A Distributed Polylogarithmic Time Algorithm for Self-Stabilizing Skip Graphs

A Distributed Polylogarithmic Time Algorithm for Self-Stabilizing Skip Graphs

Stefan Schmid & Christian ScheidelerDept. of Computer Science

University of Paderborn

Riko Jacob & Hanjo TäubigDept. of Computer Science

Technische Universität München

Andrea RichaDept. of Computer Science

Arizona State University

Overlay Network

Internet

Overlay Network

Basic question: how to organize sites in a scalable and robust overlay network???

Problem: high join/leave activity!

Overlay Network

Scalability: Every operation needs at most (poly-)logarithmic time and work

Overlay Network

Robustness: can recover from any attack ( self-stabilizing overlay network)

Overview• Basic notation• Prior work• Self-stabilizing skip graph• Conclusion

Basic Notation

Overlay network: directed graph G=(V,E):• V: set of nodes• E {(v,w) | v,wV}: set of edges

A

DB C

v knows w

Basic Notation

State of a node v:• Local variables• Neighborhood N(v)• Set of actions/rules of the form

<label>: <guard> <commands>

• The set of actions is the same for every node and is supposed to be immutable.

Basic Notation

• The nodes may only know the local state of their direct neighbors.

• Only the following overlay commands:– u.insert(v,w): u asks neighbor v to establish

edge to neighbor w

uv

wu

v

w

Basic Notation

• The nodes may only know the local state of their direct neighbors.

• Only the following overlay commands:– u.move(v,w): u asks neighbor v to establish

edge to neighbor w and removes (u,w)

– no delete operation!

uv

wu

v

w

Basic Notation

• The nodes may only know the local state of their direct neighbors.

• Only the following overlay commands:– u.move(v,w): u asks neighbor v to establish

edge to neighbor w and removes (u,w)

– sufficient to merge parallel edges

uv

wu

v

w

Basic Notation

Self-stabilizing overlay network:A network that can get back to a desired topology from any initial state in which the network is still weakly connected.

Basic Notation

During the self-stabilization process:• Node set is fixed, all nodes reliable• Messages arrive within a time unit• No message loss

Basic Notation

During the self-stabilization process:• Time proceeds in synchronous rounds.• All actions that are enabled in a round (i.e.

their guard is true) are executed.• All local and overlay commands are

correctly executed within the given round.

Simplifies analysis!

Basic Requirements

• Scalability:Only O(polylog(n)) many enabled actions per join or leave operation to get back to legal topology

• Robustness:Self-stabilization from any weakly connec-ted state

Our Requirements

• Scalability:Only O(polylog(n)) many enabled actions per join or leave operation to get back to legal topology

• Robustness:Quick self-stabilization from any weakly connected state

Overview• Basic notation• Prior work• Self-stabilizing skip graph• Conclusion

Prior Work• Self-stabilization goes back to Dijkstra´s seminal

work in 1974• Self-stabilizing algorithms known for leader

election, various communication protocols, graph theory problems, termination detection, clock synchronization, etc.

• Underlying network is usually assumed to be static (if not, then changes are not under control of algorithm but happen in a random or adversarial manner)

Prior Work• Static networks allow efficient cross-checking of

distributed computation

• Development of various techniques that convert conventional algorithms into self-stabilizing algorithms(Awerbuch, Dolev, Herman, Kutten, Patt-Shamir, Varghese,…)

uv

wSt+1(u) = f(St(u),St(v),St(w))

Prior Work

These general techniques do not work (well) for self-stabilizing overlay networks.

uv

v´

Time t Time t+1

u

w

w´

w´´

St+1(u) = f(St(u),St(v),St(v´))

Two options:• find old neighbors• reset

Prior WorkSuprisingly little known for self-stabilizingoverlay networks• Chord network [Stoica et al.]: recovery from certain degenerate

states.Problem: Chord not locally checkable

• Skip graphs [Aspnes & Shah]: recovery from certain degenerate states.Problem: skip graphs not locally checkable as well

• Self-stabilizing line/cycle/hypertree:- Angluin, Aspnes, … 2005- Cramer & Fuhrmann 2005- Shaker & Reeves 2005- Dolev & Kat 2007

Overview• Basic notation• Prior work• Self-stabilizing skip graph• Conclusion

Self-Stabilizing Skip Graph• Each node v has arbitrary unique name

v.id and random bit string v.rs• v.id and v.rs are assumed to be immutable• prei(v): first i bits of v.rs

Skip graph rule:For every node v and iIN0:• v connects to closest successor and

predecessor w (w.r.t. v.id ) with prei(w) = prei(v)

Self-Stabilizing Skip Graph

Nodes v with v.rs=0…

Nodes v with v.rs=1…

Self-Stabilizing Skip Graph

Hierarchical view:

0 1

00 01 10 11

000 001

(log n) degree, (log n) diameter, (1) expansion w.h.p.

Self-Stabilizing Skip GraphProblem: original skip graph does not allow local checking of correct

topology

10..00.. 01.. 11..11..11..10.. 00..

wv

01.. 10..

Self-Stabilizing Skip GraphProblem: original skip graph does not allow local checking of

correct topology

Solution: extend connectionsFor each node v let• succi(v,b), b{0,1}: closest successor of v with prefix

prei(v)b• predi(v,b), b{0,1}: closest predecessor of v with prefix prei(v)b• rangei(v)=[minb predi(v,b), maxb succi(v,b)]v connects to all nodes wrangei(v)with prei(w) = prei(v)

Result: SKIP+

Skip graph: rangei(v)=[predi(v),succi(v)]

Self-Stabilizing Skip Graph

(log n) degree, (log n) diameter, (1) expansion w.h.p.

Self-Stabilizing Skip Graph

• edge (u,v) stable: SKIP+ edge from viewpoint of u

• otherwise, (u,v) is temporary• flag F(v): indicator in u whether (u,v) stable

Rule 1a: create reverse edges

u v

stable

u v

Self-Stabilizing Skip Graph

Rule 1b/c: introduce stable edges

Rule 2: forward temporary edges

u v

ww.id rangei(v)

u v

wb c

u v

w (longer prefix match)

temp

stable

u v

w

Self-Stabilizing Skip Graph

Rule 3a: introduce allu changes set of stable neighbors: initiates insert(v,w) for all neighbors v,w of u

Rule 3b: linearize

u

v1 v2 v3 vk

stable

…

u

v1 v2 v3 vk

stable

…

stable

Self-Stabilizing Skip Graph

Each node u follows six rules:• Rule 1a: create reverse edges to u• Rules 1b and 1c: introduce stable edges

between neighbors of u• Rule 2: forward temporary edges• Rule 3a: introduce all when stable

neighborhood changes• Rule 3b: linearize stable neighborhood

Self-Stabilizing Skip Graph

Theorem 1: For any weakly connected graph, the rules establish SKIP+ in O(log2 n) rounds w.h.p.

Theorem 2: Any join or leave operation in a perfect SKIP+ requires at most O(log4 n) work.

Self-Stabilizing Skip Graph

Proof of Theorem 1:• Bottom-up phase: connected components

are formed for every prefix

0 1

00 01 10 11

000 001

Self-Stabilizing Skip Graph

Proof of Theorem 1:• Top-down phase: components sorted

Conclusions

Many interesting fronts to work on in contextof self-stabilizing overlay networks:• Show O(log n) runtime bound• bound/restrict number of enabled actions

to polylog at each node per round• self-stabilizing networks under adversarial

behavior• self-preserving networks• self-optimizing networks

Questions?