The Well-Separated Pair Decomposition The Well-Separated M. Farshi - PowerPoint PPT Presentation

The Well-Separated Pair Decomposition The Well-Separated M. Farshi Pair Decomposition Lab. of Combinatorial and Geometric Algorithms, M. Farshi Department of Computer Science, Definition of Yazd University WSPD Compute WSPD The Split Tree Computing WSPD SSPD February 2015 Extension to Other Metrics 7 th Winter School on Computational Geometry 1 / 28

Outline The Well-Separated Pair Decomposition Introduction 1 M. Farshi Definition of the Well-Separated Pair Decomposition 2 Definition of Computing the Well-Separated Pair Decomposition 3 WSPD Compute WSPD The split tree 4 The Split Tree Computing WSPD Extension to Other Metrics 5 SSPD Extension to Other Metrics 2 / 28

Outline The Well-Separated Pair Decomposition Introduction 1 M. Farshi Definition of the Well-Separated Pair Decomposition 2 Definition of Computing the Well-Separated Pair Decomposition 3 WSPD Compute WSPD The split tree 4 The Split Tree Computing WSPD Extension to Other Metrics 5 SSPD Extension to Other Metrics 2 / 28

Outline The Well-Separated Pair Decomposition Introduction 1 M. Farshi Definition of the Well-Separated Pair Decomposition 2 Definition of Computing the Well-Separated Pair Decomposition 3 WSPD Compute WSPD The split tree 4 The Split Tree Computing WSPD Extension to Other Metrics 5 SSPD Extension to Other Metrics 2 / 28

Outline The Well-Separated Pair Decomposition Introduction 1 M. Farshi Definition of the Well-Separated Pair Decomposition 2 Definition of Computing the Well-Separated Pair Decomposition 3 WSPD Compute WSPD The split tree 4 The Split Tree Computing WSPD Extension to Other Metrics 5 SSPD Extension to Other Metrics 2 / 28

Outline The Well-Separated Pair Decomposition Introduction 1 M. Farshi Definition of the Well-Separated Pair Decomposition 2 Definition of Computing the Well-Separated Pair Decomposition 3 WSPD Compute WSPD The split tree 4 The Split Tree Computing WSPD Extension to Other Metrics 5 SSPD Extension to Other Metrics 2 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: Extension to Other Metrics 3 / 28

Introduction Motivation P = set of n points in R d . D = {| pq | | p, q ∈ P } . The Well-Separated Pair Θ( n 2 ) | D | =? Decomposition Decomposition of P × P = { A i } i s.t. P × P = ∪ i A i , M. Farshi A i = pairs with same distance. Definition of WSPD |{ A i } i | = | D | ∈ Θ( n 2 ) . Compute WSPD What if A i = pairs with almost same distance? The Split Tree Computing WSPD Is there a decomposition with size o ( n 2 ) ? SSPD Answer: YES! Size O ( n ) exists! Extension to Other Metrics Use Well-Separated Pair Decomposition. 3 / 28

Introduction Well-Separated Pair Decomposition: The WSPD: introduced by Callahan and Kosaraju in 1995. Well-Separated Pair Decomposition It applications: To solve a large variety of proximity problems M. Farshi Definition of WSPD Compute WSPD The Split Tree Computing WSPD SSPD Extension to Other Metrics Paul B. Callahan S.Rao Kosaraju 4 / 28

Definition of WSPD Well-Separated Pair Well-Separated Pair: A, B ⊂ R d are s -well-separated pair ( s > 0 constant), The if ∃ disjoint balls, D A and D B such that Well-Separated Pair Decomposition M. Farshi Definition of WSPD Compute WSPD The Split Tree Computing WSPD SSPD Extension to Other Metrics 5 / 28

Definition of WSPD Well-Separated Pair Well-Separated Pair: A, B ⊂ R d are s -well-separated pair ( s > 0 constant), The if ∃ disjoint balls, D A and D B such that Well-Separated Pair Decomposition M. Farshi Definition of WSPD Compute WSPD B The Split Tree Computing WSPD SSPD Extension to Other Metrics A 5 / 28

Definition of WSPD Well-Separated Pair Well-Separated Pair: A, B ⊂ R d are s -well-separated pair ( s > 0 constant), The if ∃ disjoint balls, D A and D B such that Well-Separated Pair A ⊆ D A and B ⊆ D B . Decomposition M. Farshi Definition of WSPD Compute WSPD B The Split Tree Computing WSPD r B SSPD D B Extension to Other Metrics A r A D A 5 / 28

Definition of WSPD Well-Separated Pair Well-Separated Pair: A, B ⊂ R d are s -well-separated pair ( s > 0 constant), The if ∃ disjoint balls, D A and D B such that Well-Separated Pair A ⊆ D A and B ⊆ D B . Decomposition M. Farshi d ( D A , D B ) ≥ s × max(radius( D A ) , radius( D B )) . Definition of WSPD Compute WSPD The Split Tree B Computing WSPD r B SSPD D B Extension to Other Metrics ≥ s × max( r A , r B ) A r A D A 5 / 28

Property of WSP’s Property of Well-Separated Pairs If ( A, B ) is a s -well-separated, p, p ′ ∈ A , q, q ′ ∈ B , then The | pp ′ | ≤ (2 /s ) | pq | Well-Separated Pair | p ′ q ′ | ≤ (1 + 4 /s ) | pq | Decomposition M. Farshi q Definition of WSPD Compute WSPD B The Split Tree q ′ r B Computing WSPD D B SSPD ρ Extension to Other Metrics p A r A D A p ′ 6 / 28

Property of WSP’s Property of Well-Separated Pairs If ( A, B ) is a s -well-separated, p, p ′ ∈ A , q, q ′ ∈ B , then The | pp ′ | ≤ (2 /s ) | pq | Well-Separated Pair | p ′ q ′ | ≤ (1 + 4 /s ) | pq | Decomposition M. Farshi Definition of | pq | ≥ ρ q WSPD Compute WSPD ≥ s × max { r A , r B } The Split Tree B s × | pp ′ | Computing WSPD q ′ r B ≥ 2 . SSPD D B Extension to Other ρ Metrics p A r A D A p ′ 6 / 28