Ramsey Spanning Trees and their Applications

Arnold Filtser

Ben-Gurion University

Co-authors: Ittai Abraham, Shiri Chechik,Michael Elkin, Ofer Neiman

Workshop on Data SummarizationUniversity of Warwick

Arnold Filtser Ramsey Spanning Trees and their Applications 1 / 27

Metric Embeddings

(Rd, ‖·‖2)f : X → Rd

(X, dX)

Embedding f : X → Rd has distortion α if for all x , y ∈ X

dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ α · dX (x , y)

Theorem (Bourgain,85)

Every n-point metric (X , dX ) is embeddable into Euclidean spacewith distortion O(log n).

Asymptotically tight.

Arnold Filtser Ramsey Spanning Trees and their Applications 2 / 27

Metric Embeddings

(Rd, ‖·‖2)f : X → Rd

(X, dX)

Embedding f : X → Rd has distortion α if for all x , y ∈ X

dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ α · dX (x , y)

Theorem (Bourgain,85)

Every n-point metric (X , dX ) is embeddable into Euclidean spacewith distortion O(log n).

Asymptotically tight.

Arnold Filtser Ramsey Spanning Trees and their Applications 2 / 27

Metric Embeddings

(Rd, ‖·‖2)f : X → Rd

(X, dX)

Embedding f : X → Rd has distortion α if for all x , y ∈ X

dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ α · dX (x , y)

Theorem (Bourgain,85)

Every n-point metric (X , dX ) is embeddable into Euclidean spacewith distortion O(log n).

Asymptotically tight.

Arnold Filtser Ramsey Spanning Trees and their Applications 2 / 27

Metric Ramsey-Type Problem

For a fixed distortion k > 1, what is the largest subset M ⊂ X ,s.t. (M , dX ) is embeddable into Euclidean space with distortion k?

(Rd, ‖·‖2)f : X → Rd

(X, dX)

∀x , y ∈ M , dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ k · dX (x , y)

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Metric Ramsey-Type Problem

For a fixed distortion k > 1, what is the largest subset M ⊂ X ,s.t. (M , dX ) is embeddable into Euclidean space with distortion k?

(Rd, ‖·‖2)f : M → Rd

M

M

(X, dX)

∀x , y ∈ M , dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ k · dX (x , y)

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Metric Ramsey-Type Problem

For a fixed distortion k > 1, what is the largest subset M ⊂ X ,s.t. (M , dX ) is embeddable into Euclidean space with distortion k?

(Rd, ‖·‖2)f : M → Rd

M

M

(X, dX)

∀x , y ∈ M , dX (x , y) ≤ ‖f (x)− f (y)‖2 ≤ k · dX (x , y)

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Asymptotically tight.

Euclidean space can be replace here by an ultrametric U! (a.k.a HST)

Ultrametric is a spacial kind of tree which is:

1 Very useful for divide an conquer algorithms.

2 Isometrically embeds into Euclidean space (i.e. distortion 1).

Arnold Filtser Ramsey Spanning Trees and their Applications 4 / 27

Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Asymptotically tight.

Euclidean space can be replace here by an ultrametric U! (a.k.a HST)

Ultrametric is a spacial kind of tree which is:

1 Very useful for divide an conquer algorithms.

2 Isometrically embeds into Euclidean space (i.e. distortion 1).

Arnold Filtser Ramsey Spanning Trees and their Applications 4 / 27

Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into Euclidean space

with distortion O(k).

Asymptotically tight.

Euclidean space can be replace here by an ultrametric U! (a.k.a HST)

Ultrametric is a spacial kind of tree which is:

1 Very useful for divide an conquer algorithms.

2 Isometrically embeds into Euclidean space (i.e. distortion 1).

Arnold Filtser Ramsey Spanning Trees and their Applications 4 / 27

Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(k).

Asymptotically tight.

