On Robust Temporal Structures in Highly Dynamic Networks Arnaud - - PowerPoint PPT Presentation

On Robust Temporal Structures in Highly Dynamic Networks

Arnaud Casteigts

(LaBRI, University of Bordeaux)

• J. work with Swan Dubois, Franck Petit, and John Michael Robson

https://arxiv.org/abs/1703.03190

AATG@ICALP 2018

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Highly dynamic networks

Ex: How changes are perceived?

• Faults and Failures?
• Nature of the system. Change is normal.
• Possibly partitioned network

Example of scenario

Graph representations

Time-varying graphs (TVG)

G = (V, E, T , ρ, ζ)

• T ⊆ N/R (lifetime)
• ρ : E × T → {0, 1} (presence fonction)
• ζ : E × T → N/R (latency function)

[1, 2] [0] [2, 3] [0] [0, 1] [0, 2] [2, 3]

Another classical view G = G0, G1, ... G0

G1 G2 G3

Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc.

C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Graph representations

Time-varying graphs (TVG)

G = (V, E, T , ρ, ζ)

• T ⊆ N/R (lifetime)
• ρ : E × T → {0, 1} (presence fonction)
• ζ : E × T → N/R (latency function)

{t ∈ N : t prime} [0, 1] ∪ [2, 5] [1, π] [5, 7] [9999, ∞) [0, ∞) {1/i : i ∈ N}

Another classical view G = G0, G1, ... G0

G1 G2 G3

Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc.

C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Graph representations

Time-varying graphs (TVG)

G = (V, E, T , ρ, ζ)

• T ⊆ N/R (lifetime)
• ρ : E × T → {0, 1} (presence fonction)
• ζ : E × T → N/R (latency function)

{t ∈ N : t prime} [0, 1] ∪ [2, 5] [1, π] [5, 7] [9999, ∞) [0, ∞) {1/i : i ∈ N}

Another classical view G = G0, G1, ... G0

G1 G2 G3

the graph Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc.

C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Basic concepts

a b c d e G0 a b c d e G1 a b c d e G2 a b c d e G3

Basic concepts

a b c d e G0 a b c d e G1 a b c d e G2 a b c d e G3

= ⇒ Temporal path (a.k.a. Journey), e.g. a e Ex: ((ac, t1), (cd, t2), (de, t3)) with ti+1 ≥ ti and ρ(ei, ti) = 1

(can be formulated with latency)

Basic concepts

a b c d e G0 a b c d e G1 a b c d e G2 a b c d e G3

= ⇒ Temporal path (a.k.a. Journey), e.g. a e Ex: ((ac, t1), (cd, t2), (de, t3)) with ti+1 ≥ ti and ρ(ei, ti) = 1

(can be formulated with latency)

= ⇒ Temporal connectivity (∗ ∗) Satisfied here?

Basic concepts

a b c d e G0 a b c d e G1 a b c d e G2 a b c d e G3

= ⇒ Temporal path (a.k.a. Journey), e.g. a e Ex: ((ac, t1), (cd, t2), (de, t3)) with ti+1 ≥ ti and ρ(ei, ti) = 1

(can be formulated with latency)

= ⇒ Temporal connectivity (∗ ∗) Satisfied here? No, only 1 ∗.

Basic concepts

a b c d e G0 a b c d e G1 a b c d e G2 a b c d e G3

= ⇒ Temporal path (a.k.a. Journey), e.g. a e Ex: ((ac, t1), (cd, t2), (de, t3)) with ti+1 ≥ ti and ρ(ei, ti) = 1

(can be formulated with latency)

= ⇒ Temporal connectivity (∗ ∗) Satisfied here? No, only 1 ∗. = ⇒ Footprint (= underlying graph)

a b c d e

Today: Covering problems

Three ways of redefining covering problems

C., Mans, Mathieson, 2011

Ex: DOMINATINGSET

G1 G2 G3 Temporal dominating set Evolving dominating set Permanent dominating set

Today: Covering problems

Three ways of redefining covering problems

C., Mans, Mathieson, 2011

Ex: DOMINATINGSET

G1 G2 G3 Temporal dominating set Evolving dominating set Permanent dominating set

→ How about infinite time? The relation must hold infinitely often!

Classes of dynamic networks

(C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem? (based on time-varying graphs)

Classes of dynamic networks

(C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem?

(C., 2018)

Classes of dynamic networks

(C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem?

