Sparse Fault-Tolerant BFS Trees. Merav Parter and David Peleg Weizmann Institute Of Science BIU-CS Colloquium

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1 Spare Fault-Tolerant BFS Tree Merav Parter and David Peleg Weizmann Intitute Of Science BIU-CS Colloquium

2 v 5 Breadth Firt Search (BFS) Tree Unweighted graph G=(V,E), ource vertex V. Shortet-Path Tree (BFS) rooted at. Spare olution: n-1 edge. v 1 v 2 Problem: Not robut againt edge and vertex fault. v 3 v 4

3 Fault Tolerant BFS Tree Objective: Purchae a collection of edge v 1 v 2 (BFS + backup edge) that i robut v 3 v 4 againt edge fault. v 5

4 Fault-Tolerant BFS Tree Subgraph H that contain a BFS tree in G\{e} for every edge failure e in G. v 1 v 2 v 1 v 2 v 3 v 4 v 3 v 4 e 5 v 5 v 5 G {e 1 } H {e 1 }

5 Fault-Tolerant (FT) BFS Tree Subgraph H that contain a BFS tree in G\{e} for every edge failure e in G. v 1 v 2 v 1 v 2 v 3 v 4 v 3 v 4 e 5 v 5 v 5 G {e 1 } H {e 1 }

6 Fault Tolerant (FT) BFS Tree Subgraph H that contain a BFS tree in G\{e} for every edge failure e in G. v 1 v 2 v 1 v 2 v 3 v 4 v 3 v 4 e 5 v 5 v 5 G {e 1 } H {e 1 }

7 Fault Tolerant (FT) BFS Tree Subgraph H that contain a BFS tree in G\{e} for every edge failure e in G. v 1 v 2 v 1 v 2 v 3 v 4 v 3 v 4 e 5 v 5 v 5 G {e 5 } H {e 5 }

8 FT-BFS Tree - Formal Definition Conider an unweighted graph G=(V,E) and a ource vetrex. A ubgraph H i an FT-BFS of G and if for every v in V and e in E: d(,v, H\{e}) = d(,v, G\{e})

9 FT-BFS for Multiple Source (FT-MBFS) Conider an unweighted graph G=(V,E) and a ource et S in V. A ubgraph H i an FT-MBFS of G if for every in S, v in V and e in E: d(,v, H \{e}) = d(,v, G\{e})

10 The Minimum FT-BFS tree Problem Input: unweighted graph G=(V,E) ource vertex in V. Output: An FT-BFS ubgraph H G with minimum number of edge.

11 Outline Related work Lower bound contruction Upper bound Hardne and approximation algorithm.

12 Related Work Replacement Path Fault-Tolerant Spanner

13 A related problem: the replacement path problem P(,t,e) : -t hortet path in G\{e} Problem definition: Given a ource, detination t, for every e ϵ P(,t), compute P(,t,e) the hortet -t path that avoid e. Trivial algorithm: For every edge e ϵ P(,t), run Dijktra algorithm from in G\{e}. Time complexity: O(mn) P(,t,e) P(,t) e t

14 The tructure of a replacement path P(,t,e) : -t hortet path in G\{e} P(,t) e Detour t

15 The replacement path problem Better bound available for replacement path problem for Undirected graph: Time complexity: O(m+n log n) [Gupta et al. 1989] [Herhberger and Suri, 2001] Unweighted directed graph: Time complexity: O(m n ) (Randomized MonteCarlo algorithm) [Roditty and Zwick 2005]

16 Single-ource replacement path Problem definition: Given a ource, compute P(,t,e) efficiently for each t in V and every e ϵ P(,t). Time complexity: O(n ɯ ) [Grandoni and William, FOCS 12] FT-BFS tree reviited: An FT-BFS tree H contain the collection of all ingle ource replacement path. Complexity meaure: ize of H (#edge). New!

17 Spanner Graph G=(V,E) A ubgraph H i an k -panner if for every u,v in V: d(u,v,h) k d(u,v,g).

18 Fault-Tolerant Spanner A ubgraph H i an f-edge fault tolerant k panner if for every u,v in V and every et of f edge F={e 1,e 2,,e f }: d(u,v,h \F) k d(u,v,g\f).

