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

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

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The cycle

• Shortest path metric on a cycle. 1

111

1

1 11

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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

• Shortest path metric on a cycle.

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

• Extra edges don’t help

1

1

11

1 11

1

2 31

1

43

Princeton 2011

The cycle

• Shortest path metric on a cycle.

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

• 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

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QuestionCan any metric be probabilistically approximated by a

tree metric?

Approximately

Easy solution

Approximately optimal solution

(in Expectation)

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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.

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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]

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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

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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

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

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

Decomposition algorithm

Princeton 2011

1. Pick a random radius uniformly

2. Pick random permutation of vertices

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

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

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Bounding Distortion

• For any edge

• Overall

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The blaming game

• Suppose cut at level

• It blames the first center which captured but not

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𝑣𝑢

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

)

𝑢𝑣 𝑡 𝑘Δ4

𝜌

𝑡 2𝑡 1

𝑢

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Δ2

𝑣

𝑢

𝑣

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𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1

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

)

Δ4

Δ2

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

)

– should occur before in ( Pr. )

𝑢

𝑣

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𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1 Δ4

Δ2

Overall probability that separated:

Sum for

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𝑢𝑣 𝑡 𝑘

𝜌

𝑡 2𝑡 1

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

)

– should occur before in ( Pr. )

Δ4

Δ2

Thus…

• Any metric can be probabilistically approximated by tree metrics.

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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.

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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?

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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.

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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