(C., 2018)

→ ER ≡ all the edges of the footprint are recurrent → T CR ≡ temporal connectivity is recurrently achived

Classes of dynamic networks

(C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem?

(C., 2018)

→ ER ≡ all the edges of the footprint are recurrent → T CR ≡ temporal connectivity is recurrently achived

Building temporal covering structures?

→ ER: “easy” → T CR: this talk

Exploiting regularities within T CR

T CR := Temporal connectivity is recurrently achieved

(T CR := ∀t, G[t,+∞) ∈ T C)

Exploiting regularities within T CR

T CR := Temporal connectivity is recurrently achieved

(T CR := ∀t, G[t,+∞) ∈ T C) Alternative characterization: T CR ≡ Eventual footprint connected

Braud Santoni et al., 2016 a b c d e

− →

→ Can be exploited in a distributed algorithm

Kaaouachi et al., 2016

Exploiting regularities within T CR

T CR := Temporal connectivity is recurrently achieved

(T CR := ∀t, G[t,+∞) ∈ T C) Alternative characterization: T CR ≡ Eventual footprint connected

Braud Santoni et al., 2016 a b c d e

− →

→ Can be exploited in a distributed algorithm

Kaaouachi et al., 2016 → Robustness: New form of heredity asking that a property or solution holds in all connected spanning subgraphs Ex: MINIMALDOMINATINGSET (MDS) and MAXIMALINDEPENDENTSET (MIS) C., Dubois, Petit, Robson, 2017/18

EX: MAXIMAL INDEPENDENT SETS

A maximal independent set (MIS) is a maximal (= maximum) set of nodes, none of which are neighbors.

(a) (b) (c) (d)

EX: MAXIMAL INDEPENDENT SETS

A maximal independent set (MIS) is a maximal (= maximum) set of nodes, none of which are neighbors.

(e) (f) (g) (h)

Which ones are robust?

EX: MAXIMAL INDEPENDENT SETS

A maximal independent set (MIS) is a maximal (= maximum) set of nodes, none of which are neighbors.

(i) (j) (k) (l)

Which ones are robust? → Question: characterizing graphs/footprints in which

• 1. all MISs are robust: (RMIS∀)
• 2. at least one MIS is robust: (RMIS∃)
• 3. all MDSs are robust: (RMDS∀)
• 4. at least one MDS is robust: (RMDS∃)

Overview of technical results

• 1. RMDS∀ = Sputniks
• 2. RMIS∀ = Complete bipartite ∪ Sputniks
• 3. RMDS∃ bipartite + test algo
• 4. RMIS∃ bipartite + test algo

Locality:

• 1. RMDS∀ and RMIS∀

→ Robust solutions can be computed locally!

• 2. RMIS∃

→ Robust solutions cannot be computed locally!

RMDS∀ RMIS∀ RMIS∃ RMDS∃

Local algo for robust MIS in Sputniks Lower bound on the non-locality of robust MIS

RMIS∀

Graphs in which all MISs are robust? (RMIS∀)

RMIS∀

Graphs in which all MISs are robust? (RMIS∀)

Lemma

Bipartite complete (BK) graphs ⊆ RMIS∀.

RMIS∀

Graphs in which all MISs are robust? (RMIS∀)

Lemma

Bipartite complete (BK) graphs ⊆ RMIS∀. Def: A graph is a sputnik if and only if every node that belongs to a cycle also has an antenna (i.e. a pendant neighbor).

Lemma

Sputniks ⊆ RMIS∀.

RMIS∀

Graphs in which all MISs are robust? (RMIS∀)

Lemma

Bipartite complete (BK) graphs ⊆ RMIS∀. Def: A graph is a sputnik if and only if every node that belongs to a cycle also has an antenna (i.e. a pendant neighbor).

Lemma

Sputniks ⊆ RMIS∀.

Theorem

RMIS∀ = Sputniks ∪ BK

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀?

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀? N P F P: pendant node N: neighbor of a pendant node F: other

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀? N P F P: pendant node N: neighbor of a pendant node F: other

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀? N P F P: pendant node N: neighbor of a pendant node F: other

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀? N P F P: pendant node N: neighbor of a pendant node F: other

Local algorithm to find a RMIS in RMIS∀

State of the art (classical MIS)

◮ Lower bound: Ω(

• log n/ log log n) [KMW04]

◮ Best algo: 2O(√log n) [PS96] (between log n and n) ◮ Best algo in trees: O(log n/ log log n) [BE10]

Can we solve the problem locally in RMIS∀? N P F P: pendant node N: neighbor of a pendant node F: other

= ⇒ o(log n)

Not local in general graphs! (i.e. Ω(n))

Not local in general graphs! (i.e. Ω(n))

∃ Infinite family of graphs (Gk)k∈N, of diameter Θ(k) = Θ(n). . . . . . .

