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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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