Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

1
Embedding Metrics into Ultrametrics and Graphs
into Spanning Trees with Constant Average
Distortion

• Ittai Abraham, Yair Bartal, Ofer Neiman
• The Hebrew University

2
Embedding Metric Spaces

• Metric spaces MX(X,dX), MY(Y,dy)
• Embedding is a function f X?Y
• For u,v in X, non-contracting embedding f
  distf(u,v) dy(f(u),f(v)) / dx(u,v)
• Distortion dist(f) maxu,v ? X
  distf(u,v)

3
Two Schemes

1. Embedding a graph into a spanning tree of the
   graph.
2. Embedding a metric into an ultrametric

?(A)

• Metric on leaves of rooted labeled tree.
• 0 ?(D) ?(B) ?(A).
• d(x,y) ?(lca(x,y)).
• d(x,y) ?(D).
• d(x,w) ?(B).
• d(w,z) ?(A).

?(C)
?(B)
Given a weighted graph, the distance between 2
points is the length of the shortest path between
them
?(D)
x
y
z
w
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Motivation

• Simple and compact representation of a metric
  space.
• Ultrametric embedding provides approximation
  algorithms to numerous NP-hard problems.
• Constructing a spanning tree is a well studied
  network design objective.

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Previous Results

• For embedding n point metric into ultrametrics
• A single ultrametric/tree requires T(n)
  distortion. Bartal 96/BLMN 03/HM 05/RR 95.
• Probabilistic embedding with T(log n) expected
  distortion. Bartal 96,98,04, FRT 03
• Embedding into spanning trees
• Minimum Spanning Tree n-1 distortion.
• Probabilistic embedding with Õ(log2n) expected
  distortion. EEST 05

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Average Distortion

• Average distortion
• lq-distortion
• Any metric embeds into Hilbert space with
  constant average distortion ABN 06.
• Any metric probabilistically embeds into
  ultrametrics with constant average distortion
  ABN 05/06, CDGKS 05.
• Also Simultaneously tight lq-distortion for all
  q.

l8-dist distortion l1-dist average distortion.
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Our Results

• An embedding of any n point metric into a single
  ultrametric.
• An embedding of any graph on n vertices into a
  spanning tree of the graph.
• Average distortion O(1).
• l2-distortion
• lq-distortion T(n1-2/q), for 2ltq8

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Embeddings with scaling distortion

• Definition f has scaling distortion a, if for
  every e there exist at least pairs
  (u,v) such that distf(u,v) a(e).

Thm Every metric space embeds into an
ultrametric and every graph has a spanning tree
with scaling distortion

• For e¼, ¾ of pairs
• have distortion lt c2
• For e1/16, 15/16 of pairs
• have distortion lt c4
• For e1/n2, all pairs
• have distortion lt cn

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Additional Result

• Thm Any graph probabilistically embeds into a
  distribution of spanning trees with expected
  scaling distortion Õ(log2(1/e)).
• Implies that the lq-distortion is bounded by O(1)
  for any fixed 1qlt8.
• For q8 matches the EEST 05 result.

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Embedding into an ultrametric

• Partition X into 2 sets X1, X2
• Create a root labeled ? diam(X).
• The children of the root are created recursively
  on X1, X2
• Plan show for all e, at most e fraction of
  distances are distorted too much.
• Using induction, for all 0lte1 simultaneously
• Be distorted distances for current level
  and e.

X
X1
X2
A separated pair (x,y) is distorted too much if
?
X1
X2
Be eX1X2
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Partition Algorithm

• Fix some point u, such that B(u,?/2)ltn/2
  fix a constant c 1/150.
• Goal find rgt0, define X1B(u,r), X2X\X1 .
• Such that for all egt0
• (the set of possible bad pairs)

X2
X1
r
u
S1
A separated pair (x,y) is distorted if
S2
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Partition Algorithm

• Let
• Choose r from the interval
• Claim 1 The interval is sparse, contains at
  most points.
• Claim 2 Any r in the interval is good for all
• Proof
• By the maximality of ,
• Clearly S1X1.

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Small values of e

• Claim 3 There exists some r in the interval
  which is good for all simultaneously.
• While there exists uncolored r in the interval
  which is bad for some
• Take uncolored ri with largest bad .
• Color the segment of length around ri.

r is bad for e if letting X1B(u,r) will imply
BegteX1X2
r1
r2
r3
u
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Small values of e
Every point can be at most at 2 bad segments
Bound on the length of all the bad segments

• T number of points in all bad segments.

By claim 1 the interval contains at most
points
S2
S1
A bad segment contains at least
points Otherwise Be is bounded by
r1
r2
u
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Embedding into a Spanning Tree

• The spanning tree is created by a hierarchical
  star decomposition that uses ideas from EEST
  05.
• The decomposition for ultrametrics is in the
  heart of the star decomposition.
• Furthermore, the spanning tree construction
  requires some additional ideas.

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Star Decomposition
A point z is in the cone with radius r
if d(z,x1)d(x1,x0)-d(z,x0)r

• Let R be the radius for x0.
• Cut a central ball X0 with radiusR/2.
• While un-assigned points exist
• Let xi with a neighbor yi.
• Apply decompose algorithm with cone-radius akR.
• (klevel of recursion).
• Add edges (xi,yi) to the tree.
• Continue recursively inside each cluster.

x1
y1
x0
y2
x2
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Cone-radius
If u,v are separated then dT(u,v)lt2rad(TX)

• Cone-radius akR loss of 1/ak in distortion.
• Tree radius blow-up
• EEST chose a1/log n
• To ensure small blow-up and scaling distortion
  take
• as long as
• rad(X) decreases geometrically.
• Work for all eltelim

n size of original metric ? radius of
original metric
x1
y1
x0
y2
x2
Reset the parameters and k when this fails
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Conclusion

• An scaling approximation of
• Metrics by ultrametrics.
• Graphs by spanning trees.
• Implies constant approximation on average.
• Implies l2-distortion.
• A Õ(log2(1/e)) scaling probabilistic
  approximation of graphs by a random spanning
  tree.
• Implies constant lq-distortion for all fixed qlt8.