Euclidean space can be replace here by an ultrametric U! (a.k.a HST)

Ultrametric is a spacial kind of tree which is:

1 Very useful for divide an conquer algorithms.

2 Isometrically embeds into Euclidean space (i.e. distortion 1).

Arnold Filtser Ramsey Spanning Trees and their Applications 4 / 27

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(k).

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(128 · k).

The constant in the distortion important as it in the exponent.

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(128 · k).

The constant in the distortion important as it in the exponent.

Paper This Distortion Size DeterministicBFM06 O(k log log n) n1−1/k

BLMN04 O(k log k) n1−1/k

MN07 128 · k n1−1/k

BGS16 33 · k n1−1/k

NT12 2e · k n1−1/k

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(128 · k).

The constant in the distortion important as it in the exponent.

Paper This Distortion Size DeterministicBFM06 O(k log log n) n1−1/k RandomizedBLMN04 O(k log k) n1−1/k RandomizedMN07 128 · k n1−1/k RandomizedBGS16 33 · k n1−1/k RandomizedNT12 2e · k n1−1/k Randomized

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(128 · k).

The constant in the distortion important as it in the exponent.

Paper This Distortion Size DeterministicBFM06 O(k log log n) n1−1/k RandomizedBLMN04 O(k log k) n1−1/k RandomizedMN07 128 · k n1−1/k RandomizedBGS16 33 · k n1−1/k RandomizedNT12 2e · k n1−1/k RandomizedThis Paper 8 · k − 2 n1−1/k Deterministic

Our Second Result: Metric Ramsey-Type Problem

Theorem (Mendel, Naor 07, following BFM86, BLMN05)

For every n-point metric space and k ≥ 1, there exists a subset Mof size n1−1/k that can be embedded into ultrametric

with distortion O(128 · k).

The constant in the distortion important as it in the exponent.

Paper This Distortion Size DeterministicBFM06 O(k log log n) n1−1/k RandomizedBLMN04 O(k log k) n1−1/k RandomizedMN07 128 · k n1−1/k RandomizedBGS16 33 · k n1−1/k RandomizedNT12 2e · k n1−1/k RandomizedThis Paper 8 · k − 2 n1−1/k Deterministic

*Bartal had similar (deterministic) result.

Our Second Result: Metric Ramsey-Type Problem

Theorem (Our Secondary Result)

For every n-point metric space and k ≥ 1, there is adeterministic algorithm that finds a subset M of size n1−1/k thatcan be embedded into ultrametric with distortion 8 · k .

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our Second Result: Metric Ramsey-Type Problem

Theorem (Our Secondary Result)

For every n-point metric space and k ≥ 1, there is adeterministic algorithm that finds a subset M of size n1−1/k thatcan be embedded into ultrametric with distortion 8 · k .

Instead of preserving distance for M ×M ,we can preserve distances for M × X .

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our Second Result: Metric Ramsey-Type Problem

Theorem (Our Secondary Result)

For every n-point metric space and k ≥ 1, there is adeterministic algorithm that finds a subset M of size n1−1/k suchthat the hall metric can be embedded into ultrametric

with distortion 16 · k w.r.t M × X .

Instead of preserving distance for M ×M ,we can preserve distances for M × X .

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our Second Result: Metric Ramsey-Type Problem

Theorem (Our Secondary Result)

For every n-point metric space and k ≥ 1, there is adeterministic algorithm that finds a subset M of size n1−1/k suchthat the hall metric can be embedded into ultrametric

with distortion 16 · k w.r.t M × X .

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our Second Result: Metric Ramsey-Type Problem

Theorem (Our Secondary Result)

For every n-point metric space and k ≥ 1, there is adeterministic algorithm that finds a subset M of size n1−1/k suchthat the hall metric can be embedded into ultrametric

with distortion 16 · k w.r.t M × X .

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our Second Result: Metric Ramsey-Type Problem

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

U1 Ui = home(x) Uk·n1/k

(X, dX)

x

Distance Oracle

A distance oracle is a succinct data structure that (approximately)answers distance queries.

