A Centralized Local Algorithm for the Sparse Spanning Graph Problem Christoph Lenzen, Reut Levi

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1 A Centralized Local Algorithm for the Sparse Spanning Graph Problem Christoph Lenzen, Reut Levi A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, Maximilian Wötzel

2 A Centralized Local Algorithm for the Sparse Spanning Graph Problem Christoph Lenzen, Reut Levi

3 í Local Graph Algorithms Dušní Bílkova učn učn 17. listopadu listopadu 17. Křižovnická Liliová Anenská áprstkova á Širok Betlémská Kaprova Platnéřská Husova Ka rlova Žatecká Maiselov a Husova Pařížská Josefov iselova Ma iselova Jilská Staré Město Široká Dušní Ma DlouháMa sná Dlouhá Praha Rytířská Map data OpenStreetMap contributors Dlouhá T ýnská T ýnská Masná classic / global algorithm see whole input, Ω(n) time output solution Rybná Králo dvorská Panská Panská Cele Na Na Příkopě Příkopě Nekáz Nekáz local algorithm see only small parts, o(n) time provide query access to solution 1

4 The Local Sparse Spanning Graph Problem (LSSG) bounded degree graph G = (V, E) given, V = [n] LSSG algorithm provides query access to a spanning graph G = (V, E ): is (7, 18) E? 2

5 The Local Sparse Spanning Graph Problem (LSSG) bounded degree graph G = (V, E) given, V = [n] LSSG algorithm provides query access to a spanning graph G = (V, E ): is (7, 18) E? answer is computed on demand, no preprocessing, all answers consistent with one G 2

6 The Local Sparse Spanning Graph Problem (LSSG) bounded degree graph G = (V, E) given, V = [n] LSSG algorithm provides query access to a spanning graph G = (V, E ): is (7, 18) E? answer is computed on demand, no preprocessing, all answers consistent with one G local algorithm queries adjacency lists of input e. g., what is the 2nd neighbor of vertex 14? 2

7 The Local Sparse Spanning Graph Problem (LSSG) Main Result bounded degree graph G = (V, E) given, V = [n] LSSG algorithm provides query access to a spanning graph G = (V, E ): is (7, 18) E? answer is computed on demand, no preprocessing, all answers consistent with one G local algorithm queries adjacency lists of input e. g., what is the 2nd neighbor of vertex 14? An LSSG algorithm with query and time complexity O(n 2/3 ) poly(1/ε) per query. It guarantees E (1 + ε)n w.h.p. 2

8 Graph Partitions Given a graph G, 3

9 Graph Partitions Given a graph G, 1. partition G into small parts 3

10 Graph Partitions Given a graph G, 1. partition G into small parts 2. compute spanning tree inside of the parts 3

11 Graph Partitions Given a graph G, 1. partition G into small parts 2. compute spanning tree inside of the parts 3. add ɛn edges between parts to make graph connected 3

12 Voronoi Partitions 4

13 Voronoi Partitions 1. select Θ(n 2/3 ) random centers 4

14 Voronoi Partitions 1. select Θ(n 2/3 ) random centers 2. construct Voronoi cells according to path distance 4

15 Voronoi Partitions 1. select Θ(n 2/3 ) random centers 2. construct Voronoi cells according to path distance 3. sort out remote vertices: distance to center Ω(log n) 4

16 Voronoi Partitions 1. select Θ(n 2/3 ) random centers 2. construct Voronoi cells according to path distance 3. sort out remote vertices: distance to center Ω(log n) 4. sha er Voronoi cells into O(n 2/3 ) core clusters of size O(n 1/3 ) 4

17 Voronoi Partitions summary: each core cluster has a BFS spanning tree diameter O(log n) size O(n 1/3 ) 1. select Θ(n 2/3 ) random centers 2. construct Voronoi cells according to path distance 3. sort out remote vertices: distance to center Ω(log n) 4. sha er Voronoi cells into O(n 2/3 ) core clusters of size O(n 1/3 ) 4

18 Local Construction of Core Clusters 5

19 Local Construction of Core Clusters 1. each vertex lips a coin 5

20 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 5

21 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 3. cut heavy children complexity: O(n 1/3 ) 5

22 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 3. cut heavy children complexity: O(n 1/3 ) 5

23 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 3. cut heavy children complexity: O(n 1/3 ) 5

24 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 3. cut heavy children complexity: O(n 1/3 ) 5

25 Local Construction of Core Clusters 1. each vertex lips a coin 2. BFS exploration 3. cut heavy children complexity: O(n 1/3 ) Not in This Talk Partitioning of remote vertices into remote clusters Joining core clusters to reduce number of cluster pairs ( number of edges needed to connect clusters) to εn 5

26 A Sublinear Tester for Outerplanarity (& Other Forbidden Minors) With One-Sided Error Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, Maximilian Wötzel

27 í í Sublinear Graph Algorithms Dušní Bílkova učn učn Dušní Bílkova učn učn 17. listopadu listopadu 17. Křižovnická Liliová Anenská áprstkova á Širok Betlémská Kaprova Platnéřská Husova Ka rlova Žatecká Maiselov a iselova Ma iselova Husova Pařížská Josefov Jilská Staré Město Široká Dušní Ma DlouháMa sná Dlouhá Praha Rytířská Map data OpenStreetMap contributors Dlouhá T ýnská T ýnská Masná classic / global algorithm see everything, Ω(n) time output solution Rybná Králo dvorská Panská Panská Cele Na Na Příkopě Příkopě Nekáz Nekáz 17. listopadu listopadu 17. Křižovnická Liliová Anenská áprstkova Širok á Betlémská Kaprova Platnéřská Husova Ka rlova Žatecká Maiselov a Husova Pařížská Josefov iselova Ma iselova Jilská Staré Město Široká Dušní Ma DlouháMa sná Dlouhá Praha Rytířská Map data OpenStreetMap contributors Dlouhá T ýnská T ýnská Masná Rybná Králo dvorská Panská Panská sublinear algorithm see only small parts, o(n) time estimate solution s value Cele Na Na Příkopě Příkopě Nekáz Nekáz 6

28 Testing Outerplanarity With One-Sided Error x # < d x = 0 : accept always 0 < x ɛdn : don't care ɛdn < x : reject w.p. 2/3 7

29 Testing Outerplanarity With One-Sided Error x x = 0 : accept always 0 < x ɛdn : don't care ɛdn < x : reject w.p. 2/3 # < d ɛ-far ɛ-close 7

30 Testing Outerplanarity With One-Sided Error x Main Result # < d x = 0 : accept always 0 < x ɛdn : don't care ɛdn < x : reject w.p. 2/3 ɛ-far ɛ-close An F -minor freeness tester for every family F of forbidden minors that contains either the K 2,k, (k 2)-grid or k-circus graph with query complexity / running time O(n 2/3 /ε 5 ) 7

31 Partitioning Revisited ɛdn 2 How about the cut in Voronoi partitions? number of cut edges involving a remote cluster is εdn/4 number of cut edges between core clusters might be > εdn/4 8

32 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors 9

33 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) 9

34 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) 9

35 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors at most d incident edges per vertex idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) 9

36 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors (k+1) log(n) cut vertices on le side idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) at most d incident edges per vertex 9

37 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors log(n) forces branching O(log n) (k+1) log(n) cut vertices on le side idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) at most d incident edges per vertex 9

38 Construction of K 2,k Theorem: cuts of size > ƒ between clusters imply K 2,k -minors log(n) forces branching O(log n) (k+1) log(n) cut vertices on le side idea: always have BFS tree, enforce more structure by large cut size ƒ Θ(d k log(n)) at most d incident edges per vertex 9

39 Summary Result: Local Spanning Graphs An LSSG algorithm with query and time complexity O(n 2/3 ) poly(1/ε) per query. It guarantees E (1 + ε)n w.h.p. Result: Minor-Freeness Testing An F -minor freeness tester for every family F of forbidden minors that contains either the K 2,k, (k 2)-grid or k-circus graph with query complexity / running time O(n 2/3 /ε 5 ) recent progress by Kumar et al. (2018) for arbitrary F : O(n 1/2+o(1) ) 10

40 Additional Slides

41 Algorithm 1. sample O(ƒ / ɛ) edges 2. for every sampled edge (u,v): i) explore cluster(s) of u,v ii) compute cut sizes between core cluster and remaining Voronoi cell of u,v iii) compute cut sizes between core / core cluster of u / v 3. reject i minor found or some cut > ƒ

42 Super Clusters Problem: ƒ #(core clusters) 2 O(ɛdn)

43 Super Clusters Problem: ƒ #(core clusters) 2 O(ɛdn) 1. mark each Voronoi cell w.p. 1/n 1/3

44 Super Clusters Problem: ƒ #(core clusters) 2 O(ɛdn) 1. mark each Voronoi cell w.p. 1/n 1/3 2. mark each core cluster of marked cells

45 Super Clusters Problem: ƒ #(core clusters) 2 O(ɛdn) 1. mark each Voronoi cell w.p. 1/n 1/3 2. mark each core cluster of marked cells 3. join unmarked core clusters with marked neighboring core clusters

46 Super Clusters Problem: ƒ #(core clusters) 2 O(ɛdn) 1. mark each Voronoi cell w.p. 1/n 1/3 2. mark each core cluster of marked cells 3. join unmarked core clusters with marked neighboring core clusters locally reconstructable local membership queries ƒ #(core clusters) #(super clusters) O(ɛdn)

47 Tester With Super Clusters 1. sample O(ƒ / ɛ) edges 2. for every sampled edge (u,v): i) explore cluster(s) of u,v ii) compute cut sizes between core cluster and remaining Voronoi cell of u,v iii) compute cut sizes between core / core and core / super cluster of u / v 3. reject i minor found or some cut > ƒ

48 Remote Clusters [Elkin, Neiman, 2017] 1. each remote vertex picks random delay 2. a er delay, start BFS: one level per time 3. construct remote clusters from BFS

Local Algorithms for Sparse Spanning Graphs

Local Algorithms for Sparse Spanning Graphs Reut Levi Dana Ron Ronitt Rubinfeld Intro slides based on a talk given by Reut Levi Minimum Spanning Graph (Spanning Tree) Local Access to a Minimum Spanning