19 Fault-Tolerant Spanner d(u,v,h \F) (2k 1) d(u,v,g\f) for all u,v in V Robut to f-vertex fault: Stretch: 2k-1 #edge: O f 2 k f+1 n 1+1 k [Chechik et al., 2009] O f 2 1 k n 1+1 k [Dinitz and Krauthgamer, 2011]

20 Fault-Tolerant Spanner d(u,v,h \F) (2k 1) d(u,v,g\f) for all u,v in V Robut to f-edge fault: Stretch: 2k-1 #edge: O f n 1+1 k [Chechik et al., 2009]

21 FT-Spanner v. FT-BFS tree FT-Spanner FT-BFS tree All-pair V V Single ource V FT-BFS eaier approximate exact FT-BFS harder

22 Outline Related work Lower bound contruction Upper bound Hardne and approximation algorithm.

23 Lower Bound Theorem [Single ource]: For every integer n 1, there exit an n-vertex graph G=(V,E) and a ource vertex ϵ V uch that every FT-BFS tree H ha Ω(n n) edge.

24 Generalization to multiple ource (FT-MBFS) Theorem [Multiple ource]: For every integer n 1, there exit an n-vertex graph G=(V,E) and a ource et S V uch that every FT-BFS tree H ha Ω(n S n) edge.

25 The Lower Bound Contruction Complete bipartite graph B(X,Z): X =Ω(n), Z =Ω( n) Path of length Z Collection of Z path which are - Vertex dijoint - of monotone increaing length. Z Z X

26 Total number of vertice: n Ω( n) vertex dijoint path of increaing length contain Ω(n) vertice. Total number of edge: Ω(n n) The Contruction Z Z X Ω(n n) edge

27 The Contruction Cl. : Every FT-BFS tree H mut contain ALL the edge of the bipartite graph. By contradiction: Aume there exit an edge e i,j that i not in H. f i X Conider the cae where f i fail. z i e i,j x j Z Ω(n n) edge

28 The Contruction d(,x j, H \{f i }) >d(,x j, G\{f i }) Contradiction ince H i an FT-BFS tree. v i f i v i+1 X x j z i Z Ω(n n) edge

29 Outline Related work Lower bound contruction Upper bound Hardne and approximation algorithm.

30 Matching Upper Bound Theorem: For every graph G=(V,E) and every ource ϵ V there exit a (polynomially contructible) FT-BFS tree H with O(n n) edge.

31 Algorithm for contructing FT-BFS Input: unweighted graph G=(V,E), ource vertex. Output: FT-BFS tree H G. *Aume that all hortet path in G are unique. T 0 := BFS(, G) T e := BFS(, G \{e}) v 1 v 2 H T T e T 0 e 0 v 3 e 5 v 5 v 4

32 Algorithm for contructing FT-BFS T 0 := BFS(, G) T e := BFS(, G \{e}) T 0 v 1 v 2 v 3 v 4 e 5 v 5

33 Algorithm for contructing FT-BFS T 0 := BFS(, G) T e := BFS(, G \{e}) T e1 v 1 v 2 v 3 v 4 e 5 v 5

34 Algorithm for contructing FT-BFS T 0 := BFS(, G) T e := BFS(, G \{e}) T e2 v 1 v 2 v 3 v 4 e 5 v 5

35 Algorithm for contructing FT-BFS T 0 := BFS(, G) T e := BFS(, G \{e}) H v 1 v 2 v 3 v 4 v 5

36 Correctne Recall: P(,t,e) i the -t hortet path in G\{e}. H contain the collection of all ingle ource replacement path. T e1 The replacement path v 1 v 2 P(,v 5,e 1 ) i the -t path in T e1 =BFS(, G\{e 1 }). v 3 e 5 v 4 v 5

37 Size Analyi Baic Intuition An edge e in H i new if it i not in T 0. Lemma: Every vertex t ha at mot O( n) new edge in H. v 1 v 2 v 3 v 4 v 5

38 v 5 Size Analyi Baic Intuition π(,t,t): -t path in tree T New(t) ={ Lat edge of π(,t,t e ), e T 0 } \T 0 H T New( t), 0 t V v 1 v 2 H T T e T 0 e 0 v 3 v 4

39 v 5 Size Analyi Firt Bound π(,t,t): -t path in tree T New(t) ={ Lat edge of π(,t,te), e T 0 } \T 0 Cl. 1: New(t) dit(,t,g) Proof: If lat edge of π(,t,t e ) v 1 v 2 i new then e π(,t,t 0 ) v 3 v 4