The properties of interest are size, distortion and query time.

Arnold Filtser Ramsey Spanning Trees and their Applications 7 / 27

Distance Oracle

A distance oracle is a succinct data structure that (approximately)answers distance queries.

The properties of interest are size, distortion and query time.

Arnold Filtser Ramsey Spanning Trees and their Applications 7 / 27

Distance Oracle

A distance oracle is a succinct data structure that (approximately)answers distance queries.

The properties of interest are size, distortion and query time.

Arnold Filtser Ramsey Spanning Trees and their Applications 7 / 27

Distance Oracles: State of the ArtDO Distortion Size Query Deterministic?TZ05 2k − 1 O(k · n1+1/k) O(k) noMN07 128k O(n1+1/k) O(1) noW13 (2 + ε)k O(k · n1+1/k) O(1/ε) noC14 2k − 1 O(k · n1+1/k) O(1) noC15 2k − 1 O(n1+1/k) O(1) no

RTZ05 2k − 1 O(k · n1+1/k) O(k) yesW13 2k − 1 O(k · n1+1/k) O(log k) yes

Arnold Filtser Ramsey Spanning Trees and their Applications 8 / 27

Our contribution: Deterministic Distance OraclesDistance Oracle Distortion Size QueryRTZ05 2k − 1 O(k · n1+1/k) O(k)W13 2k − 1 O(k · n1+1/k) O(log k)This paper 8(1 + ε)k O(n1+1/k) O(1/ε)This paper+C14 2k − 1 O(k · n1+1/k) O(1)

Arnold Filtser Ramsey Spanning Trees and their Applications 9 / 27

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

U1 Ui = home(x) Uk·n1/k

(X, dX)

x

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Theorem (Tree Distance Oracle, HT84, BFC00)

For every tree metric, there is an exact distance oracle of linearsize and constant query time.

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Theorem (Tree Distance Oracle, HT84, BFC00)

For every tree metric, there is an exact distance oracle of linearsize and constant query time.

Theorem (Ramsey based Deterministic Distance Oracle)

For any n-point metric space, there is a distance oracle with :

Distortion Size Query time16 · k O(k · n1+1/k) O(1)

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Theorem (Ramsey based Deterministic Distance Oracle)

For any n-point metric space, there is a distance oracle with :

Distance Oracle k Distortion Size QueryThis paper 16 · k O(k · n1+1/k) O(1)This paper 8(1 + ε)k O(n1+1/k) O(1/ε)This paper+C14 2k − 1 O(k · n1+1/k) O(1)

Our contribution: Deterministic Distance Oracles

CorollaryFor every n-point metric space and k ≥ 1,there is a set U of k · n 1

k ultrametrics and a mappinghome : X → U , such that for every x , y ∈ U ,

dhome(x)(x , y) ≤ (16 · k) · dX (x , y)

Theorem (Ramsey based Deterministic Distance Oracle)

For any n-point metric space, there is a distance oracle with :

Distance Oracle k Distortion Size QueryThis paper 16 · k O(k · n1+1/k) O(1)This paper 8(1 + ε)k O(n1+1/k) O(1/ε)This paper+C14 2k − 1 O(k · n1+1/k) O(1)C15 (Randomized) 2k − 1 O(n1+1/k) O(1)

Ramsey Spanning Tree Question

Given a weighted graph G = (V ,E ,w), and a fixed distortionk > 1, what is the largest subset M ⊂ V , such that:there is a spanning tree T of G with distortion k w.r.t M × V ?

Arnold Filtser Ramsey Spanning Trees and their Applications 11 / 27

Ramsey Spanning Tree Question

Given a weighted graph G = (V ,E ,w), and a fixed distortionk > 1, what is the largest subset M ⊂ V , such that:there is a spanning tree T of G with distortion k w.r.t M × V ?

For all v ∈ M and u ∈ V , dT (v , u) ≤ k · dG (v , u).