Not local in general graphs! (i.e. Ω(n))

∃ Infinite family of graphs (Gk)k∈N, of diameter Θ(k) = Θ(n). . . . . . . Lemma: ∀k, Gk admits only two robust MISs M1 (in red) and M2 = V \ M1.

Not local in general graphs! (i.e. Ω(n))

∃ Infinite family of graphs (Gk)k∈N, of diameter Θ(k) = Θ(n). . . . . . . Lemma: ∀k, Gk admits only two robust MISs M1 (in red) and M2 = V \ M1. (1) Anonymous case (easy): Both extremities have same view up to distance Θ(n), but they must decide differently.

Not local in general graphs! (i.e. Ω(n))

∃ Infinite family of graphs (Gk)k∈N, of diameter Θ(k) = Θ(n). βk γk αk β1 γ1 α1 β0 γ0 α0 c0 a0 b0 c1 a1 b1 ck ak bk . . . . . . Lemma: ∀k, Gk admits only two robust MISs M1 (in red) and M2 = V \ M1. (1) Anonymous case (easy): Both extremities have same view up to distance Θ(n), but they must decide differently. (2) Identified networks: let L1, L2, L3 be disjoint labeling functions that assign identifiers to n/3 nodes starting at one extremity (left or right). Let the whole graph be labeled either (1) L1·x·L2; (2) L1·y·L3; (3) L2·z·L3, with x, y, and z arbitrary. Unless using information within Ω(n) hops, βk and bk will decide identically in some cases, whatever the algorithm.

Not local in general graphs! (i.e. Ω(n))

∃ Infinite family of graphs (Gk)k∈N, of diameter Θ(k) = Θ(n). βk γk αk β1 γ1 α1 β0 γ0 α0 c0 a0 b0 c1 a1 b1 ck ak bk . . . . . . Lemma: ∀k, Gk admits only two robust MISs M1 (in red) and M2 = V \ M1. (1) Anonymous case (easy): Both extremities have same view up to distance Θ(n), but they must decide differently. (2) Identified networks: let L1, L2, L3 be disjoint labeling functions that assign identifiers to n/3 nodes starting at one extremity (left or right). Let the whole graph be labeled either (1) L1·x·L2; (2) L1·y·L3; (3) L2·z·L3, with x, y, and z arbitrary. Unless using information within Ω(n) hops, βk and bk will decide identically in some cases, whatever the algorithm. → Essentially as bad as collecting all information at one node and use offline algo.

Centralized algorithm to find RMISs in general (in P)

Objective: Finds a RMIS if one exists, rejects otherwise.

12

C D

14

E F 15 G

16

H I J K L M N O

17 18 19 20 22 21 23 24 25 26 27 28 29 30 31 5 4 3 1 6 7 8 9 10 11 13 2

A B

{14,15} 21 {8,21} 15 16 17 18 20 {18,20}

H

{16,17} {8,14} 11 14

M

{3,4} 3 {2,3} {3,5} 4 7 {6,7} 5 2 6

J

8 {15,16} 24

K

22 28 12 10 {11,12} {10,28}

N

{22,24}

↑ Decomposition into biconnected components ABC-tree ր

Polynomial-time algorithm to find RMISs (2)

PO PI PE PI PE PI PO PI PE PI PO PI PO PE PE PI PI PO PI PO PI PO PI PO PI PE PI PO PI PO PI PE PI PO PI PO PI PE PI PE PI PE PI PO PI PO PI PO PI PO PI PO PI PE PI PO PI PO PE PE PI PI PI 16 17 18 20 {18,20}

H

{16,17} {8,14} {14,15}

root

PO 11 14

M

{3,4} 3 {2,3} {3,5} 4 7 {6,7} 5 2 6

J

8 {15,16} 24

K

22 28 12 10 {11,12} {10,28}

N

{22,24} 21 {8,21} 15 5 4 3 1 6 7 8 9 10 11 13 2 14 15 16 17 18 19 20 22 21 23 24 25 26 27 28 29 30 31 12

↑ Tagging Resulting RMIS ր

Dˇ ekuji !