Download presentation

Presentation is loading. Please wait.

Published byDakota Deakin Modified over 2 years ago

1
Competitive fault tolerant Distance Oracles and Routing Schemes Weizmann Open U Weizmann Bar Ilan Shiri Chechik Michael Langberg David Peleg Liam Roditty

2
Formal framework A network G A service S(G) Desired properties represented by requirement predicate P(S,G) cost(S) of constructing / maintaining S Optimization problem: Select cheapest service S satisfying P(S,G)

3
Example: Connectivity oracle S = database maintained over graph G Requirement predicate P conn on S: For every two vertices u,w in G: in response to a query Q conn (u,w), answer whether u and w are connected in G. Costs: cost 1 (S) = total / max memory for S cost 2 (S) = query time of S

4
Connectivity oracles The graph G Centralized connectivity oracle for G u u0 w v w v 1 S1S1.............

5
Connectivity oracles The graph G Distributed connectivity oracle for G 〈 w,3 〉 1 2 3 〈 u,1 〉 〈 v,3 〉 Assign each vertex v in connected component G i a label L(v) = 〈 v,i 〉 On query Q conn (L(u),L(w)): check labels 0 1

6
Cost of distributed connectivity oracle Memory = O(log n) per vertex Query time = O(1)

7
Fault Tolerance Assumption: vertices / edges may occasionally fail or malfunction Failure event: set F of failed vertices/edges

8
Fault Tolerance As a result of a failure event: G and S(G) are partially destroyed Surviving network: G' = G \ F Surviving service: S' = S \ F Requirement predicate P might no longer hold

9
Coping with failures in services Goal: Make S(G) competitive fault tolerant, i.e., ensure that the requirement predicate P(S’,G’) still holds subsequent to a failure event. Relaxed variant of problem: Construct service S(G) that can guarantee a weakened form of desired properties (i.e., some relaxed predicate P‘(S’,G’))

10
Rigid vs. competitive fault tolerance Rigid fault tolerance: Ensure that S(G) satisfies the requirement predicate w.r.t. the original G, i.e., that P(S’,G) still holds after a failure event. Competitive fault tolerance: S(G) satisfies the requirement predicate w.r.t. the surviving G’, i.e., P(S’,G’) must hold after a failure event. ⇨ Useful for structures (subgraphs,…) ⇨ Useful for services (oracles, routing,…)

11
Back to connectivity example Goal: Construct a competitive fault-tolerant connectivity oracle S capable of withstanding f edge failures (|F|≤f) (a.k.a. f-sensitivity connectivity oracle) Connectivity oracle: receives query Q conn (s,t) for vertices s, t answers whether s and t are connected

12
Dynamic connectivity oracle [Patrascu-Thorup- 07] Maintain a dynamic data structure Update after each edge deletion / failure Space O(|E(G)|) Query time O(f log 2 n loglogn) (following a batch of f edge deletions)

13
F-T connectivity oracles Note: Unlike dynamic connectivity oracles, in an F-T connectivity oracle, the faulty sets F for two consecutive queries could be very different. receive a query Q conn (s,t,F) for vertices s,t, and failure event F ⊆ E (for |F|≤f) answer whether s and t are connected in G’ = G \ F

14
F-T connectivity oracles Dynamic connectivity oracle: failures come one by one

15
F-T connectivity oracles F-T connectivity oracle: failure events F 1, F 2, … are separate F1F1 F2F2 F3F3 Q(s 1,t 1 )? s3s3 t3t3 t1t1 s1s1 t2t2 s2s2 Q(s 2,t 2 )? Q(s 3,t 3 )?

16
F-T connectivity oracles Claim: The dynamic connectivity oracle of [PT-07] can be modified into F-T connectivity oracle against f edge failures.

17
F-T connectivity oracles Given query Q conn (s,t,F) (with vertices s,t and failure event F ⊆ E) : Update the data structure, deleting all the edges of F While updating, record the changes made Answer the connectivity query Q conn (s,t) Undo all changes to the data structure Operation of the F-T connectivity oracle:

18
F-T connectivity oracles Claim: The resulting F-T connectivity oracle has: Space O(|E(G)|) Query time O(f log 2 n loglogn)

19
F-T connectivity oracles The connectivity oracle of [CLPR-10] : Space O(fkn 1+1/k log(nW)) Query time O(f log 2 n loglogn) (using the sparse spanner construction of [Chechik-Langberg-P-Roditty-10] )

20
(Approximate) Distance Oracle: Data-structure that can answer (approximate) distance queries Q dist (s,t) for any two vertices s, t (with a stretch of at most k) Distance oracles

