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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)

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

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

- 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

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

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