CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. Lecture 8 • Topics: • Time Synchronization Lower Bounds • Sources: • Fan, Chakraborty & Lynch paper • Fan & Lynch paper • Lenzen, Locher & Wattenhofer paper • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

3. Clock Synchronization Model (Again) • Nodes have physical (hardware) clocks Hi(t), with bounded drift rate r: • Rate of physical clock, relative to real time, always in the range [1-r, 1+r]. • Some nodes may have periodic access to external sources of reliable real time signals (GPS). • Nodes communicate by passing messages (point-to-point or broadcast). • Message delay from i to j is variable, within [0, di,j]. • Call di,j the distance from i to j; may be related to Euclidean distance. • Adversary controls clock drift and message delay. Discrete Algs for Mobile Wireless Sys

4. Clock Synchronization Problem (Again) • Nodes compute logical clock Li(t) based on physical clock and messages received. • Logical clocks should be synchronized with each other and with real time: • (Well-behaved) Don’t move backwards. • (Agreement) Minimize difference (skew) between nodes’ logical clocks. • (Internal validity) Logical clocks remain (approximately) within range of physical clocks. • (External validity) Logical clocks remain close to real time (closeness depends on r). Discrete Algs for Mobile Wireless Sys

5. Gradient Clock Synchronization • Nearby nodes should have small worst-case logical clock skew, nodes that are further away may have larger clock skews. • E.g., can we ensure that all i,j always have O(di,j) skew ? • Or find some small function f such that, for every i, j, and t, | Li(t) – Lj(t) | £ f(di,j) ? • Gradient synchronizaton good enough for many uses: • Nearby nodes comparing observations at nearby points in time, e.g., for determining speed of object observed by two nodes. • For collision-avoidance algorithms based on time, e.g., reservation-based protocols, or TDMA. Discrete Algs for Mobile Wireless Sys

6. 1 1 1 … 1 1 2 3 D Gradient Clock Synchronization Motivation • Anyway, it’s impossible to guarantee too-close synchronization between distant network nodes: • Consider a line network G with D nodes. • Distance between neighbors is 1, diameter is D-1. • Can’t achieve better than (D-1) / 2 worst-case logical clock skew between nodes 1 and D. Discrete Algs for Mobile Wireless Sys

7. Lower Bound on Skew for a Line • First let's consider a simpler case when there is no clock drift but clocks start unsynchronized • Use a simple shifting argument: • nodes on the line are 1, 2, 3, …, D • In reference execution , message delay from k to k+1 is 0 and from k+1 to k is 1 • Shift  to create new execution ': • shift node 1 by 0, node 2 by 1, node 3 by 2, …, and node D by (D-1) Discrete Algs for Mobile Wireless Sys

8. Lower Bound on Skew for a Line • Check that msg delays in ' are in range: • msg from i to i+1: new delay = old delay – i's shift + (i+1)'s shift = 0 – (i – 1) + i = 1 • msg from i+1 to i: new delay = old delay – (i+1)'s shift + i's shift = 1 – i + (i – 1) = 0 • After algorithm terminates in both executions: • L1' = L1 since node 1 shifted by 0 ≥ LD –  since  is max skew = (LD' + D – 1) –  since node D shifted by D – 1 ≥ (L1' – ) + D – 1 –  since  is max skew • Implies  ≥ (D – 1)/2 Discrete Algs for Mobile Wireless Sys

9. Lower Bound on Skew for a Line • Can modify the previous argument for the case when clocks start synchronized but there is clock drift: • Execution : • Delays from left-to-right are always 0. • Delays from right-to-left are always 1. • Physical clocks don’t drift. • ': Shift , keeping drift rate within bounds, so that after a while: • Delays from left-to-right are always 1. • Delays from right-to-left are always 0. • Then for large t, | L1(t) – LD(t) | ≤ in both executions. • But LD(t) differs by D-1 in the two executions, contradiction. Discrete Algs for Mobile Wireless Sys

10. Slow down the clocks, until enough skew is created Shifting to Get' 1 D 1 D time a a' Discrete Algs for Mobile Wireless Sys

11. Gradient Clock Synchronization • So, it seems we are forced to settle for gradient clock synchronization. • Now describe: • A simple almost-gradient algorithm • A surprising impossibility result for gradient synchronization---says that worst-case skews between neighbors must depend on the size of the network! • A new algorithm that matches the lower bound. Discrete Algs for Mobile Wireless Sys

12. An Algorithm[Fan, Chakraborty, Lynch] • A good attempt at a practical wireless network algorithm that “usually” satisfies the gradient property. • Best/newest version in [Fan, PhD thesis 2008]. • Accommodates external clock inputs. • Tolerates node failures/recoveries, changing network. • Assume symmetric link delays: dij = dji. • GPS service. • Periodically (at least every time G) delivers correct real times to nodes in a designated subset VG of the nodes V. • Propagate through network (as part of the GPS service) using local broadcasts. • Stability: Analyze what happens when network stops changing (for sufficiently long). Discrete Algs for Mobile Wireless Sys

13. Algorithm Guarantees • External accuracy lower bound: • In any execution, sufficiently long after stabilization, each node i’s logical clock is at least the real time, minus a quantity that depends on the GPS period G and closest distance from i to a GPS source. • External accuracy upper bound: • Sufficiently long after stabilization, i’s logical clock is at most the real time, plus a quantity that depends on G and the diameter of the network. • Need diameter here, because time information could arrive via roundabout paths. • Gradient accuracy: • After stabilization, except during certain “synchronization periods”, logical clocks satisfy gradient property. • Can translate this claim to say that, under reasonable assumptions, the gradient property holds “most of the time”. Discrete Algs for Mobile Wireless Sys

