Brief Announcement: A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids

LIPICS - Leibniz International Proceedings in Informatics, Sep 2018

In this paper we show a tight closed-form expression for the optimal clock synchronization in k-ary m-cubes with wraparound, where k is odd. This is done by proving a lower bound of 1/4um (k-1/k), where k is the (odd) number of processes in each of the m dimensions, and u is the uncertainty in delay on every link. Our lower bound matches the previously known upper bound.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://drops.dagstuhl.de/opus/volltexte/2018/9836/pdf/LIPIcs-DISC-2018-47.pdf

Brief Announcement: A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids

LIPIcs.DISC. Brief Announcement: A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids Jennifer L. Welch 1 2 0 Texas A&M University, College Station , TX, USA https://orcid.org/0000-0002-0423-1071 , USA 1 Reginald Frank 2 Texas A&M University, College Station , TX , USA In this paper we show a tight closed-form expression for the optimal clock synchronization in k-ary m-cubes with wraparound, where k is odd. This is done by proving a lower bound of 1 um k ? k1 , where k is the (odd) number of processes in each of the m dimensions, and u is 4 the uncertainty in delay on every link. Our lower bound matches the previously known upper bound. 2012 ACM Subject Classification Theory of computation ? Distributed algorithms and phrases Clock synchronization; Lower bound; k-ary m-toroid - Introduction Synchronizing clocks in a distributed system in which processes communicate through messages with uncertain delays is subject to inherent errors. A body of work has sought bounds on how closely the clocks can be synchronized when there is no drift in the hardware clocks and there are no failures. Lundelius and Lynch [5] showed that, in an n-process clique with the same uncertainty u on every link, the best synchronization possible is u 1 ? n1 . Subsequently, Halpern et al. [4] considered arbitrary topologies in which each link may have a different uncertainty and showed that the optimal clock synchronization is the solution of an optimization problem. This work was generalized by [1, 6] in which algorithms were given for finding the optimal clock synchronization in any given execution. In contrast to the more general lower bounds of [4, 1, 6], Biaz and Welch [3] gave a collection of closed-form upper and lower bounds on the optimal clock synchronization in the worst case for k-ary m-cubes (m-dimensional hypercubes with k processes in every dimension), both with and without wraparound, in which every link has the same uncertainty, u. When there is no wraparound, the tight bound is 12 um (k ? 1). When there is wraparound and k is even, the tight bound is 1 umk. However, when there is wraparound and k is odd, there is a gap between the upper 4 bound of 14 um k ? k1 and the lower bound of 14 um (k ? 1). 1 Supported in part by CRA-W and CDC?s DREU program and NSF grant CNS-0540631. 2 Supported in part by NSF grant 1526725. A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids In this paper, we consider k-ary m-cubes with wraparound (?m-toroids?) and odd k. We show a lower bound of 14 um k ? k1 , which matches the previously known upper bound. We use the same shifting technique from previous lower bounds for clock synchronization (e.g., [5, 4, 3]). The key insight in our improved lower bound is to exploit the fact that the graph is a collection of rings in each dimension and to use multiple shifted executions instead of one. 2 Preliminaries We first present our model and problem statement (following [5, 2, 3]). We consider a graph of km processes, where k ? 3 is odd and m ? 1, in which each process id is a tuple hp0, p1, ..., pm?1i where each pi ? {0, 1, ..., k ? 1}. There are links in both directions between any two processes p~ and ~q if and only if their ids differ in exactly one component, say the i-th, such that pi = qi + 1 (addition on process id components is modulo k throughout). Each process p~ has a hardware clock modeled as a function Hp~ from reals (real time) to reals (clock time). We assume there is no drift, so Hp~(t) = t + cp~ for some constant cp~. Each process is modeled as a state machine whose transition function takes as input the current state, current value of the hardware clock, and current event (receipt of a message or some internal occurrence), and produces a new state and a message to send over each incident link. A history of process p~ is a sequence of alternating states and pairs of the form (event, hardware clock value), beginning with p~?s initial state. Each state must follow correctly from the previous one according to p~?s transition function and the hardware clock values must increase. A timed history of p~ is a history together with an assignment of a real time t to each pair (e, T ) in the history such that Hp~(t) = T . An execution is a set of km timed histories, one per process, with a bijection for each link between the set of messages sent over