Ramsey Problem ADO ADO construction Conclusion
Ramsey Partitions & Proximity Data Structures Manor Mendel1
1
Assaf Naor2
The Open University of Israel 2 Microsoft Research
November ’06
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Outline
1
Metric Ramsey-type Problem
2
Approximate Distance Oracles
3
Construction of approximate distance oracles
4
Conclusion
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Outline
1
Metric Ramsey-type Problem
2
Approximate Distance Oracles
3
Construction of approximate distance oracles
4
Conclusion
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Metric Embedding & Distortion 2
1
2
1
1 2 X = K1,3
Definition (Best distortion) cH (X ) = inf{dist(e)| e : X → H} D
Notation: cH (X ) ≤ D ⇔ X ,→ H. cH (X ) measures the faithfulness of the representing X by H. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Metric Embedding & Distortion z 2
1
2
1
1
y
x
2 X = K1,3
R3 (3D Euclidean Space)
Definition (Best distortion) cH (X ) = inf{dist(e)| e : X → H} D
Notation: cH (X ) ≤ D ⇔ X ,→ H. cH (X ) measures the faithfulness of the representing X by H. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Metric Embedding & Distortion z 2
1
2
e:X→H
√
1
2
1
1
1
y
x
2 X = K1,3 dist(e) =
√
R3 (3D Euclidean Space) 2
Definition (Best distortion) cH (X ) = inf{dist(e)| e : X → H} D
Notation: cH (X ) ≤ D ⇔ X ,→ H. cH (X ) measures the faithfulness of the representing X by H. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Metric Embedding & Distortion z 2
1
2
e:X→H
√
1
2
1
1
1
y
x
2 X = K1,3 dist(e) =
√
R3 (3D Euclidean Space) 2
Definition (Best distortion) cH (X ) = inf{dist(e)| e : X → H} D
Notation: cH (X ) ≤ D ⇔ X ,→ H. cH (X ) measures the faithfulness of the representing X by H. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ramsey type Problem
[Bourgain Figiel Milman ’85]
Given size n, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. Y ,→ `2 .
In general, k = min{3, n}. Not interesting. Given size n and distortion D, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. D
Y ,→ `2 .
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ramsey type Problem
[Bourgain Figiel Milman ’85]
Given size n, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. Y ,→ `2 .
In general, k = min{3, n}. Not interesting. Given size n and distortion D, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. D
Y ,→ `2 .
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ramsey type Problem
[Bourgain Figiel Milman ’85]
Given size n, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. Y ,→ `2 .
In general, k = min{3, n}. Not interesting. Given size n and distortion D, find the largest k satisfying: Every n-point metric X contains a subset Y ⊂ X such that |Y | ≥ k. D
Y ,→ `2 .
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Motivation
Metric analog to Dvoretzky theorem (Ramsey type result for finite dimensional normed spaces) [BFM ’85] “Universal” lower bounds for the “k-server problem” (online optimization problem). [Karloff Rabani Ravid ’91] [Blum Karloff Rabani Saks ’94] [Bartal Bollobas M ’01]
Proximity data structures. [this talk ]
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Motivation
Metric analog to Dvoretzky theorem (Ramsey type result for finite dimensional normed spaces) [BFM ’85] “Universal” lower bounds for the “k-server problem” (online optimization problem). [Karloff Rabani Ravid ’91] [Blum Karloff Rabani Saks ’94] [Bartal Bollobas M ’01]
Proximity data structures. [this talk ]
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Motivation
Metric analog to Dvoretzky theorem (Ramsey type result for finite dimensional normed spaces) [BFM ’85] “Universal” lower bounds for the “k-server problem” (online optimization problem). [Karloff Rabani Ravid ’91] [Blum Karloff Rabani Saks ’94] [Bartal Bollobas M ’01]