Arnold Filtser Ramsey Spanning Trees and their Applications 11 / 27

Main Result

Ramsey Spanning Tree QuestionGiven a weighted graph G = (V ,E ,w), and a fixed distortionk > 1, what is the largest subset M ⊂ V , such that:there is a spanning tree T of G with distortion k w.r.t M × V ?

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Theorem (Mendel, Naor 07)

For every n-point metric space (X , dX ) and k ≥ 1,there exists a subset M of size n1−1/k and an ultrametric U over X

with distortion O(k)

· log log n

w.r.t M × X .

Arnold Filtser Ramsey Spanning Trees and their Applications 12 / 27

Main Result

Ramsey Spanning Tree QuestionGiven a weighted graph G = (V ,E ,w), and a fixed distortionk > 1, what is the largest subset M ⊂ V , such that:there is a spanning tree T of G with distortion k w.r.t M × V ?

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Theorem (Mendel, Naor 07)

For every n-point metric space (X , dX ) and k ≥ 1,there exists a subset M of size n1−1/k and an ultrametric U over X

with distortion O(k)

· log log n

w.r.t M × X .

Arnold Filtser Ramsey Spanning Trees and their Applications 12 / 27

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Corollary

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

G T1 Ti = home(v) Tk·n1/k

v

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Corollary

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

G T1 Ti = home(v) Tk·n1/k

v

Corollary

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

G T1 Ti = home(v) Tk·n1/k

v

The union of all the trees in T creates an O(k · log log n)-spanner

with O(k · n1+ 1k ) edges.

Arnold Filtser Ramsey Spanning Trees and their Applications 14 / 27

Application: Compact Routing Scheme

Huge network

Arnold Filtser Ramsey Spanning Trees and their Applications 15 / 27

Application: Compact Routing Scheme

Huge network

There is a server in each node.

Arnold Filtser Ramsey Spanning Trees and their Applications 15 / 27

Application: Compact Routing Scheme

%$#32

Huge network

There is a server in each node.

Task: route packages throughout the network.

Arnold Filtser Ramsey Spanning Trees and their Applications 15 / 27

Application: Compact Routing Scheme

%$#32

Huge network

There is a server in each node.

Task: route packages throughout the network.

Store the whole network in each node is unfeasible.

Arnold Filtser Ramsey Spanning Trees and their Applications 15 / 27

Compact Routing Scheme

DestinationLabelContent

Initiatingnode

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Decisiontime

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Destination

Decisiontime

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Destination

Decisiontime

In order to keep other parameters small, we will allow stretch.

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Destination

Decisiontime

In order to keep other parameters small, we will allow stretch.Stretch k : the length of a route from v to u will be ≤ k · dG (v , u).

Compact Routing Scheme

Table001011100010010. . . . . .. . . . . .

LabelContent

DestinationLabelContent

Initiatingnode Routing

Destination

Decisiontime

In order to keep other parameters small, we will allow stretch.Stretch k : the length of a route from v to u will be ≤ k · dG (v , u).

Theorem (Thorup, Zwick, 01)

For any n-vertex tree T = (V ,E ), there is a routing scheme with :

Stretch Label Table Decision time1 O(log n) O(1) O(1)

Routing using Ramsey Spanning TreesFor every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

To route a package from u to v , we will simply route on home(v)!

The label of v will consist of:(

home(v), Labelhome(v)(v)

).

The table of v will consist of union of all tables in T .

G T1 Ti = home(v) Tk·n1/k

v

Routing using Ramsey Spanning TreesFor every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

To route a package from u to v , we will simply route on home(v)!

The label of v will consist of:(

home(v), Labelhome(v)(v)

).

The table of v will consist of union of all tables in T .

G T1 Ti = home(v) Tk·n1/k

v

Routing using Ramsey Spanning TreesFor every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

To route a package from u to v , we will simply route on home(v)!

The label of v will consist of:(

home(v), Labelhome(v)(v)

).

The table of v will consist of union of all tables in T .