21
Distance Oracles Thm [Thorup-Zwick-05]: For every weighted undirected graph G it is possible to construct a data structure of size O(n 1+1/k ) that is capable of answering distance queries Q dist (s,t) in O(k) time, with multiplicative approximation factor ≤ 2k-1 (returns a distance estimate đ such that dist(s,t,G) đ (2k-1)dist(s,t,G) )

22
Distributed distance oracles – Distance labeling schemes [P-99] Data structure stored in pieces at the graph vertices (as vertex labels) To answer distance query Q dist (s,t), it suffices to consult the labels L(s), L(T)

23
Distributed distance oracles – Distance labeling schemes [P-99] Thm [P-99]: For every weighted undirected graph G it is possible to construct an approximate distance labeling scheme of size Õ (n 1/k ) capable of answering distance queries Q dist (s,t) with multiplicative approximation factor O(k)

24
F-T distance oracle: Data structure capable of answering distance queries Q dist (s,t,F) between vertex pairs s,t, on the surviving graph G\F, subsequent to a failure event F F-T distance oracles

25
Related work [Demetrescu et al-08]: It is possible to preprocess a weighted graph in time Õ(mn 2 ) to produce a data structure of size O(n 2 logn) for answering 1-failure distance queries Q dist in O(1) time F-T distance oracles

26
Related work [Karger-Bernstein-09] improved preprocessing time for 1-failure queries, to O(n 2 √m), then to Õ(mn), with same size and query time. [Duan-Pettie-09] oracle for 2-failure queries, of size O(n 2 log 3 n) and query time in O(log n) F-T distance oracles

27
Thm [CLPR-10] There exists a polynomial-time constructible data structure of size O(kn 1+8(f+1)/(k+2(f+1)) log(nW)), that given a set of edges F of size |F|≤f and two vertices s,t, returns in time O(f log 2 n loglogn loglogd) a distance estimate đ such that dist(s,t,G\F) đ (2k-1)dist(s,t,G\F) F-T distance oracles

28
Neighborhoods B ρ (v) = ρ -neighborhood of v = vertices at distance ρ or less from v B 0 (v) B 1 (v) B 2 (v)

29
Tree covers Basic notion: A tree T covering the ρ -neighborhood of v v B 2 (v) covering T

30
G = weighted undirected graph A tree cover TC( ,k) : collection of trees {T 1,…,T l } such that 1. For every v there exists a tree T i such that B (v) T i 2. For every T i and every v T i, dist(v,r i,T i ) (2k-1) 3. For every v, the number of trees that contain v is O(kn 1/k ). Tree Covers TiTi B (v) (2k-1) v O(kn 1/k )

31
Lemma [Awerbuch-Kutten-P-91, Cohen-93] A tree cover TC( ,k) can be constructed in time Õ(mn 1/k ) Tree Covers

32
Lemma (F-T connectivity oracle [CLPR-10]) There exists a poly time constructible data structure of size O(fkn 1+1/k log(nW)), that given a set of failed edges F of size f and two vertices s,t, replies in time O(f log 2 n loglogn) whether s and t are connected in G\F Connectivity Oracle (reminder)

33
Construction The algorithm has log(nW) iterations Iteration i handles distances ≤ 2 i G i = graph obtained by removing from G all edges of weight > 2 i f-Sensitivity Distance Oracles

34
Iteration i: Construct a tree cover TC i with ρ= 2 i and k G i | T = subgraph of G i induced by T vertices For each T TC i, construct connectivity oracle Conn_Or T on G i | T For each vertex v store pointer to tree T i (v) TC i containing B ρ (v) f-Sensitivity Distance Oracles

35
Lemma The size of the data structure is O(fkn 1+1/k log(nW)) f-Sensitivity Distance Oracles

36
Q dist (s,t,F) For i 1 to log(nW) if Conn_Or Ti(s) (s,t,F) = true, then return đ=(8k-2)(f+1)2 i-1 Return Answering Queries

37
Lemma Consider vertices s,t and failure set F. Let d=dist(s,t,G\F). The estimated distance đ returned by Q dist (s,t,F) satisfies d đ (8k-2)(f+1)d f-Sensitivity Distance Oracles

38
Lemma The f-sensitivity distance query (s,t,F) can be implemented to return a distance estimate in time O(f log 2 n loglogn loglogd) f-Sensitivity Distance Oracles

39
Main result [CLPR-10] 2-sensitive compact routing scheme: Given a message M at a source vertex s and a destination t, in the presence of failed edges F={e 1,e 2 }, routes M from s to t over a path of length O(k ⋅ dist(s,t,G\{e 1,e 2 }). Total information stored in vertices: O(kn 1+1/k log(nW)logn). 2-sensitivity routing

