Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with...

Approximating metrics by tree metrics

Kunal TalwarMicrosoft Research Silicon Valley

Joint work with

Jittat FakcharoenpholKasetsart University

Thailand

Satish RaoUC Berkeley

Metric

Metric

(shortest path distances in a graph)

Show up in various optimization problems– often as solutions to relaxations

0 10 15 5

0 25 15

0 20

0

10

20

5 2515

15

a

d c

b

Princeton 2011

BatB Network design

Given source-sink pairsBuild a network so as to route one unit

of flow from each to .

Cost of building edge with capacity

Concave Cost Function

𝑐 ( 𝑓 )

𝑓T1

Optical fiber

Princeton 2011

Tree metrics

• Shortest path metric on a weighted tree

• Simple to reason about

• Easier to design algorithms which are simple and/or fast.

10

515

a

d c

b

Princeton 2011

BatB Network design

Given source-sink pairsBuild a network so as to route one unit

of flow from each to .

Cost of building edge with capacity

Unique pathsEasy on trees

𝑐 ( 𝑓 )

𝑓T1

Optical fiber

Princeton 2011

Question

Can any metric be approximated by a tree metric?

Approximately

Easy solution

Approximately optimal solution

Princeton 2011

The cycle

• Shortest path metric on a cycle. 1

111

1

1 11

Princeton 2011

The cycle

• Shortest path metric on a cycle.

• [Rabinovich-Raz98, Gupta01] Any embedding of this metric into a tree incurs distortion .

1

1

11

1 11

Princeton 2011

The cycle



• Extra edges don’t help

1

1

11

1 11

1

2 31

1

43

Princeton 2011

The cycle



• Extra vertices don’t help either

1

1

11

1 11

22

2

2

2

2

2

2

Princeton 2011

[Karp 89] Cut an edge at random !

…but Dice help

1

111

1

1 11

u v

…but Dice help

[Karp 89] Cut an edge at random !

• Expected stretch of any fixed edge is at most 2.

1

111

1

1 11

u v

Probabilistic Embedding

1

111

1

1 11

u v

Probabilistic Embedding

Embed into a probability distribution over trees

such that:

• For each tree

• Expected value of

Distortion

Princeton 2011

QuestionCan any metric be probabilistically approximated by a

tree metric?

Approximately

Easy solution

Approximately optimal solution

(in Expectation)

Princeton 2011

Why?

• Several problems are easy (or easier) on trees:

Network design, Group Steiner tree, k-server, Metric labeling, Minimum communication cost spanning tree, metrical task system, Vehicle routing, etc.

Princeton 2011

History

• [Alon-Karp-Peleg-West-92] Defined the problem; upper bound; lower bound.

• [Bartal96] upper bound; several applications

• [Bartal98] upper bound

• [Fakcharoenphol-Rao-T-03] upper bound

Princeton 2011

Approximating by tree metrics

High level outline:

1. Hierarchically decompose the points in the metric– Geometrically decreasing

diameters

2. Convert clustering into tree

Distances Increase

High level outline:

1. Hierarchically decompose the points in the metric– Geometrically decreasing

diameters

2. Convert clustering into tree

Suppose

Then separated when cluster diameter is

Thus

Dia

Dia

Dia

2𝑖− 1

2𝑖− 2

Bounding Distortion

If separated at level

Dia

Dia

Dia

2𝑖− 1

2𝑖− 2

Low Diameter Decomposition

• Thus main problem: decomposition

• Given a set of points, break into clusters of diameter at most

• Ensure small compared to

Princeton 2011

Our techniques

• Techniques used in approximating 0-extension problem by [Calinscu-Karloff-Rabani-01]

• Improved algorithm and analysis used in [Fakcharoenphol-Harrelson-Rao-T.-03]

Princeton 2011

Decomposition algorithm

1. Pick a random radius uniformly

2. Pick random permutation of vertices

3. For , captures all uncaptured vertices in a ball of radius

Princeton 2011


Princeton 2011




Bounding Distortion

• For any edge

• Overall

Princeton 2011

The blaming game

• Suppose cut at level

• It blames the first center which captured but not

Princeton 2011

𝑣𝑢

For to cut – falls in a range of length ( Pr.

)

𝑢𝑣 𝑡 𝑘Δ4

𝜌

𝑡 2𝑡 1

𝑢

Princeton 2011

Δ2

𝑣

𝑢

𝑣

Princeton 2011

𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1


)

Δ4

Δ2


)

– should occur before in ( Pr. )

𝑢

𝑣

Princeton 2011

𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1 Δ4

Δ2

Overall probability that separated:

Sum for

Princeton 2011

𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1


)

– should occur before in ( Pr. )

Δ4

Δ2

Thus…

• Any metric can be probabilistically approximated by tree metrics.

Princeton 2011

Few terminals case

[GNR10] Given a set of terminals, we can find a distribution over trees such that

Leads to approximation to BatB when we have k source-sink pairs.Leads to capacity-approximating a graph by a tree when we care about terminals.

E.g. for Steiner linear arrangement.[CLLM10, EKGRTT10, MM10]

Princeton 2011

Remarks

Given metric , weights on pairs of vertices, find one tree such that

Can be phrased as a dual of the probabilistic embedding problem [CCGGP98]

Allows us to get trees in our distribution.Duality very useful. E.g. to get capacity maps.

Princeton 2011

More remarks

Tree has geometrically decreasing edge lengths (HST)useful for some problems

Simultaneous padding at all levels [GHR06]

Decompositions useful in other settings.[KLMN04] Volume respecting embeddings[GKL04] Decomposition of doubling metrics

Probabilistic embeddings into spanning trees[EEST05,ABN08] Distortion

Can we get the optimal bound?

Princeton 2011

BatB Network Design

• Let be the optimal solution on G• Expected Cost of on the tree is • Thus

• Alg produces optimal solution on tree. Thus

• Embedding was deterministically expanding. Thus cost of on the original metric is only smaller.

Princeton 2011

Summary

• Any metric can be probabilistically approximated by expanding HSTs with distortion

• Useful for approximation and online algorithms

• Decomposition lemma has many applications

• Bottom up embedding?• Other useful abstractions of graph properties?

• approximation for the best tree embedding for a given metric?

Princeton 2011

Princeton 2011

Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with...

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