G T1 Ti = home(v) Tk·n1/k

v

Routing using Ramsey Spanning TreesFor every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

To route a package from u to v , we will simply route on home(v)!

The label of v will consist of:(

home(v), Labelhome(v)(v)

).

The table of v will consist of union of all tables in T .

G T1 Ti = home(v) Tk·n1/k

v

Routing using Ramsey Spanning Trees

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,

there is a set T of k · n 1k spanning trees and a mapping

home : V → T , such that for every u, v ∈ V ,

dhome(v)(v , u) ≤ O(k · log log n) · dG (v , u)

To route a package from u to v , we will simply route on home(v)!

The label of v will consist of:(

home(v), Labelhome(v)(v)

).

The table of v will consist of union of all tables in T .

Theorem (Ramsey based Compact Routing Scheme)

For any n-vertex graph, there is a routing scheme with :

Stretch Label Table Decision time

O(k · log log n) O(log n) O(k · n 1k ) O(1)

Theorem (Ramsey based Compact Routing Scheme)

For any n-vertex graph, there is a routing scheme with :

Stretch Label Table Decision time

O(k · log log n) O(log n)

· k

O(k · n 1k ) O(1)

(initial: O(k))

Theorem (Thorup, Zwick 01, Chechik 13)

For any n-vertex graph, there is a routing scheme with :

Stretch Label Table Decision time

3.68k = O(k) O(k · log n) O(k · n 1k ) O(1) (initial: O(k))

By choosing k = log n, we get:

Stretch Label Table D. timeHere O(log n · log log n) O(log n) O(log n) O(1)

[TZ01] O(log n) O(log2 n) O(log n) O(1) (O(log n))

Arnold Filtser Ramsey Spanning Trees and their Applications 18 / 27

Theorem (Ramsey based Compact Routing Scheme)

For any n-vertex graph, there is a routing scheme with :

Stretch Label Table Decision time

O(k · log log n) O(log n)

· k

O(k · n 1k ) O(1)

(initial: O(k))

Theorem (Thorup, Zwick 01, Chechik 13)

For any n-vertex graph, there is a routing scheme with :

Stretch Label Table Decision time

3.68k = O(k) O(k · log n) O(k · n 1k ) O(1) (initial: O(k))

By choosing k = log n, we get:

Stretch Label Table D. timeHere O(log n · log log n) O(log n) O(log n) O(1)

[TZ01] O(log n) O(log2 n) O(log n) O(1) (O(log n))

Arnold Filtser Ramsey Spanning Trees and their Applications 18 / 27

Technical Ideas

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Framework: Petal decomposition.

Hierarchically padded decompositions.

Region growing.

Arnold Filtser Ramsey Spanning Trees and their Applications 19 / 27

Technical Ideas

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Framework: Petal decomposition.

Hierarchically padded decompositions.

Region growing.

Arnold Filtser Ramsey Spanning Trees and their Applications 19 / 27

Technical Ideas

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Framework: Petal decomposition.

Hierarchically padded decompositions.

Region growing.

Arnold Filtser Ramsey Spanning Trees and their Applications 19 / 27

Technical Ideas

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Framework: Petal decomposition.

Hierarchically padded decompositions.

Region growing.

Arnold Filtser Ramsey Spanning Trees and their Applications 19 / 27

Petal Decomposition

x0

Petal Decomposition

x0

x1

y1 = t0

t1

X1

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

y2

X1 X2

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

t3y2

y3

X1 X2

X3

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Petal Decomposition

Each cluster X (petal) has acenter vertex x .

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 21 / 27

Petal Decomposition

Each cluster X (petal) has acenter vertex x .

The radius ∆ defined w.r.tthe center.

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 21 / 27

Petal Decomposition

Each cluster X (petal) has acenter vertex x .

The radius ∆ defined w.r.tthe center.

The radius decrease by 34

factor in each hierarchi. step.

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

≤ 34

≤ 34

≤ 34

Arnold Filtser Ramsey Spanning Trees and their Applications 21 / 27

Petal Decomposition

Each cluster X (petal) has acenter vertex x .

