localized self- healing using expanders gopal pandurangan nanyang technological university,...

Localized Self-healing using

Expanders

Gopal PanduranganNanyang Technological University, Singapore

Amitabh Trehan Technion - Israel Institute of Technology, Haifa, IL

TTI-C

healheal

PODC’11 G. Pandurangan, A. Trehan

Epic Fail

• Adsense Mar 2010, Google, May

15, 2009

•Twitter, August 6, 2009

•Facebook, August 6, 2009

•Skype, August 15, 2007

PODC’11 G. Pandurangan, A. Trehan

How to self-heal?

•Brain: component fails, brain rewires and does without it

•Computer networks: components fail, network fails until components fixed.

An autonomic system

•Self-managing:

•Self-configuring

•Self-healing

•Self-optimizing

•Self-protecting

PhD Dissertation’10 Amitabh Trehan

PODC’11 G. Pandurangan, A. Trehan

Autonomic Computing•IBM’s autonomic computing

initiative

•Self-CHOP

PODC’11 G. Pandurangan, A. Trehan

Self-healing

•A self-healing system, starting from a correct state, under attack from an adversary, goes only temporarily out of a correct state.

•Our work: Under attack from powerful adversary, maintain certain topological properties within acceptable bounds.

PODC’11 G. Pandurangan, A. Trehan

Ensuring Robustness

•Want to ensure that our network is robust to node failures.

•Idea: build some redundancy into the network?

•Example: Connectivity

•Use k-connected graph.

•Price: degree must be at least k.

PODC’11 G. Pandurangan, A. Trehan

Ensuring Robustness

•Want to ensure that our network is robust to node failures.

•Idea: build some redundancy into the network?

•Example: Connectivity

•Use k-connected graph.

•Price: degree must be at least k.

Expensive!Expensive!

PODC’11 G. Pandurangan, A. Trehan

Model

•Start: a network G.

•An adversary inserts or deletes nodes .

•After each node addition/deletion, we can add and/or drop some edges between pairs of nearby nodes, to “heal” the network.

PODC’11 G. Pandurangan, A. Trehan

Challenge 1: properties conflict

Low degree increase => high diameter/stretch/ poorer expansion?

PODC’11 G. Pandurangan, A. Trehan

Challenge 2: local fixing of global

properties

Low diameter => high degree increase?

✴ Limited global Information with nodes✴ Limited resources and time constraints

PODC’11 G. Pandurangan, A. Trehan

Our Self-healing Goals

•Healing should be fast, local and distributed.

•Certain topological properties should be maintained within bounds:

- Connectivity

- Degree

- Stretch

- Spectral properties (~Expansion/Conductance)

PODC’11 G. Pandurangan, A. Trehan

A series of unfortunate events

PODC’11 G. Pandurangan, A. Trehan

Xheal Goals•Maintain connectivity.

•Edge Expansion of graph not much worse than ‘original’ graph.

•Distance between any two nodes shouldn’t increase by too much (low stretch).

•If vertex v starts with degree d, then its degree should never be much more than d.

•Healing should be fast and localized.

Comparing resultsG: healed network

G’: graph of only insertions and original

nodes

PODC’11 G. Pandurangan, A. Trehan

Main Result•A distributed algorithm, Xheal such

that:

•Degree increase: Degree of node in G ≤ times degree in G’

G’

3

G

5v v

PODC’11 G. Pandurangan, A. Trehan

Main Result (Contd..)

•Stretch: Distance between any two nodes in G = O(log n) times their distance in G’

G

u vd(u,v) =

5

G’

uv

d(u,v) = 3

PODC’11 G. Pandurangan, A. Trehan

Main Result (Contd..)

•Spectral properties:

- h(Gt) ≥ min( , h(G′t)), for constant ≥ 1 (If G′t is a (bounded degree) expander, so is Gt )

- (Bounded 2nd smallest eigen value): Put equation here?

PODC’11 G. Pandurangan, A. Trehan

Main Result (Contd..)•Costs:

- Deletions (by Law-Siu implementation):

‣ O(log n) rounds per deletion.

‣ Amortized O(k.(log n)A(p)*) messages for healing by k-degree expander.

*A(p) is average degree of deleted nodes over p deletions i.e (put in formula).

PODC’11 G. Pandurangan, A. Trehan

Xheal: Outline•Node inserted without restrictions.

• On node deletion, its neighbors reattach to form a primary expander cloud (k-degree expander).

Over further deletions....

•Multiple primaries joined by secondary expander clouds using ‘free’ nodes*.

•If no ‘free’ nodes (happens over a large number of deletions), clouds merged into new primary expander cloud. *`Free’ nodes: nodes not participating in

secondary clouds.

PODC’11 G. Pandurangan, A. Trehan

Xheal: Outline (Contd..)

•Each node of degree d part of at most d primary clouds and one secondary cloud.

•All clouds maintained as expanders.

• Efficient distributed implementation dependant on distributed expander construction (Using Law-Siu construction in this paper).

PODC’11 G. Pandurangan, A. Trehan

Healing by expanders

• Lemma: At end of any timestep t, h(Gt) min(c’,h(G’t)), c’ 1, a fixed constant

generalization of base case:

• Assume deletion at timestep t =1 , h(G1) min(c’,h(G’1)), c’ 1, a fixed constant

PODC’11 G. Pandurangan, A. Trehan

Proof๏ h(G1) min(c’,h(G0))

- h(G) = ES,S’(G) / S(G), S(G) ≤ n/2

• I: k-reg expander subgraph replacing deleted node

PODC’11 G. Pandurangan, A. Trehan

Proof (contd)

• E(I) intersection ES,S’(G_1) is null : latex equations here

PODC’11 G. Pandurangan, A. Trehan

Proof (contd..)

• E(I) intersection ES,S’(G_1) not null : latex equations here, 2 parts

PODC’11 G. Pandurangan, A. Trehan

Future Directions•Improving distributed construction of expander graphs (will enhance Xheal):

- Deterministic, or improved randomized.

•Self-healing routing

•Load-balanced self-healing: Chord like structures? Small world models?

•Extend model and algorithms: Byzantine faults, multiple failures, sensor networks, social networks, self-*.

PODC’11 G. Pandurangan, A. Trehan

Summary•Efficient, distributed algorithm Xheal for

self-healing spectral properties (expansion), stretch, degree and connectivity.

•Xheal ensures maintainance of good expansion, stretch of at most log n, with constant degree increase, low latency and messages.

•Distributed implementation using distributed expander construction techniques; better techniques can improve implementation.

PODC’11 G. Pandurangan, A. Trehan

Thank You

PODC’11 G. Pandurangan, A. Trehan

More future directions

•Behavioral self-healing in social networks

•Self-* problems

•Network evolution and group formation

•Byzantine agreement: byzantine faults

PODC’11 G. Pandurangan, A. Trehan

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