Proximity data structures. [this talk ]
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Previous Quantitative Bounds Theorem (Phase transition [Bartal Linial M Naor ’03] [BFM ’85]) For fixed distortion D ≥ 1, k = k(n, D) behaves as follows D 1 (1, 2) 2 (2, ∞)
k 3 Θ(log n) ? nδ(D)
Theorem (Large distortions [BLMN ’03]) Positive result: c log D D ∀D > const, ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D , and Y ,→ `2 . Negative result: c ∀D∃∞ X , ∀(Y ⊂ X ), |Y | ≥ |X |1− D ⇒ c`2 (Y ) ≥ D. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Previous Quantitative Bounds Theorem (Phase transition [Bartal Linial M Naor ’03] [BFM ’85]) For fixed distortion D ≥ 1, k = k(n, D) behaves as follows D 1 (1, 2) 2 (2, ∞)
k 3 Θ(log n) ? nδ(D)
Theorem (Large distortions [BLMN ’03]) Positive result: c log D D ∀D > const, ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D , and Y ,→ `2 . Negative result: c ∀D∃∞ X , ∀(Y ⊂ X ), |Y | ≥ |X |1− D ⇒ c`2 (Y ) ≥ D. Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Improved Ramsey-type Theorem
Theorem c
D
∀D > const, ∀metric X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D , and Y ,→ `2 . Remark 1 This bound is tight. 2
The proof is different — much simpler (next hour).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Improved Ramsey-type Theorem
Theorem c
D
∀D > const, ∀metric X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D , and Y ,→ `2 . Remark 1 This bound is tight. 2
The proof is different — much simpler (next hour).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Outline
1
Metric Ramsey-type Problem
2
Approximate Distance Oracles
3
Construction of approximate distance oracles
4
Conclusion
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Approximate Distance Oracle (ADO)
[Thorup Zwick ’01]
Definition Given a metric space (or a graph) X . Preprocess X “quickly” and obtain a “compact” data structure S. Answer in “constant time” queries of the form Given x, y ∈ x, compute dX (x, y ) “approximately”.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Space / Approximation Trade-off
uu d
Example (Distance Matrix) Constant query time. Exact answer. storage space of S is Θ(n2 ) words.
Proposition ([TZ ’01]) 1
If ADO S estimates distances with distortion < 3, then space(S) = Ω(n2 ) bits.
2
A wildely believed girth conjecture [Erd¨ os ’63] implies: If DS S estimates distances with distortion < 2h + 1, then space(S) = Ω(n1+1/h ) bits.
3
The conjecture is verified for h = 1, 2, 3, 5.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Space / Approximation Trade-off
uu d
Example (Distance Matrix) Constant query time. Exact answer. storage space of S is Θ(n2 ) words.
Proposition ([TZ ’01]) 1
If ADO S estimates distances with distortion < 3, then space(S) = Ω(n2 ) bits.
2
A wildely believed girth conjecture [Erd¨ os ’63] implies: If DS S estimates distances with distortion < 2h + 1, then space(S) = Ω(n1+1/h ) bits.
3
The conjecture is verified for h = 1, 2, 3, 5.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Theorem ([TZ ’01]) Let h ∈ N. ∃ ADO with the parameters 2h − 1 approximation. Storage: O(hn1+1/h ) words. Query time O(h). Preprocessing time O(n2 ). Remark 1 Space / approximation trade-off almost optimal. 2
Oracle: For constant approximations, constant query time.
3
Follow-ups concentrated on improving preprocessing time.
4
Are there “true oracles” with universal constant query time?
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Theorem ([TZ ’01]) Let h ∈ N. ∃ ADO with the parameters 2h − 1 approximation. Storage: O(hn1+1/h ) words. Query time O(h). Preprocessing time O(n2 ). Remark 1 Space / approximation trade-off almost optimal. 2
Oracle: For constant approximations, constant query time.
3
Follow-ups concentrated on improving preprocessing time.
4
Are there “true oracles” with universal constant query time?