the link and the set of messages received over the link. The delay of a message is the difference between the real time when it is received and the real time when it is sent. An execution is admissible if every message has delay in [0, u] where u is a fixed value called the uniform uncertainty. We assume each process p~ has a local variable adjp~ as part of its state and we define its adjusted clock Ap~(t) to be equal to Hp~(t) + adjp~(t). An execution has terminated once all processes have stopped changing their adj variables. We say the algorithm achieves synchronized clocks if every admissible execution eventually terminates with |Ap~(t)?Aq~(t)| ? for all processes p~ and ~q and all times t after termination. ?Shifting? an execution changes the real times at which events occur [5]. Let x be an m-dimensional matrix of real numbers with k elements in each dimension, which we call a shift matrix; elements of x are indexed by process ids. Define shift(?,x) be the result of adding xp~ to the real time associated with each event in p~ ?s timed history in ?. Shifting changes the hardware clocks and message delays as follows [5, 2]: I Lemma 1. Let ? be an execution with hardware clocks Hp~ and let x be a shift matrix. Then shift(?,x) is a (not necessarily admissible) execution in which (a) the hardware clock of each p~, denoted Hp~0(t), equals Hp~(t) ? xp~ and (b) every message from p~ to ~q has delay ? ? xp~ + xq~, where ? is the message?s delay in ?. 3 Lower Bound I Theorem 2. For any algorithm that achieves -synchronized clocks in a k-ary m-toroid with uniform uncertainty u, where k is odd, it must be that ? 41 um k ? k1 . The complete proof and an example for the k = 5 case are in the full paper. Proof sketch. Let A be any algorithm that achieves -synchronized clocks in a k-ary mtoroid with uniform uncertainty u, where k = 2r + 1 for some integer r ? 1. Let ? be the admissible execution of A in which Hp~(t) = t for each process p~, every message from p~ to ~q, where ~q is p~?s neighbor in the h-th dimension such that qh = ph + 1, has the same fixed delay ?p~,q~, which is 0 if 0 ? ph < r and is u if r ? ph < k, and every message from ~q to p~ has the same fixed delay ?q~,p~ = u ? ?p~,q~. For 0 ? i < k, define ?i = shift(?,xi), where the p~-th element of the shift matrix xi, denoted xip~, is defined as Pjm=?01 Wipj , where W is defined as follows: range of i ? {0, . . . , m ? 1} 0 ? i < r range of pj 0 ? pj ? i 0 i < pj ? r (pj ? i)u r < pj ? r + i + 1 (r ? i)u r + i + 1 < pj ? 2r (2r ? pj + 1)u r ? i < k range of pj pju (i ? r)u (i ? pj)u 0 The idea behind the shift amounts in W is to cause two processes that are farthest apart in the graph to be shifted as far apart in real time as possible ? thus achieving a large skew between their adjusted clocks ? while maintaining valid message delays between all neighbors. By considering multiple shifted executions, we can cancel out terms involving adjusted clocks, leaving behind only terms that involve the system parameters and u, and the graph parameters k and m. In the full paper we show that all shifted executions are admissible, i.e., that all message delays are in [0, u]: I Lemma 3. For all i, 0 ? i < k, ?i is admissible. Fix any i with 0 ? i < r. We focus on two processes that are maximally far away from each other. Since ?i is admissible by Lemma 3, A must ensure that Aihi,...,ii ?Aihi+r+1,...,i+r+1i ? , where Aip~ denotes the adjusted clock of process p~ after termination in ?i. By definition of ?i and Lemma 1(a), Aihi,...,ii = Ahi,...,ii and Aihi+r+1,...,i+r+1i = Ahi+r+1,...,i+r+1i ? m(r ? i)u. Thus by substituting we get Ahi,...,ii ? Ahi+r+1,...,i+r+1i ? ?m(r ? i)u + , for 0 ? i < r. Similarly, we can show Ahi,...,ii ? Ahi?r,...,i?ri ? ?m(i ? r)u + , for r ? i < k. Adding together these k inequalities and simplifying gives ? 41 um k ? k1 . J 1 2 3 4 5 6 Hagit Attiya , Amir Herzberg, and Sergio Rajsbaum . Optimal clock synchronization under different delay assumptions . SIAM J. Comput. , 25 ( 2 ): 369 - 389 , 1996 . Hagit Attiya and Jennifer L. Welch . Distributed Computing: Fundamentals, Simulations, and Advanced Topics, Second Edition . John Wiley & Sons, Hoboken, NJ, 2004 . Saad Biaz and Jennifer L. Welch . Closed form bounds for clock synchronization under simple uncertainty assumptions . Inf . Process. Lett., 80 ( 3 ): 151 - 157 , 2001 . Joseph Y. Halpern , Nimrod Megiddo, and Ashfaq A. Munshi . Optimal precision in the presence of uncertainty . J. Complexity , 1 ( 2 ): 170 - 196 , 1985 . Inform. Control , 62 ( 2 /3): 190 - 204 , 1984 . Boaz Patt-Shamir and Sergio Rajsbaum . A theory of clock synchronization (extended abstract) . In Proc. 26th Annual ACM Symp. Theory of Comput. , pages 810 - 819 , 1994 .


This is a preview of a remote PDF: http://drops.dagstuhl.de/opus/volltexte/2018/9836/pdf/LIPIcs-DISC-2018-47.pdf

Reginald Frank, Jennifer L. Welch. Brief Announcement: A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids, LIPICS - Leibniz International Proceedings in Informatics, 2018, 47:1-47:3, DOI: 10.4230/LIPIcs.DISC.2018.47