CS603 Clock Synchronization

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# CS603 Clock Synchronization - PowerPoint PPT Presentation

CS603 Clock Synchronization. February 4, 2002. What is the best we can do? Lundelius and Lynch ‘84. Assumptions: No failures No drift Fully connected network of n nodes Uncertainty of ε in message delivery time Best guarantee: ε (1 – 1/ n ) This is a tight lower bound.

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### CS603Clock Synchronization

February 4, 2002

What is the best we can do?Lundelius and Lynch ‘84
• Assumptions:
• No failures
• No drift
• Fully connected network of n nodes
• Uncertainty of ε in message delivery time
• Best guarantee:
• ε(1 – 1/n)
• This is a tight lower bound
Lower bound proof
• Idea: Based on view of each node
• Views indistinguishable even if real time not the same
• Shift execution of a node relative to real time
• Shift of global view and local view equivalent if message delays changed
• Can always shift by at least ε(1 – 1/n) without changing local views
Proof: Induction
• Clocks synchronized to within γ
• Assume messages one way take time μ, return takes time μ+ε (e1)
• Induction: Assume node i-1 sends with delay μ, receives with delay μ+ε
• Shift processes < i by ε
• Let V1,…,Vn be local times at termination of e1.
• In e1, Vn ≤ V1 + γ
• In ei, Vi-1 ≤ Vi + y – ε
• ∑ Vi ≤ ∑ Vi+nγ – (n-1) ε
• (n-1) nγ
• γ ≥ ε(1-1/n)
• Problem: What if some sites are really bad?
• Notation
• C: Logical clock
• D: Physical clock
• C = D + TAR
• Δ: Uncertainty in message delay
• C(t), D(t) – value of clock at REAL time t
Assumptions
• Fully connected, but not necessarily complete
• Recipient knows source of message
• Given nodes p,q; H(p,q) and L(p,q) are upper/lower bounds on transmission time
• ρ is min(H/L)
• A real time frame (not directly observable)
• Correct physical clock has bounded drift rate: R such that time u>v, (1/R)(u-v) ≤ D(u)-D(v) ≤ R(U-v)
• Correct processor has correct clock, implements algorithm
• No assumptions on behavior of faulty processor
• Don’t care if faulty processor knows correct time
• All processors start within time B (can easily show B ≤ R(n-1)H)
Weak Synchronization
• Weak Clock Synchronization Condition: Constants PER, DMAX, ADJ such that:
• TAR changes only at times that are multiples of PER by amount less than ADJ
• Difference between clocks bounded by DMAX
• Theorem: There is an algorithm that achieves WCSC, independent of faults, for which C(t) is unbounded
• Proof: Set TAR(t’) = logPER(D(t))-D(t)
Real clock synchronization
• Changes occur only first time C reads iPER
• If change when C(t)=iPER, then C(t’) ≠ iPER  t’<t
• Gives Linear Envelope Synchronization:
• at+b < C(t) < ct+d, a>0
• Theorem:Linear Envelope Synchronization impossible if  1/3 processors faulty
Proof Sketch
• Construct algorithm that forces a correct processor to run at rate greater than aρn
• Idea: faulty processor p uses one algorithm for processor q, other for others
• Two-faced behavior
• Can’t tell which is two-faced
• Correct processor caught in the middle – follow fast clock or slow clock?
Three-processor case (p, q, r)
• Assume algorithm A synchronizes in time N and tolerates one fault
• F0 = A
• Fm+1: p pretends its clock runs at ρ times q’s rate
• p pretends r sends messages so Cp(t) > aρmDp(t)+b-mDMAX
• Fm gives these messages
• q cannot distinguish from case where p’s clock is fast, r is sending p messages according to Fm
• Cq(t) > Cp(t) – DMAX > aρmDp(t) + b – (m+1) DMAX = aρm+1Dq(t)+b-(m+1) DMAX (since Dp(t) = ρDq(t)
Possibility(Fischer, Lynch, Merritt)
• If no uncertainty in message delay, f faulty, can do with 2f+1 processors
• Send messages to all neighbors
• Send all messages back
• Round trip gives time
• Faulty processor will be detected if it tries to be worse than round-trip time
• Messages out of order
Possibility(Dolev Halpern Simons Strong)
• We CAN do better
• Requires authentication
• Assumptions:
• Messages will be received with bounded delay
• Bounded drift
• Digital signature
• If p has set of messages M at time t with more than f distinct signers, one signer was correct at time signed
• 2ρ(f+1) < 1
• Key: Synchronization time known in advance
• At time, send signed “time is now”
• If receive f+1 messages saying “time is now” before getting to that time, update local time
Recruiting Bulletin
• Harris Corporation is in the CS lobby until 3pm today