A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics

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A Tight Bound on Approximating Arbitrary Metrics
by Tree Metrics

• J. Fakcharoenphol, S. Rao and K. Talwar
• Presented By
• Noam Arkind and Liah Kor

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Paper Result

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

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Outline

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

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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

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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

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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

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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

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The Algorithm Assumptions and Definitions cont.

• r-cut decomposition of (V, d) is a partitioning
  of V into clusters, each centered around a vertex
  and having radius at most r (diameter at most
  2r).
• hierarchicalcut-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)

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The Algorithm Assumptions and Defintions cont.

• Hierarchical decomposition defines a laminar
  family and corresponds to a rooted tree

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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 its
  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

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The Algorithm Partition(V,d)

1. Choose random permutation on
2. Choose randomly from the distribution
4. While has non singleton clusters, do
6. For do
7. For every cluster in
8. Create a new cluster consisting of all unassigned
   vertices in S closer than to

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Analysis of the algorithm

• Observation
• For any
• Next well show that the expected value for
  is bounded by

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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,v get assigned
• Center w is said to cut edge (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

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Analysis of the algorithm cont.

• When w cuts (u,v) at level i, it adds
  to the tree length of the edge (u,v)
• Define , to be the indicator
  function for the above event
• Define

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Analysis of the algorithm cont.

• Arrange the vertices of V in order of increasing
  distance from the edge (u,v)
• Let be the sth vertex
• w.l.o.g
• For to cut (u,v), it must be
• for some
• settles (u,v) at level
• The contribution to is at most

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Analysis of the algorithm cont.

• Let
• The probability that some falls in
  is at most
• The probability of event (b) conditioned on (a)
  is

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Analysis of the algorithm cont.
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Analysis of the algorithm cont.

• By using linearity of expectation

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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

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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

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Region Growing Lemma cont.

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

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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

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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

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Applications