The radius ∆ defined w.r.tthe center.

The radius decrease by 34

factor in each hierarchi. step.

The radius of T is at most 4times larger than in G .

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0≤ 4·

Arnold Filtser Ramsey Spanning Trees and their Applications 21 / 27

Petal Decomposition

Each cluster X (petal) has acenter vertex x .

The radius ∆ defined w.r.tthe center.

The radius decrease by 34

factor in each hierarchi. step.

The radius of T is at most 4times larger than in G .

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

uv

≤ 8·

CorollarySuppose v , u were separated while being in cluster of radius ∆. ThendT (v , u) ≤ 8 ·∆.

Arnold Filtser Ramsey Spanning Trees and their Applications 21 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

x0

t

Wlo Wr Whi

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

x0

t

Wlo Wr Whi

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

Set δ = ∆/(k · log log n).

x0

t

Wlo Wr Whi

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

Set δ = ∆/(k · log log n).

Vertex v s.t. B(v , δ) ⊆ Wr is padded.

x0

t

Wlo Wr Whi

vδ

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

Set δ = ∆/(k · log log n).

Vertex v s.t. B(v , δ) ⊆ Wr is padded.

All vertices out of Wr+δ \Wr−δ(restricted area) are padded.

x0

t

Wlo Wr Whi

vδ

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

Set δ = ∆/(k · log log n).

Vertex v s.t. B(v , δ) ⊆ Wr is padded.

All vertices out of Wr+δ \Wr−δ(restricted area) are padded.

x0

t

Wlo Wr Whi

vu

dG(u, v) = Ω(δ)

dT (u, v) = O(∆)

dT (u,v)dG(u,v) = O(∆

δ ) = O(k · log log n)

u′

δ

Padded vertices suffer distortion at most ∆/δ = O(k · log log n)!

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Growth

Degree of freedom:

parameter R ∈ [lo, hi] (hi− lo = ∆8

).

Wr denotes the petal (cluster)created for R = r .

Monotonicity: r ′ ≤ r ⇒ Wr ′ ⊆ Wr .

Set δ = ∆/(k · log log n).

Vertex v s.t. B(v , δ) ⊆ Wr is padded.

All vertices out of Wr+δ \Wr−δ(restricted area) are padded.

x0

t

Wlo Wr Whi

vu

dG(u, v) = Ω(δ)

dT (u, v) = O(∆)

dT (u,v)dG(u,v) = O(∆

δ ) = O(k · log log n)

u′

δ

Padded vertices suffer distortion at most ∆/δ = O(k · log log n)!

Goal: find r , with many padded vertices! (sparse restricted area).

Arnold Filtser Ramsey Spanning Trees and their Applications 22 / 27

Petal Decomposition

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 23 / 27

Petal Decomposition

A vertex which is padded in allthe levels will have smalldistortion w.r.t all other vertices.

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 23 / 27

Petal Decomposition

A vertex which is padded in allthe levels will have smalldistortion w.r.t all other vertices.

Goal: choose parameters(r ∈ [lo, hi ]) s.t. at least n1− 1

k

vertices will be padded in alllevels.

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 23 / 27

Petal Decomposition

A vertex which is padded in allthe levels will have smalldistortion w.r.t all other vertices.

Goal: choose parameters(r ∈ [lo, hi ]) s.t. at least n1− 1

k

vertices will be padded in alllevels.

A vertex is called active if it ispadded in all levels up till now.

x0

x1

y1 = t0

t1t2

x2

x3

x8

x7x5

x6

x4

t3

t8

t7

t5

t6

t4y2

y8y7

y5

y6

y4

y3

X1 X2

X3

X8

X7X5

X6

X4

X0

Arnold Filtser Ramsey Spanning Trees and their Applications 23 / 27

Region Growing

For petal Wr :

Active x ∈ Wr−δ remains active.