40
Use hierarchy of tree covers as in distance oracle. To route from s to t, try increasingly higher levels Suppose dist(s,t,G) ≤ 2 i There is a tree T TC i that contains B (s) for =2 i ⇨ T contains also the destination t 2-sensitivity routing T B (s) t

41
T is of depth ≤ (2k-1) ⋅ 2 i The route from s to t is at most k ⋅ 2 i+2 = O(k ⋅ dist(s,t,G)) Only remaining problem: handle edge disconnections… 2-sensitivity routing T B (s) t

42
Consider this tree T u v 2-sensitivity routing Each edge e=(u,v) T – if failed - T u (e) T v (e) disconnects T into two connected components, T u (e) and T v (e)

43
A recovery edge of e is any edge e’ of G that connects T u (e) and T v (e). Define for each edge e T a recovery edge rec(e). u v rec(e) 2-sensitivity routing T u (e) T v (e)

44
Slight simplification for analysis: Assume the graph edges are sorted - e 1,…, e m , and choose rec(e) for every e to be the first recovery edge e i of e 2-sensitivity routing

45
Consider the recovery edge rec(e) = (u',v‘) of the edge e. Denote by P(u,u‘) (resp., P(v,v‘)) the path connecting u and u’ (resp., v and v’) in the tree T u (e) (resp., T v (e)) Denote the entire alternative path for e=(u,v) by P(e) = P(u,u‘) ∙ (u',v') ∙ P(v',v). rec(e) P(u,u’) P(v,v’) e 2-sensitivity routing

46
Failed edges: e 1 = ( u 1, v 1 ) and e 2 = ( u 2, v 2 ) Simplest case: e 1 and e 2 are not in T : just route on T. 2-sensitivity routing

47
Still simple case: e 1 T, e 2 T. 2-sensitivity routing ⇨ T ⋃ rec(e 1 ) \ { e 1, e 2 } is broken into two connected components only when rec(e 1 ) = e 2 To overcome this, it suffices to store, for each edge e T, one additional recovery edge.

48
Hard case: both e 1, e 2 T. Subcase 1 (not too hard): e 2 does not appear on the alternative path P(e 1 ) via rec(e 1 ) (and s,t are connected in G i | T \{e 1,e 2 }): 2-sensitivity routing ⇨ rec(e 1 ), rec(e 2 ) suffice for routing from s to t

49
Subcase 2 (hardest): both e 1 is on P(e 2 ) and e 2 is on P(e 1 ). 2-sensitivity routing ⇨ rec(e 1 ) = rec(e 2 )

50
Store for e 1 (similarly, for each edge e T) two additional recovery edges rec u1 (e 1 ) and rec v1 (e 1 ). Choose rec u1 (e 1 ) to be the edge that “bypasses” as many edges on P(u 1,u 1 ‘) as possible. 2-sensitivity routing

51
Specifically: Consider the path P’ from u 1 to any other recovery edge e’ ≠ rec(e 1 ) P’ has some common prefix with P(u 1,u 1 ‘). rec u1 (e 1 ) ← recovery edge e’ of e 1 that minimizes the length of the common prefix rec v1 (e 1 ) chosen analogously w.r.t P(v 1,v 1 ‘) 2-sensitivity routing

52
These choices ensure successful routing (requires detailed case analysis) The routing process

53
T1T1 T2T2 T3T3

54
T1T1 T2T2 T3T3 First subcase: The routing process

55
T1T1 T2T2 T3T3 Second subcase: The routing process

56
Thm [CLPR-10] There exists a 2-sensitive compact routing scheme that given a message M at a source vertex s and a destination t, in the presence of failed edges e 1 and e 2, routes M from s to t in a distributed manner over a path of length at most O(k ⋅ dist(s,t,G\{e 1,e 2 }). The total amount of information stored in vertices of G is O(kn 1+1/k log(nW)logn). The routing process

Similar presentations

OK

An optimal dynamic spanner for points residing in doubling metric spaces Lee-Ad Gottlieb NYU Weizmann Liam Roditty Weizmann.

An optimal dynamic spanner for points residing in doubling metric spaces Lee-Ad Gottlieb NYU Weizmann Liam Roditty Weizmann.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on personality development for college students Appt online Ppt on electrical power generation system using railway track slides Download ppt on oxidation and reduction calculator Ppt on books are my best friends Ppt on industrial revolution in england Animated ppt on magnetism facts Ppt on human resources planning Ppt on marie curie's husband Free ppt on physical features of india