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CS603 Clock Synchronization

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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CS603 Clock Synchronization

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  1. CS603Clock Synchronization February 4, 2002

  2. 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

  3. 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

  4. 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)

  5. Synchronization with Faulty Clocks(Dolev, Halpern, Strong ‘84) • Problem: What if some sites are really bad? • Bad clocks • Don’t follow protocol • Notation • C: Logical clock • D: Physical clock • TAR: Time Adjustment Register • C = D + TAR • Δ: Uncertainty in message delay • C(t), D(t) – value of clock at REAL time t

  6. 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)

  7. 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)

  8. Real clock synchronization • Clock Synchronization Condition: Add • PER > ADJ • 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

  9. 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?

  10. 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)

  11. 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

  12. 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

  13. Recruiting Bulletin • Harris Corporation is in the CS lobby until 3pm today

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