40 v 5 Size Analyi Second Bound π(,t,t): -t path in tree T New(t) ={ Lat edge of π(,t,t e ), e T 0 } \T 0 Cl. 2: New(t) 2n v 1 v 2 v 3 v 4

41 Size Analyi Second Bound π(,t,t): -t path in tree T New(t) ={ Lat edge of π(,t,t e ), e T 0 } \T 0 A replacement path P(,t,e) whoe lat edge i new v 1 v 2 v 3 v 4 Count the number of new ending path. v 5

42 New Ending Replacement Path P(,t,e) i the -t path in T e =BFS(, G\{e}). P(,t) P(,t) e e t P(,t,e) Non-New Ending Path t P(,t,e) New Ending Path

43 Analyi Second Bound Strategy: Count the number of new ending path. Conider the et of L new ending replacement path P 1 =P(,t,e 1 ), P 2 =P(,t,e 2 ),., P L =P(,t,e L ) where each P i end with a ditinct new edge of t. Show that L 2n

44 The tructure of a new ending replacement path P(,t) Divergence point T 0 =BFS(,G) t e P(,t,e) Detour Lemma: The detour egment i edge dijoint from P(,t)

45 Analyi Baic Intuition Cl. 1: The detour egment i edge dijoint from P(,t) P(,t) By Contradiction: e P(,t,e) There are two v-t hortet path in G\{e}. Contradiction! v t New edge

46 Analyi Baic Intuition P(,t) P(,t,e 1 ) New edge t e 1 e 2 Claim 2: The detour egment are vertex dijoint! P(,t,e 2 )

47 Analyi Baic Intuition Claim 2: the detour are vertex dijoint! P(,t,e 1 ) P(,t) P(,t,e 2 ) e 1 b 2 there are two v-t hortet path in G\ {e 1, e 2 }. v e 2 t

48 New Ending Replacement Path Notation: b i := unique divergence point of P(,t,e i ) and P(,t). d i D i :=detour egment of P(,t,e i ). b i P(,t) e i D i t P(,t,e i )

49 Analyi Baic Intuition Set of new ending replacement path P 1, P 2,., P L. d(, b 1 ) d(, b 2 ) d(, b L ) b L The divergence point b i are ditinct! d(, b 1 )>d (, b 2 )> >d(, b L ) b i b 3 b 2 b 1 t

50 Analyi Baic Intuition Set of new ending replacement path P 1, P 2,., P L. Toward contradiction aume L > 2n The total #vertice in the detour i: L i=1 D i = L D i L i=1 i=1 i > L2 > n D i Detour are vertex dijoint Contradiction! Divergence point are are ditinct i t b i

51 Generalization to multiple ource (FT-MBFS) Theorem [upper bound] For every graph G=(V,E) and every ource et S V there exit a (polynomially contructible) FT-MBFS tree H with O(n S n) edge.

52 Outline Related work Lower bound contruction Upper bound Hardne and approximation algorithm.

53 The Minimum FT-BFS tree Problem Theorem [Hardne] The Minimum FT-BFS problem i NP-hard and cannot be approximated to within a factor of (log n) unle NP TIME(n ploylog(n) ). (By a gap preerving reduction from Set-Cover)

54 The Minimum FT-BFS tree Problem Theorem [Approximation] The Minimum FT-BFS problem can be approximated within a factor of O(log n).

55 O(log n) Approximation algorithm for the Min-FT BFS problem Solve n-1 intance of Set-Cover. A Set-Cover intance of vertex t: Univere of vertex t: U t = E(P(,t)) P(,t) e 1 e 2 Every neighbor v of t i a et S vt : e P(,t) i in the et S vt if dit(, t, G\{e})=dit(,v, G\{e})+1 e 5 t

56 Summary FT-BFS with O(n n) edge (tight!). FT-MBFS (S ource) with O(n S n) edge (tight!). The Minimum FT-MBFS problem i NP-hard. O(log n)-approximation (tight!).

57 What about approximate FT-BFS tructure? Multiplicative tretch =3: Upper bound: 4n edge. Additive tretch : Lower bound: Ω(n 1+ε β) edge. P, Peleg, SODA 14

58 Thank! Happy Tu-bihvat!