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Theorem ([TZ ’01]) Let h ∈ N. ∃ ADO with the parameters 2h − 1 approximation. Storage: O(hn1+1/h ) words. Query time O(h). Preprocessing time O(n2 ). Remark 1 Space / approximation trade-off almost optimal. 2
Oracle: For constant approximations, constant query time.
3
Follow-ups concentrated on improving preprocessing time.
4
Are there “true oracles” with universal constant query time?
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Theorem Let h ∈ N. ∃ ADO with the parameters O(h) approximation. Storage: O(n1+1/h ) words. Query time is a universal constant. Preprocessing time is O ∗ (n2+1/h ). Remark 1 Construction is based on the metric Ramsey-type theorem. 2
The technique is different from all previous approaches to this problem.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Theorem Let h ∈ N. ∃ ADO with the parameters O(h) approximation. Storage: O(n1+1/h ) words. Query time is a universal constant. Preprocessing time is O ∗ (n2+1/h ). Remark 1 Construction is based on the metric Ramsey-type theorem. 2
The technique is different from all previous approaches to this problem.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Outline
1
Metric Ramsey-type Problem
2
Approximate Distance Oracles
3
Construction of approximate distance oracles
4
Conclusion
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ultrametrics Definition A metric (X , d) is called ultrametric if it satisfies ∀x, y , z ∈ X , d(x, z) ≤ max{d(x, y ), d(y , z)} .
Definition (Alternative) ∆ = 10 ∆=8
∆=8
∆=9
∆=7
x1
xn
X
d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ultrametrics Definition A metric (X , d) is called ultrametric if it satisfies ∀x, y , z ∈ X , d(x, z) ≤ max{d(x, y ), d(y , z)} .
Definition (Alternative) ∆ = 10 ∆=8
∆=8
∆=9
∆=7
x1
xn
X
d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ultrametrics Definition A metric (X , d) is called ultrametric if it satisfies ∀x, y , z ∈ X , d(x, z) ≤ max{d(x, y ), d(y , z)} .
Definition (Alternative) ∆ = 10 ∆=8
∆=8
∆=9
∆=7
x1
xn
X
d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ultrametrics Definition A metric (X , d) is called ultrametric if it satisfies ∀x, y , z ∈ X , d(x, z) ≤ max{d(x, y ), d(y , z)} .
Definition (Alternative) ∆ = 10 ∆=8
∆=8
∆=9
∆=7
x1
xn
X
d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Ultrametrics Definition A metric (X , d) is called ultrametric if it satisfies ∀x, y , z ∈ X , d(x, z) ≤ max{d(x, y ), d(y , z)} .
Definition (Alternative) ∆ = 10 ∆=8
∆=8
∆=9
∆=7
x1
xn
X
d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Properties of UM
1
Ultrametrics are tree metrics.
2
Ultrametrics are isometrically embeddable in `2 .
3
Hence, Y ,→ UM implies that Y ,→ `2 .
4
All known proofs of “positive” metric Ramsey-type results use embedding into ultrametrics.
D
D
Reforumlating the metric Ramsey theorem: Theorem c
D
∀(D > const), ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D and Y ,→ UM.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Properties of UM
1
Ultrametrics are tree metrics.
2
Ultrametrics are isometrically embeddable in `2 .
3
Hence, Y ,→ UM implies that Y ,→ `2 .
4
All known proofs of “positive” metric Ramsey-type results use embedding into ultrametrics.
D
D
Reforumlating the metric Ramsey theorem: Theorem c
D
∀(D > const), ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D and Y ,→ UM.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Properties of UM
1
Ultrametrics are tree metrics.
2
Ultrametrics are isometrically embeddable in `2 .
3
Hence, Y ,→ UM implies that Y ,→ `2 .
4
All known proofs of “positive” metric Ramsey-type results use embedding into ultrametrics.
D
D
Reforumlating the metric Ramsey theorem: Theorem c
D
∀(D > const), ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D and Y ,→ UM.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Properties of UM
1
Ultrametrics are tree metrics.