Active x ∈ Wr+δ \Wr−δceases to be active.

x0

Whi

r ∈ [lo, hi]

Wr = Wlo+δ

Wlo

t

Arnold Filtser Ramsey Spanning Trees and their Applications 24 / 27

Region Growing

For petal Wr :

Active x ∈ Wr−δ remains active.

Active x ∈ Wr+δ \Wr−δceases to be active.

x0

Whi

r ∈ [lo, hi]

Wr = Wlo+2δ

Wlo+δ

t

Arnold Filtser Ramsey Spanning Trees and their Applications 24 / 27

Region Growing

For petal Wr :

Active x ∈ Wr−δ remains active.

Active x ∈ Wr+δ \Wr−δceases to be active.

x0

Whi

r ∈ [lo, hi]

Wr = Wlo+3δ

Wlo+2δ

t

Arnold Filtser Ramsey Spanning Trees and their Applications 24 / 27

Region Growing

For petal Wr :

Active x ∈ Wr−δ remains active.

Active x ∈ Wr+δ \Wr−δceases to be active.

x0

Whi

r ∈ [lo, hi]

Wr = Wlo+iδ

Wlo+(i−1)δ

t

Arnold Filtser Ramsey Spanning Trees and their Applications 24 / 27

Region Growing

For petal Wr :

Active x ∈ Wr−δ remains active.

Active x ∈ Wr+δ \Wr−δceases to be active.

x0

Whi

r ∈ [lo, hi]

Wr = Wlo+iδ

Wlo+(i−1)δ

t

IntuitionThere is r ∈ [lo, hi ] such that Wr−δ is

large enough compared to Wr+δ.

Arnold Filtser Ramsey Spanning Trees and their Applications 24 / 27

IntuitionThere is r ∈ [lo, hi ] such that Wr−δ is

large enough compared to Wr+δ.

IntuitionThere is r ∈ [lo, hi ] such that Wr−δ is

large enough compared to Wr+δ.

IntuitionThere is r ∈ [lo, hi ] such that Wr−δ is

large enough compared to Wr+δ.

Corollary

At least n1−1/k vertices remainactive at the end of the process.

IntuitionThere is r ∈ [lo, hi ] such that Wr−δ is

large enough compared to Wr+δ.

Corollary

At least n1−1/k vertices remainactive at the end of the process.

Theorem (Main Result)

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n) w.r.t M × V .

Open Questions

1 Remove the log log n factor.

Conjecture

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n//////////) w.r.t M × V .

2 Improve construction for deterministic distance oracle.

Distance Oracle Distortion Size QueryThis paper+C14 2k − 1 O(k · n1+1/k) O(1)C15 (Randomized) 2k − 1 O(n1+1/k) O(1)

3 Find more applications to Ramsey spanning trees!

Arnold Filtser Ramsey Spanning Trees and their Applications 26 / 27

Open Questions

1 Remove the log log n factor.

Conjecture

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n//////////) w.r.t M × V .

2 Improve construction for deterministic distance oracle.

Distance Oracle Distortion Size QueryThis paper+C14 2k − 1 O(k · n1+1/k) O(1)C15 (Randomized) 2k − 1 O(n1+1/k) O(1)

3 Find more applications to Ramsey spanning trees!

Arnold Filtser Ramsey Spanning Trees and their Applications 26 / 27

Open Questions

1 Remove the log log n factor.

Conjecture

For every n-vertex weighted graph G = (V ,E ,w) and k ≥ 1,there exists a subset M of size n1−1/k and spanning tree T of G

with distortion O(k · log log n//////////) w.r.t M × V .

2 Improve construction for deterministic distance oracle.

Distance Oracle Distortion Size QueryThis paper+C14 2k − 1 O(k · n1+1/k) O(1)C15 (Randomized) 2k − 1 O(n1+1/k) O(1)

3 Find more applications to Ramsey spanning trees!

Arnold Filtser Ramsey Spanning Trees and their Applications 26 / 27