Clock Synchronization with Bounded Global and Local Skew

Transcription

1 Clock Synchronization with ounded Global and Local Skew Distributed Computing Christoph Lenzen, ETH Zurich Thomas Locher, ETH Zurich Roger Wattenhofer, ETH Zurich October 2008 Motivation: No Global Clock Many tasks in distributed systems require a common notion of time Sometimes not all devices can be connected to a "global" clock Equip each device with its own clock! Problem 1: Different clocks have different clock rates Even worse, these clock rates may vary over time! Communication is required to synchronize the clocks! Problem 2: What if the message delays vary? Clock drifts! Each message has a different delay 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) How well can distributed clocks be synchronized? Group Philadelphia, PA, USA Thomas Locher, ETH FOCS Overview Model: Clocks Each device has a hardware clock H H(t):= 0t h(τ) dτ. The hardware clock rate h is bounded t: h(t) [1-²,1+²] Each device computes a logical clock value L based on: H(t) Its hardware clock H and its message history (the messages it received) Messages are required to correct clock skews! Minimize clock skew of logical clocks! A clock synchronization algorithm specifies how the logical clock value L is adapted! And triggers synch messages! L(t) Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS

2 Model: Graph & Communication Distributed system = Graph G of diameter D Node = Computational device Simple Edge = idirectional communication link normalization! Nodes communicate via reliable, but delayed messages Each message may be delayed by any value [0,1] D=3 Model: Optimization Criteria Good real time approximation: v V, t: L v (t)-t ²t t = 0 Minimum progress: v V, t 2 > t 1 : L v (t 2 )-L v (t 1 ) (1-²)(t 2 -t 1 ) Minimize the skew among all nodes: max v,w,t L v (t)-l w (t) L(t) (1+²)t t (1-²)t Minimize the global skew! L cannot go back in time! Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS Model: Optimization Criteria II Model: Importance of Local Skew More importantly: We want a small clock skew between v and w, if the distance between v and w is short! Allow more skew with increasing distance! For many applications, locally well synchronized clocks are more important! Monitoring applications (record <event, timestamp>) Minimize the skew among neighboring nodes: N(v) = Neighbors of v max v,w N(v),t L v (t)-l w (t) Minimize the local skew! Tracking applications Use <event,time> recordings to determine movement/speed etc. More fundamental: E.g., TDMA requires (locally) synchronized clocks! Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS

3 Model: Old Results Overview A well-known result is that the skew between two nodes at distance d is Ω(d) in the worst case! Ω(D) lower bound on global skew! Guaranteeing a global skew of Θ(D) is easy Always set L to largest clock value! ounding the local skew is hard(er): Many (reasonble) algorithms O(D) est known bound O( D) Lower bound Ω(log D / log log D) Diameter determines the local skew!!! True bound probably Ω(log D) Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS Algorithm: Simple Strategies Algorithm: etter Strategies Strategy I: Always set L to largest clock value! Problem: Ο(D) local skew! Strategy II: Take the average clock value! Problem: O(D 2 ) global skew! ( O(D) local skew) L= 2 Strategy III: Always increase the clock value L UNLESS a neighbor is behind. Problem: Tolerate skew! v w skew! skew! skew! Length of this chain O(D/) v can built up skew to w at rate O(²) for O(D/) time O(² D/) = O(D) skew!!! How can we fix this?!? Choose O( D) O( D) local skew!!! L= 3 = 2 L= 2 L= 5 Ok, but can we do better? Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS

4 Algorithm: Increase Tolerance Algorithm: Intuition Strategy III+: Tolerate skew, but if v experiences a skew of i Tolerate i skew! uild up 2 skew! 2 Tolerance increases! For any i {2,3,} Thomas Locher, ETH FOCS Skew moves away! If the adversary wants to build up 3 skew A chain with 2 skew between neighbors is needed! The longer the better! + skew at rate O(²) between l nodes! Only O(D/) time to build chain! If l is the length of the chain Ω( l/²) time is needed Ω( l/²) O(D/) l O(² D/ 2 ) O(D/ 2 ) Inductively: A skew of (i+1) requires a chain with i skew between nodes l i O(D/ i ) Lose a factor of! Local Skew O( log D)! Thomas Locher, ETH FOCS Algorithm: Why It Fails Algorithm: How bad is it? How can we fix it? That s it? Progress = x v thinks w is behind! We get the following picture: Unfortunately, no. The message delays cause problems: -x Skew < -x! v w i -x (i-1) -x (i-2) -x -x... Local skew O( D) Since global skew O(D)! -x -x -x uild up skew! 2+x Increase tolerance! -x! +x 2-x Sees < 2 -x -x skew!!! No increase! Thomas Locher, ETH FOCS How can we fix this?!? React earlier! If a neighbor w is > i -x behind, ask w to increase its clock value!!! That s it? Fortunately, yes. If i-x+r behind, increase by r! Thomas Locher, ETH FOCS

5 Overview Conclusion: Results Probably Local skew O(log D) asymptotically optimal! L v L w O(d(v,w) log(d/d(v,w)) Global skew O(D) L v -L w (1+O(²))D it complexity O( log 2 D) In fact, only a factor 2 larger than the lower bound! = Maximum node degree Space complexity O( log log D + log 2 D) Thomas Locher, ETH FOCS Thomas Locher, ETH FOCS Conclusion: Outlook Questions and Comments? Open problems? ound the logical clock rate! Ideally: l(t) [1-O(²),1+O(²)] Reduce the bit complexity! Send less bits per message Reduce the message frequency Enable piggybacking! Prove tight bounds for global/local skew! Clocks should behave normally even when correcting clock skew! Clock skew is built up at a low rate! Thank you for your attention! Thomas Locher Distributed Computing Group ETH Zurich, Switzerland lochert@tik.ee.ethz.ch Thomas Locher, ETH FOCS