2
Ultrametrics are isometrically embeddable in `2 .
3
Hence, Y ,→ UM implies that Y ,→ `2 .
4
All known proofs of “positive” metric Ramsey-type results use embedding into ultrametrics.
D
D
Reforumlating the metric Ramsey theorem: Theorem c
D
∀(D > const), ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D and Y ,→ UM.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Properties of UM
1
Ultrametrics are tree metrics.
2
Ultrametrics are isometrically embeddable in `2 .
3
Hence, Y ,→ UM implies that Y ,→ `2 .
4
All known proofs of “positive” metric Ramsey-type results use embedding into ultrametrics.
D
D
Reforumlating the metric Ramsey theorem: Theorem c
D
∀(D > const), ∀met X , ∃(Y ⊂ X ), |Y | ≥ |X |1− D and Y ,→ UM.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X T1
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X T1
T1
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X T1
Te1
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
Te1
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
Te1
T2
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
Te1
Te2
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1 T3
Te1
Te2
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1 T3
Te1
Te2
Manor Mendel, Assaf Naor
T3
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1 T3
Te1
Te2
Manor Mendel, Assaf Naor
Te3
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
T4 T3
Te1
Te2
Manor Mendel, Assaf Naor
Te3
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
T4 T3
Te1
Te2
Manor Mendel, Assaf Naor
Te3
T4
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
T4 T3
Te1
Te2
Manor Mendel, Assaf Naor
Te3
Te4
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
T4 T3
Te1
Te2
x1
Manor Mendel, Assaf Naor
Te4
Te3
x23
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Construction of “Ramsey Chain” / ADO X
T2
T1
T4 T3
Te1
Te2
x1
Manor Mendel, Assaf Naor
Te4
Te3
x23
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Storage
Storage is dominated by the trees. |Ti | ≥ |X \ (T1 ∪ · · · ∪ Ti−1 )|1−c/D . Therefore, the number of trees is O(D nc/D ). Each trees is complemented to a full tree on n leaves. Therefore total storage is O(D n1+c/D ). A more carefull accounting improves the storage to O(n1+c/D ).
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Query Algorithm Query x, y ∈ X . Use the array to locate the UM tree Tix in which x is represented by a “black point”. Let xˆ, yˆ ∈ Tix the represetatives of x and y in Tix . Find u = lcaTix (ˆ x , yˆ ). Return ∆(u). Least Common Ancestor The only non-trivial step: Finding LCA. Use a data structure of [Harel Tarjan ’84]: Preprocess an n-vertex tree in O(n) time. Answer LCA queries in O(1) time.
Practical implemetation: [Bender Farach-Colton ’00] Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Outline
1
Metric Ramsey-type Problem
2
Approximate Distance Oracles
3
Construction of approximate distance oracles
4
Conclusion
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
More Proximity Data-structures
DS for approximate ranking according to distances. DS for estimation of kf kLip , for a query f : X → Y . Spanners
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Replacement for Well Seperated Pair Decomp? Well seperated pair decomposition (WSPD): A linear size DS for estimating distances upto (1 + ) factor in fixed dimensional spaces. In high dimension storage becomes Ω(n2 ). Ramsey chain has a similar role in “high dimensional space”. Some common applications! Ramsey chain compared to WSPD
ud d
Representation seems to have more structure. Approximation is only D > const comapred with 1 + . Storage is O(n1+c/D ), comapred with linear.
(In many cases the Ramsey chain’s approximation/Storage tradeoff is unavoidable in general metrics)
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Topic for Future Research
Obtain truly ADO with near optimal storage/approximation tradeoff. Improve the construction time of the Ramsey chain. More applications?
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
Ramsey Problem ADO ADO construction Conclusion
Next: Proof of the metric Ramsey theorem.
Manor Mendel, Assaf Naor
Ramsey Partitions & Proximity Data Structures
