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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Scalable and Distributed Self-HealingAlgorithms for Reconfigurable Networks
Amitabh Trehan Jared Saia
Department of Computer ScienceUniversity of New Mexico
IEEE International Parallel and Distributed ProcessingSymposium
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Motivation
Skype network crashesAugust 15, 2007. Service disrupted to about200 million users. Outage attributed to failureof ‘self-healing mechanisms’.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Self-healing
Self-healing: A process of recovery motivated by anddirected by the patient.Our Goal:Make this concept concrete.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Brain vs. Computer
Components fail in both systems.Brain rewires automatically to maintain functionality.A computer does not rewire automatically.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline
1 Problem
2 Algorithm (DASH)
3 Theorems
4 Experiments
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline
1 Problem
2 Algorithm (DASH)
3 Theorems
4 Experiments
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Our Problem
Given: A connected network.Goal: Keep the network connected and "small".Adversary: deletes nodes.Algorithm: adds edges.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH-imation
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Amitabh Trehan, Jared Saia Heal thy self
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Our Model
The Adversary:Omniscient: knows our network and algorithms.Deletes one node at a time.
The Home team (Nodes):Have a small time to recover after each attack.Can add new links (reconfigure).Maintain neighbour-of-neighbour information.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Reconfigurable Networks.
Reconfigurable: can add new edges.Examples:
Peer-to-Peer (P2P) networksSocial NetworksAd-hoc networks
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Applications
P2P NetworksNode: PeerEdge: Communication link
Social NetworksNode: PersonEdge: Social connection
Ad-hoc NetworksNode: SensorEdge: Communication link
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Goals
Self-healing goals:Maintain connectivityKeep degree increase smallThe algorithm must be efficient: latency, bandwidth “small”Keep pair-wise distance increase small
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline
1 Problem
2 Algorithm (DASH)
3 Theorems
4 Experiments
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
One approach
Reconnect neighbours of deleted nodes in a line. [BASS’06].
da
b c
x
e
b c e da
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Pluses
Keeps degrees smallEnsures connectivitySimple algorithm
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Problems
Not distributed.Too many messages exchanged O(n).Too slow O(n).Distances may increase dramatically.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Some definitions
For a fixed time t :G(V , E): The network.E ′: The edges added by our algorithm.(E ′ ⊆ E).G′ = (V , E ′): G′ will be a forest.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Definitions (Continued)
N(v , G′): neighbors of v in G′.UN(v , G) (Unique Neighbours): Set of neighbours of v in Gsuch that no tree in G′ has more than one representative.
v
a f
d
cg
e
h
ji
k lm
E − E’
E’
b
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH: Degree-Based Self-Healing.
1 Init: Initialise each vertex with a random number IDbetween [0,1] selected uniformly at random.
2 When a vertex v is deleted do:1 Nodes in UN(v , G) ∪ N(v , G′) are reconnected into a
complete binary tree sorted top-down in increasing order ofdegree increase.
2 Let MINID be the minimum ID of any node inUN(v , G) ∪ N(v , G′). Propagate MINID to all the nodes inthe tree of UN(v , G) ∪ N(v , G′) in G′.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Demo
v
ab
c
de
f
+1
+0
+5
+1
+2
+3
a
b
e
cf+3
+2
+5
+3
+1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 10
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 30
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 50
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 70
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DASH Timeline: Deletion 90
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline
1 Problem
2 Algorithm (DASH)
3 Theorems
4 Experiments
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
DaSH
Theorem
DASH guarantees the following properties even if up to all thenodes in the network are deleted:
The network stays connected.The degree of any vertex is increased by at most 2 log n.Number of messages any node of initial degree d sendsout and receives is no more than 2(d + 2 log n) ln n whp.The latency to reconnect is O(1) after attack; and theamortized latency to update the state of the network overθ(n) deletions is O(log n) with high probability.The algorithm is completely distributed.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower Bound
Theorem
Consider any locality-aware algorithm that increases thedegree of any node after an attack by at most a fixed constant.Then there exists a graph and a strategy of deletions on thatgraph that will force the algorithm to increase the degree ofsome node by at least log n.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline of the proof
Observation
For a tree, deletion of a node of degree d increases the sumtotal of degrees of its neighbors by d − 2 for a locality-awareacyclic healing strategy.
v
a
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d
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline of the proof
Prune (v,x) : Given node v and its subtree headed by node x ,deletion of all the nodes in that subtree includingx , despite self-healing. Accomplished by repeateddeletion of leaf nodes in the subtree.
d
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline of the proof
Graph: M+2-ary tree for a M-bounded self-healingalgorithmAttack Strategy: LEVELATTACK
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower bound illustration (Ternary Tree, 2-Degreebounded (DASH))
L0
L2
L3
L1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower bound illustration
L0
L2
L3
L1+1
+1 +1+1 +1+1 +1 +1+1 +1
+1
+1+1 +1+1 +1
+1
+1 +1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower bound illustration
L0
L2
L3
L1
+2+1 +1+1 +1 +1+2 +2
+1
+1+1 +1 +2 +1 +1+2
+2
+1 +1x
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower bound illustration
L0
L2
L3
L1
+2+1 +1+1 +1 +1+2 +2
+1
+1+1 +1 +2 +1 +1+2
+2
+1 +1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Lower bound illustration
L0
L2
L3
L1
+2+1 +1+1 +1 +1+3 +2
+1
+1+1 +1 +2 +1 +1+2 +2 +1
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Outline
1 Problem
2 Algorithm (DASH)
3 Theorems
4 Experiments
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Strategies and Heuristics
Attack strategies:Max Degree (MD) delete: Delete node of maximum degree.MD Neighbor delete: Keep deleting neighbours ofmaximum degree node.
Healing strategies:Binary Graph: reconnect all neighbours; naive.Binary Tree: reconnect neighbours keeping G′ as forest.Degree based Binary Tree (DASH)SDASH (Surrogate DASH): Let a node surrogate for thedeleted node whenever feasible.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Degree increase: MD neighbor delete
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Stretch and SDASH
Stretch: maxu,v ,t
(δt (u,v)δ0(u,v)
), where δt is distance in graph Gt ,
δ distance in original graph G0.Surrogation: A neighbor of the deleted node takes over all the
deleted edges.SDASH: If degree increase using surrogation < degree
increase using DASH, do surrogation.
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Stretch: Max degree delete
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Summary
Concrete definition of self-healing: maintaining an invariantover multiple attacksProvably efficient algorithm for self-healingProvably ensure: connectivity and small degree increaseEmpirically small stretch
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Future Work
Keeping stretch of the network low- The Forgiving Tree: A Self-Healing Distributed DataStructure (PODC 2008).
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Future Work
Self-healing in sensor networks.Self-healing in social networks.Functional self-healing: maintaining functionality (circuitboards, the brain).
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Question Time
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ProblemAlgorithm (DASH)
TheoremsExperimentsConclusions
Handling insertions
Easily handled. The new node and neighbors simpyupdate their data structures.True degree no more than degree if only insertions + 2log n+ 1.
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