Mobicom2010_S3
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Mobicom2010_S3

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Presented on Mobicom 2010 S3 workshop

Presented on Mobicom 2010 S3 workshop

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Mobicom2010_S3 Mobicom2010_S3 Presentation Transcript

  • Capacity of Byzantine Agreement:Summary of Recent Results
    Guanfeng Liang
    NitinVaidya
    University of Illinois at Urbana-Champaign
  • Byzantine Agreement: Broadcast
    Source S wants to send message to n-1 receivers
    Fault-free receivers agree on identical message
    S fault-free agree on its message
    Up to f < n/3 failures
    Impossible with more failures
  • Bringing in the Networking View
    Rate region determines how fast information can be communicated
    • How does the rate region affect broadcast performance?
    • Has never been looked at by the distributed algorithm community
  • Throughput of Agreement
    Borrow notion of throughput from communications literature
    b(t) = number of bits agreed upon in [0,t]
    Long timescale measure
  • Capacity of Agreement
    Supremum of achievable throughputs for a given rate region
    Point-to-point rate region
    “Ethernet” rate region
  • Point-to-Point (P2P) Network Rate Region
    Each directed link has its own capacity
    Load ij≤ Cij
    S
    A
    C
    B
  • Necessary Conditions for Achievable Throughput
    NC1: If any f receivers are removed, the min-cut from S to any remaining receiver is ≥ R
    S
    A
    C
    SA + SB ≥ R
    SA + BA ≥ R
    SB + AB ≥ R
    B
  • Necessary Conditions for Achievable Throughput
    NC2: If S and any f-1 receivers are removed, the incoming rate of any remaining receiver is ≥ R
    S
    A
    BA + CA ≥ R
    AB + CB ≥ R
    AC + BC ≥ R
    B
    C
  • Results for 4-Node P2P Networks
    NC1 and NC2 together are tight if at least one incominglink of S > 0
    є
    S
    4
    4
    2
    A
    C
    3
    3
    Agreement
    Capacity = 6
    4
    4
    B
    3
    3
  • Results for 4-Node P2P Networks
    NC3: in a 4-node network, if S has no incoming link, every outgoinglink of S must ≥ R
    S
    4
    4
    2
    A
    C
    3
    3
    Agreement
    Capacity = 2
    4
    4
    B
    3
    NC1-3 are tightfor
    arbitrary 4-node networks
    3
  • Achieving Capacity with Coding
    Divide the message into generations
    Error detection network code
    Any misbehavior will be detected by at least 1 good node
    Is error detected?
    No  next generation
    Yes  extra communication, then adapt the code for next generation
    Won’t occur too many times
  • Data = a, b
    S
    a
    a+b
    b
    3
    1
    2
    a, b non-binary
    Achieving Capacity with Coding (example)
  • Achieving Capacity with Coding
    Data = a, b
    S
    a
    a+b
    b
    3
    1
    2
    b
    a+b
  • Achieving Capacity with Coding
    Data = a, b
    S
    a
    a+b
    b
    3
    1
    2
    b
    a+b+e
    At least one “bad link”
    removed after detection
  • Point-to-Point Networks with > 4 Nodes
    Symmetric networks: each link with identical capacity
    Result: NC1 and NC2 are tight for symmetric networks with arbitrary size
    Open problem:Everything else
  • Additional Necessary Conditions for Achievable Throughput
    NC4: if any f nodes are removed, any remaining undirected cut must be ≥ R
    (AC+CA) + (AD+DA) +
    (BC+CB) + (BD+DB) ≥ R
    A
    C
    S
    NC4 is implied by NC1-3 in
    4-node networks and symmetric networks
    But not in other networks
    B
    D
  • “Ethernet” Rate Region
    Sum of private link capacities ≤ C
    We achieve agreement throughput =
    _______
    All previous known algorithms ≤ C/Ω(n2)
    (n-f)
    C = C/ϴ(n)
    n(n-1)
    Conjecture: this is capacity
  • Conclusion
    Capacity of agreement
    Point-to-point rate region:
    Necessary conditions as upper bound on capacity
    Capacity achieving algorithms for 4-node networks and symmetric networks
    “Ethernet” rate region:
    We believe we’ve found a capacity achieving algorithm
  • Open Problems
    Capacity of agreement for general rate regions
    What if a broadcast channel to some nodes at lower rate?
  • Open Problems
    Capacity of agreement for general rate regions
    E.g. a broadcast channel to some nodes at lower rate?
    Even the multicast capacity problem with Byzantine nodes is unsolved
    For multicast, sources fault-free
    The same error detect & adapt approach is capacity achieving if enough connectivity
    Nonlinear network codes!
  • Thank you!