Lecture 2: Overlay Networks for Wired Systems

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Lecture 2Overlay Networks for Wired Systems

• Christian Scheideler
• Institut für Informatik
• Technische Universität München

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Overlay Network

• Every distributed system needs a connect-ed
  knowledge graph among its sites.
• Such a graph overlay network

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Open Distributed Systems

• In open distributed systems, the set of sites may
  frequently change over time.
• Problem how to maintain overlay network in a
  scalable and robust way??

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Distributed Systems

• Three basic approaches

server-based
supervised
peer-to-peer
structured overlay networks
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Structured Overlay Networks

• Many structured overlay networks based on concept
  of virtual space
• Elegant approach Naor Wieder 03

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Continuous-discrete Approach

• V set of peers, U virtual space
• Each v 2 V mapped to region R(v) ½ U
• Family F of functions fU ! U
• v,w edge , F(R(v)) Å R(w) F(R(w)) Å R(v)
  

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Continuous-discrete Approach

• Basic questions
• How to map peers to regions?
• What family F to choose?

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Continuous-discrete Approach

• Take a classical family of networks(Hypercube,
  de Bruijn graph,)
• Convert it into continuous form by interpreting
  node labels as points in U,edges as a family of
  functions F
• Mapping peers to regions will then convert
  continuous form back into discrete graph.

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Hypercube

• Classical hypercube
• V nodes with labels (x1,,xd) 2 0,1d
• For all i (x1,,xd) ! (x1,..,1-xi,..,xd)
• Continuous version of hypercube
• Interpret (x1,,xd) as z?i xi/2i
• d ! 1 U0,1)
• F fi(x) x1/2i, fi-(x) x-1/2i 8 igt0

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De Bruijn Graph

• Classical de Bruijn graph
• V nodes with labels (x1,,xd) 2 0,1d
• E (x1,,xd) ! (0,x1,,xd-1), (1,x1,,xd-1)
• Continuous de Bruijn graph
• Interpret (x1,,xd) as z?i xi/2i
• d ! 1 U0,1)
• F f0(x) x/2, f1(x) (1x)/2

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Gabber-Galil Graph

• Classical Gabber-Galil graph
• Node set (x,y) 2 0,,n-12
• (x,y) ! (x,xy),(x,xy1), (xy,y),
  (xy1,y)
• Continuous Gabber-Galil graph
• N ! 1 U0,1)2
• F f1(x,y)(x,xy), f2(x,y)(xy,y)

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How to map peers to regions?

• Supervised networksfocus on U0,1)d, d
  fixeduse recursive labeling hierarchical
  decomposition
• Pure peer-to-peer systemsfocus on U0,1)use
  consistent hashing

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Supervised Overlay Networks

• Supervisor handles all join/leave requests.
• All other operations handled by peers.
• Central requirements
• Supervisor stores only polylog information about
  system.
• Join and leave operations with constant work and
  time.

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Supervised Overlay Networks

• General approach
• Place peers into points of 0,1)-ring
• Connect neighboring peers to cycle

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Recursive Labeling

• Assign labels to peers in following order0 1 01
  11 001 011 101 111 0001 0011
• Interpret them as binary encodings of real
  numbers in 0,1) xv ?i bi/2i

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Recursive Labeling

• Node wants to join (n nodes in system)give it
  nth label in list
• Node v wants to leavemove node with nth label
  to vs position

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Maintaining a Cycle

• v node with nth label
• supervisor stores pred(v), v, succ(v),
  succ(succ(v))
• join and graceful leave operation

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Hierarchical decomposition

• How to use continuous-discrete approach?
• Consider any space U0,1)d
• Hierarchical decomposi-tion tree

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Hierarchical Decomposition
0
1
0
001
01
10
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Hierarchical Decomposition

• Fact
• Volumes of subcubes assigned to nodes differ by
  factor of at most 2.
• Subcubes pairwise disjoint.
• Union of subcubes gives U.
• Combine this with family F of functions.

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Join Operation
0
1
0
001
01
10
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Leave Operation
0
1
0
001
01
10
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Conclusion

• For any supervised network based on
  continuous-discrete approach
• Supervisor only has to store constant information
  about network.
• Join/leave can be performed with constant time
  and work for supervisor.

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Pure Peer-to-Peer Network

• We focus on U0,1).
• Every peer mapped to random point in 0,1).
• Peer v owns region v,succ(v)).
• Chord cryptographic hash function
• CAN random number

0
1
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Join Operation

• New node chooses random position x.
• Route to node owning position.

0
1
x
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Leave Operation

• Node that wants to leave transfers its
  connections to its predecessor.

0
1
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Scalability and Robustness

• Scalability
• Network has (poly-)logarithmic diameter
• Peers have (poly-)logarithmic degree
• Robustness
• Network can handle large fraction of adversarial
  peers (i.e. honest peers form single connected
  component)! join-leave attacks

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Join-Leave Model

• n honest peers
• ?n adversarial peers, ?lt1
• Operations
• Join(v) peer v joins the system
• Leave(v) peer v leaves the system
• Goal maintain scalability and robustness for
  any sequence of polynomially many adversarial
  rejoin (leavejoin) requests

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More specific goal

• n honest peers, ?n adversarial peers
• U0,1), region selection via Karger et al.(
  R(v) xv, succ(xv)) )
• For any interval I ½ 0,1) of size (c log n)/n
• Balancing condition ?(log n) peers in I
• Majority condition honest peers in majority

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How to satisfy conditions?

• Chord uses cryptographic hash function to map
  peers to points in 0,1)
• randomly distributes honest peers
• does not randomly distribute adversarial peers

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How to satisfy conditions?

• CAN map peers to random points in 0,1)

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How to satisfy conditions?

• Group spreading AS04
• Map peers to random points in 0,1)
• Limit lifetime of peers

Too expensive!
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How to satisfy conditions?

• Rule that works k-cuckoo rule

n honest ?n adversarial
evict k/n-region
? lt 1-1/k
Rejoin leave and join via k-cuckoo rule
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Analysis of k-cuckoo rule

• k-region region of size k/n starting at integer
  multiple of k/n
• R fixed set of c log n consec. k-regions
• New node not yet replaced after joining
• ?gt0 small constant
• Lemma R has at most c log n new nodes.
• Lemma Sum of ages of k-regions in R in (1 ?)
  (c log n)n/k, w.h.p.

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Analyis of k-cuckoo rule

• R fixed set of c log n consecutive k-regions
• T(?/?)log3 n
• ?gt0 small constant
• Lemma In any time interval of size T, (1?)kT
  honest nodes and (1?)?kT adv. nodes evicted,
  w.h.p.
• Lemma R has (1 ?)(c log n)k old honest and
  lt(1?)(c log n)?k old adv. nodes, w.h.p.

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Analysis of k-cuckoo rule

• honest nodes in R gt(1-?)(c log n)k
• adversarial nodes in Rlt(1?)(c log n)?k (c
  log n)
• Theorem When using the k-cuckoo rule with
  ?lt1-1/k, the balancing and majority conditions
  are satisfied for poly many adversarial rejoin
  requests, w.h.p.

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Limitation of k-cuckoo rule

• Only works for any sequence of rejoin requests of
  adversarial peers.
• Does not work for any sequence of rejoin
  requests.
• Example adversary orders all peers in a region
  of size O(log n / n) to leave

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Questions?