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Approximating metrics by tree metrics
Kunal TalwarMicrosoft Research Silicon Valley
Joint work with
Jittat FakcharoenpholKasetsart University
Thailand
Satish RaoUC Berkeley
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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
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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
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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
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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
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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
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Question
Can any metric be approximated by a tree metric?
Approximately
Easy solution
Approximately optimal solution
Princeton 2011
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The cycle
• Shortest path metric on a cycle. 1
111
1
1 11
Princeton 2011
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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
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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 .
• Extra edges don’t help
1
1
11
1 11
1
2 31
1
43
Princeton 2011
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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 .
• Extra vertices don’t help either
1
1
11
1 11
22
2
2
2
2
2
2
Princeton 2011
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[Karp 89] Cut an edge at random !
…but Dice help
1
111
1
1 11
u v
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…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
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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
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QuestionCan any metric be probabilistically approximated by a
tree metric?
Approximately
Easy solution
Approximately optimal solution
(in Expectation)
Princeton 2011
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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.
Princeton 2011
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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
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Approximating by tree metrics
High level outline:
1. Hierarchically decompose the points in the metric– Geometrically decreasing
diameters
2. Convert clustering into tree
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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
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Bounding Distortion
If separated at level
Dia
Dia
Dia
2𝑖− 1
2𝑖− 2
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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
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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
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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
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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
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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
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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
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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
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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
![Page 29: Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with Jittat Fakcharoenphol Kasetsart University Thailand](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d485503460f94a23641/html5/thumbnails/29.jpg)
Bounding Distortion
• For any edge
• Overall
Princeton 2011
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The blaming game
• Suppose cut at level
• It blames the first center which captured but not
Princeton 2011
𝑣𝑢
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For to cut – falls in a range of length ( Pr.
)
𝑢𝑣 𝑡 𝑘Δ4
𝜌
𝑡 2𝑡 1
𝑢
Princeton 2011
Δ2
𝑣
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𝑢
𝑣
Princeton 2011
𝑢𝑣 𝑡 𝑘
𝜌
𝑡 2𝑡 1
For to cut – falls in a range of length ( Pr.
)
Δ4
Δ2
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For to cut – falls in a range of length ( Pr.
)
– should occur before in ( Pr. )
𝑢
𝑣
Princeton 2011
𝑢𝑣 𝑡 𝑘
𝜌
𝑡 2𝑡 1 Δ4
Δ2
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Overall probability that separated:
Sum for
Princeton 2011
𝑢𝑣 𝑡 𝑘
𝜌
𝑡 2𝑡 1
For to cut – falls in a range of length ( Pr.
)
– should occur before in ( Pr. )
Δ4
Δ2
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Thus…
• Any metric can be probabilistically approximated by tree metrics.
Princeton 2011
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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
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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
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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?
Princeton 2011
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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.
Princeton 2011
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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
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Princeton 2011