d) • Choose random permutation on • Choose randomly from the distribution • While has non singleton clusters, do: • For do: • For every cluster in : • Create a new cluster consisting of all unassigned vertices in S closer than to

Analysis of the algorithm • Observation: For any • Next we’ll show that the expected value for is bounded by

Analysis of the algorithm cont. • Let (u,v) be an arbitrary edge • Center w is said to settle edge (u,v) at level if it is the first center to which at least one of u,vget assigned • Center w is said to cutedge (u,v) at level , if it settles the edge at this level but exactly one of u,v is assigned to w at this level

Derandomization Original problem: find distribution over tree metrics with small edge stretch. Dual problem: find a single tree with small (weighted) average edge stretch. Given weights ,find tree metric such that

Region Growing Lemma • We define the volume of an edge as • Let ,we assume • We imagine placing vertices arbitrary close to each other along the edges, call them volume elements. • W(t,r) is the volume of the neighborhood B(t,r) around a volume element, each egde (u,v) cut by B(t,r) contributes • Let

Region Growing Lemma cont. • Lemma: for any volume element t, there exists a series of radiisuch that • We can also relax the assumption that

The Deterministic Algorithm • Let be an upper bound on the diameter of G. • Let t be the volume element that maximizes , we cut out where is defined with respect to t, and recurse on the sub pieces. • We get a tree from the same laminar family as before. • Each edge in this cut has tree length

The Deterministic Algorithm cont. • We charge the cost of this cut, i.e. to the volume in • Each unit of volume t’ in get charged (by the lemma) • The total charge to t’ is bounded by

Applications

A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics