How to Grow Your Balls (Distance Oracles Beyond the Thorup Zwick Bound) FOCS 10

Transcription

1 How to Grow Your Balls (Distance Oracles Beyond the Thorup Zwick Bound) Mihai Pătrașcu Liam Roditty FOCS 10

2 Distance Oracles Distance oracle = replacement of APSP matrix [Thorup, Zwick STOC 01+ For k=1,2,3, : Preprocess undirected, weighted graph G to answer: query(s,t): return đ with d G (s,t) đ (2k-1) d G (s,t) Space O(n 1+1/k ) with O(1) query time Approximation Space 1 n 2 3 n n 4/3 2k-1 n 1+1/k

3 Two Sides of Distance Oracles Compression question: Encode dense graphs with O(n ~ 1+1/k ) bits, such that (2k-1)-apx distances can be retrieved *Matoušek 96+ ( ) finite metric space on n points l with dimension O(k n 1+1/k lg n) with distortion 2k-1 Spanners: ( ) unweighted graph G ( ) H G with O(n 1+1/k ) edges: d G d H (2k-1) d G Data structures question: A data structure of size O(n ~ 1+1/k ) can answer (2k-1)-apx distance queries in constant time

4 Two Sides of Distance Oracles Compression question: Encode dense graphs with O(n ~ 1+1/k ) bits, such that (2k-1)-apx distances can be retrieved *Matoušek 96+ ( ) finite metric space on n points l with dimension O(k n 1+1/k lg n) with distortion 2k-1 Spanners: ( ) unweighted graph G ( ) Optimal H G with space, O(nassuming: 1+1/k ) edges: d G d H (2k-1) d G Girth conjecture: [Erdős] et al. Data structures question: ( ) graphs with Ω(n A data structure of size O(n ~ 1+1/k 1+1/k ) edges and girth 2k+2 ) can answer (2k-1)-apx distance queries in constant time

5 Two Sides of Distance Oracles Compression [Sommer, question: Verbin, Yu FOCS 09+ Encode dense graphs with O(n ~ 1+1/k ) bits, Assume m=o(n) ~ sparse graphs. such that (2k-1)-apx distances can be retrieved If c-apx distance queries take O(1) time *Matoušek 96+ the ( ) space finite is metric n 1+Ω(1/c) space on n points l with dimension O(k n 1+1/k lg n) with distortion 2k-1 Spanners: Beat ( ) [Thorup, unweighted Zwick] graph for graphs G with n 1+1/k edges? ( ) H G with O(n 1+1/k ) edges: d G d H (2k-1) d G Data structures question: A data structure of size O(n ~ 1+1/k ) can answer (2k-1)-apx distance queries in constant time

6 Two Sides of Distance Oracles Many other spanners: Compression question: Encode *BKMP 05+ dense d H graphs d G +6 with with O(n ~ O(n 1+1/k 4/3 ) edges bits, *EP 01+ such that d(2k-1)-apx H (1+ε) ddistances G + O(1) with can O(n be retrieved 1+δ ) edges *Matoušek 96+ ( ) finite metric space on n points Use l additive with dimension approximation O(k n 1+1/k in lg distance n) with oracles? distortion 2k-1 Spanners: ( ) unweighted graph G ( ) H G with O(n 1+1/k ) edges: d G d H (2k-1) d G Data structures question: A data structure of size O(n ~ 1+1/k ) can answer (2k-1)-apx distance queries in constant time

7 New Upper Bounds Unweighted graphs: Preprocess any graph G distance oracle of size O(n 5/3 ) that finds distance 2d G + 1 Weighted graphs: Preprocess G with m=n 2 /α edges distance oracle of size O(n 2 / α 1/3 ) with approximation 2 Can report path in O(1) / edge.

8 The Algorithm A = { } = sample n 2/3 vertices

9 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices

10 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) E[ C ]=n 2/3 Data structure: distances C V

11 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) s t Data structure: distances C V E[ C ]=n 2/3 Query(s,t):

12 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) E[ C ]=n 2/3 Data structure: distances C V Query(s,t): R s, R t = distance to NN in C If Ball(s, R s ) Ball(t, R t ) = min { R s, R t - ½ d(s,t)

13 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) E[ C ]=n 2/3 Data structure: distances C V Query(s,t): R s, R t = distance to NN in C If Ball(s, R s ) Ball(t, R t ) = min { R s, R t - ½ d(s,t)

14 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) Data structure: distances C V E[ C ]=n 2/3 pairs when Ball(s) Ball(t) Query(s,t): R s, R t = distance to NN in C If Ball(s, R s ) Ball(t, R t ) = min { R s, R t - ½ d(s,t) If Ball(s, R s ) Ball(t, R t )

15 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) E[ C ]=n 2/3 Key observation: Data structure: E[#pairs] = O(n 5/3 ) distances C V pairs when Ball(s) Ball(t) Query(s,t): R s, R t = distance to NN in C If Ball(s, R s ) Ball(t, R t ) = min { R s, R t - ½ d(s,t) If Ball(s, R s ) Ball(t, R t )

16 The Algorithm A = { } = sample n 2/3 vertices B = { } = sample n 1/3 vertices C = u B Ball(u A) Data structure: distances C V E[ C ]=n 2/3 pairs when Ball(s) Ball(t) Query(s,t): This is a lie. R s, R t = distance to NN in C If Ball(s, R s ) Ball(t, R t ) = min { R s, R t - ½ d(s,t) If Ball(s, R s ) Ball(t, R t )

17 Geometric balls Balls in graphs Weighted graphs: Ball(s,r)=,edges adjacent to vertices at distance r} To bound ball, sample edges sparsity matters! s t

18 Geometric balls Balls in graphs Weighted graphs: Ball(s,r)=,edges adjacent to vertices at distance r} To bound ball, sample edges sparsity matters! Unweighted graphs: d(s,u) = d(t,v) = ½ d(s,t) Just accept additive 1 s t u v

19 Upper Bounds Can we get rid of +1 Unweighted graphs: Preprocess any graph G for not-too-dense graphs? distance oracle of size O(n 5/3 ) that finds distance 2d G + 1 Better bounds? Weighted graphs: Preprocess G with m=n 2 /α edges distance oracle of size O(n 2 / α 1/3 ) with approximation 2 Milder dependence on m? E.g. O(m + n 5/3 )

20 Set-Intersection Hardness Preprocess S 1,, S n [X] query(i,j): is S i S j =? Conjecture: Let X = lg O(1) n. If query time = O(1), space = Ω(n ~ 2 ) Even in sparse graphs, approximation < 2 requires Ω(n ~ 2 ) space S 1 1 X S n

21 Set-Intersection Hardness Preprocess S 1,, S n [X] query(i,j): is S i S j =? Conjecture: Let X = lg O(1) n. If query time = O(1), space = Ω(n ~ 2 ) Even in sparse graphs, approximation < 2 requires Ω(n ~ 2 ) space Apx. (2-ε)d G + O(1) in unweighted graphs requires Ω(n ~ 2 ) space

22 Set-Intersection Hardness Preprocess S 1,, S n [X] query(i,j): is S i S j =? Conjecture: Let X = lg O(1) n. If query time = O(1), space = Ω(n ~ 2 ) Even in sparse graphs, approximation < 2 requires Ω(n ~ 2 ) space Apx. (2-ε)d G + O(1) in unweighted graphs requires Ω(n ~ 2 ) space

23 New Lower Bounds Preprocess S 1,, S n [X] query(i,j): is S i S j =? Conjecture: Let X = lg O(1) n. If query time = O(1), space = Ω(n ~ 2 ) Actually, randomized conjecture In unweighted graphs with m=n 2 /α edges, constant-time 2-approximation needs space Ω(n ~ 2 / α) Constant-time approximation 2d G +O(1) needs space Ω(n ~ 1.5 )

24 Lower Bound In graphs with O(n) ~ edges, 2-apx needs space Ω(n ~ 1.5 ) (a,b) S ab T a (a,x,y) (c,x,y) T c S ab * n+ X 2 * n+ X 2 (c,d) (S ab T c ) (T a S cd ) distance = 3 3-apx distance 5 * n+ * n+ * n+ * n+

25 Lower Bound In graphs with O(n) ~ edges, 2-apx needs space Ω(n ~ 1.5 ) S ab T a (a,x,y) (a,b) (c,d) (c,x,y) T c S (a,b ) ab * n+ X 2 * n+ X 2 * n+ * n+ * n+ * n+ (S ab T c ) (T a S cd ) distance = 3 3-apx distance 5 either T a S cd or S ab T c

26 My flight is at 4: Recent progress: space Ω(n 5/3 ) for 2-apx many result for apx > 2