14. The Algorithm • [Fan’s thesis, p. 62] • Based on earlier algorithm of [Kaynar et al. 2005]: • Each node i estimates time using a local clock, which is the same as its physical clock. • Estimates maximum of all nodes’ local clocks using a global clock. • Increases global clock, but at a slower rate than the physical clock, to be sure it never exceeds the maximum local clock. • Everyone sends around maximum known local clock. • Logical clock = max (local clock, global clock) • Modify the algorithm to incorporate GPS inputs, tolerate network changes: • For each known GPS input value g, node i maintains: • local(g), for the best local estimate of real time based on GPS input g. • global(g), for the best estimate of the maximum of all nodes’ local(g). Discrete Algs for Mobile Wireless Sys

15. The Gradient Lower Bound[Fan, Lynch 04] • A surprising lower bound result expressing an inherent limitation for gradient algorithms. • Best/newest version in [Fan, PhD thesis 2008]. • System model (simpler setting than previous algorithm): • Static graph network, no failure, no GPS inputs • Symmetric link delays: dij = dji • Hardware clock, Hi(t) • Initially 0, rate in [1-r, 1+r], for non-zero drift r • Adversary controls hardware clock rates (drift) and message delays. Discrete Algs for Mobile Wireless Sys

16. The Gradient Lower Bound • Logical clock, Li(t), determined by the algorithm. • Requirements: • Well-behavedness: Logical clocks always advance at rate  ½. • Agreement (f-gradient property): For every i, j, and t, Li(t) – Lj(t) £ f(di,j) Discrete Algs for Mobile Wireless Sys

17. The Gradient Lower Bound Theorem • Theorem: For any f-gradient clock synchronization algorithm, we have: f(1) = W(log D / log log D). • More precisely, for every D, there is some graph G of diameter D, and some execution in which two neighbors in G exhibit skew W(log D / log log D). • Thus, the skew depends on the total size of the network: as the network grows, neighboring nodes will have weaker synchronization guarantees. • Surprising that the skew between neighbors should depend on the size of the entire network Discrete Algs for Mobile Wireless Sys

18. Discussion of Assumptions • Requires a non-zero drift • In the case of no drift, with a one-shot algorithm, this result does not hold • Only applies to a class of algorithms in which logical clocks always increase at some minimum rate • rules out algorithms which set clocks backwards (e.g., average algorithms) • but also rules out algorithms that can "smooth" out negative adjustments Discrete Algs for Mobile Wireless Sys

19. 1 1 1 … 1 1 2 3 D Overview of the Proof • Construct explicit executions in which we adversarially control the hardware clock rates and message delays. • Uses “shifting” ideas as in the earlier example. • Use line graph for nodes 1,…,D, with dij = 1 for all pairs of neighbors; actually, diameter = D -1, but ignore this detail here. • Begin with execution  in which hardware clock rates and message delays have specially-chosen values. • Then transform  in stages into another execution with high skew between two neighbors, using two lemmas: • Add Skew Lemma (ASL) • Bounded Increase Lemma (BIL) Discrete Algs for Mobile Wireless Sys

20. Overview of the Proof • Create a series of executions 0, 1,… s.t. for each k, there are 2 nodes ik and jk: • distance b/w ik and jk is (D–1)/(c*f(1))k = nk • skew b/w ik and jk at end of k is at least k*nk/24 • k satisfies preconditions of Add Skew Lemma (ASL) • k satisfies preconditions of Bounded Increase Lemma (BIL) Discrete Algs for Mobile Wireless Sys

21.  k-1 k apply ASL to scrunch ASL implies skew b/w ik and jk has increased  k-1 k '  k-1 k+1 Inductive Construction • Start with carefully chosen 0 • Suppose k satisfying the above properties has been constructed. To create k+1: then add a suffix to k to get: k+1 satisfies precond. of BIL, so skew b/w ik and jk decreases in ' only half as much as it increased Discrete Algs for Mobile Wireless Sys

22. Limits to the Construction • We can continue creating new executions in the inductive process as long as nk, the distance b/w ik and jk, is greater than 1, i.e., the two nodes identified are not the same • Set 1 = nk and solve for k: 1 = (D–1)/(c*f(1))k k = logc*f(1) (D–1) Discrete Algs for Mobile Wireless Sys

23. distance is nk, skew is ≥ k*nk/24 jk ik ik+1 i i+1 distance is 1, skew is ≥ k/24 Finishing Up • In the final k, there are nodes ik and jk: • So there are nodes i and i+1 such that the skew between them is at least k/24 • So f(1) ≥ k/24 • Since k = (logf(1)D), algebra gives us that f(1) = (log D / log log D). Discrete Algs for Mobile Wireless Sys

24. Significance • Lower bound result says that, for any gradient clock synchronization algorithm, the worst-case skew between neighbors must depend on the diameter D of the network, W(log D / log log D). • Significance in practice? • Maybe not so serious, because the constants in the W(log D / log log D) lower bound are extremely small. • Also, how long does the large skew persist? Discrete Algs for Mobile Wireless Sys

25. A New Gradient Algorithm[Lenzen, Locher, Wattenhofer 08] • Algorithm ensures close-to-optimal gradient and network-wide bounds. • O(log D) local skew (cf. lower bound O(log D/log log D) • O(D) global skew • Uses an interesting strategy of sending very carefully crafted demands for neighbors to increase their clocks. • Can this algorithm be a practical synchronization strategy for real ad hoc networks? Discrete Algs for Mobile Wireless Sys