Self-Stabilizing Symmetry Breaking in Constant-Space extended abstract y z x Alain Mayer Yoram Ofek Rafail Ostrovsky Moti Yung Abstract 1.1 Hardware-Based Proto col Design: an emerging technology Weinvestigate the problem of self-stabilizi ng round-robin Our motivation for considering the ab ove mo del comes from token managementscheme on an anonymous bidirection al practical high-sp eed b er optic media communication net- ring of identical pro cessors, where each pro cessor is an asyn- works, where a constant-size messages can go through many chronous probabili stic coin- ippi ng nite state machine high-sp eed hardware switches i.e. nite-state machines which sends and receives messages. We show that the so- and through many communication links during a single soft- lution to this problem is equivalent to symmetry breaking ware clo ck-tick. i.e., leader election. Requiring only constant-size messages In many such systems, the way fair access to the network and message-passing mo del has practical implications: our is regulated is by passing a single token i.e. a constant- solution can b e implemented in high-sp eed networks using a bit control signal in a round-robin fashion on a physical or universal fast hardware switches i.e., nite state machines emb edded ring. When a network no de needs a p ermission of size independent of the size of the network. for a certain action, it waits for a token, holds it for a single Our automata-based message-passing mo del has inherent unit of time and then passes it on to ensure fairness in case deadlo ck p ossibility i.e., when all pro cessors are waiting for other pro cessors are waiting. a message whichwe assume is detected by an external time- It must b e stressed that the eciency of the ab ove out mechanism. Provided that there is no deadlo ck to b egin hardware-based approach heavily relies on the fact that the with, we showhow starting from an arbitrary con guration, token is a control signal comprising of only a constantnum- the system never enters a deadlo ck state and further stabi- b er of bits and that switching hardware is a nite state ma- lizes in p olynomial time. We note that Dijkstra showed that chine which can decide even b efore the token arrives if there the last problem do es not have a deterministic solution even is a need to just let it pass through or to hold it | de- when the identical pro cessors p ossess an arbitrary p ower: p ending only on its lo cal state and the requirements of the starting from a ring with a multitude of tokens, any de- network no de. For example, let us consider the case when terministic system will either not stabilize or will enter a only a single pro cessor on a ring comp etes for a resource deadlo ck state. access control. The fact that the pro cessor must giveupa token to ensure fairness i.e. to check if there is some other 1 Intro duction pro cessor waiting does not incur a slowdown prop ortional to the size of the ring. That is, when the pro cessor gives up We consider an arbitrary ring of anonymous identical pro- a token it do es not wait for the numb er of software clo ck- cessors, where each pro cessor is a probabili stic nite-state ticks prop ortional to the size of the ring: the constant-bit machine whose size is xed and indep endent of the size of control message makes a full round without substantial de- the ring. We adopt asynchronous message-passing commu- layby going through high-sp eed digital hardware switches at nication, where each pro cessor can change state and send each no de and not waiting for software clo ck-ticks at anyof messages only up on receiving messages from one of its two them. Moreover, the constant-bit control signal essentially adjacent neighb ors. do es not decrease the bandwidth of the communication links, making the ab ovescheme very much useful in practice. Computer Science Department, Brown University, Providence, RI, 02912. The research of this author was partly supp orted by ONR grant N00014-91-J-1613.Part of this work was p erformed while the author was at the IBM T.J. Watson Research Center. Author's e-mail GENERAL−PURPOSE wn.edu address: a [email protected] SOFTWARE WITH y atson Research Center, Yorktown Heights, NY 10598. IBM T.J. W SLOW CLOCK Author's e-mail address: [email protected] z NETWORK echnology Square, MIT Lab oratory for Computer Science, 545 T NODE Cambridge, MA 02139. Supp orted by IBM Graduate Fellowship. Part ork was done at IBM T.J. Watson Research Center. Author's of this w SIMPLE AND FAST .lcs.mit.edu e-mail address: raf@theory HARDWARE x atson Research Center, Yorktown Heights, NY 10598. IBM T.J. W (FINITE STATE MACHINE) Author's e-mail address: [email protected] COMMUNICATION CHANNEL Hardware−Based Protocol Design 1.2 Towards Hardware-Based Fault-Tolerant tokens meet, one dies then eventually only one token will Token Management remain [IJ90a]. Notice, however, that their solution gives up the round-robin prop erty | it uses random-walk on the An imp ortant concern in practice is to make the hardware- ring which enables token-management, but not round-robin. based round-robin token-managementscheme fault-tolerant This relaxation has heavy implication on the actual use of where the kind of fault-tolerance needed is an ability to re- the token after stabilizati on, since the token continues its cover from an arbitrary transient fault which puts the system drunkard walk at all times | thus its use is mainly for in any or even maliciously chosen state. In fact, round- mutual exclusion to solve contention, but not for regu- robin token-management in the shared memory mo del was lation and fair-co ordinati on of the network communication considered in 1974 by Dijkstra in the seminal pap er which medium, which is the usual task of token systems. Indeed, intro duced the notion self-stabilization [Dij74 ]. Informally, Lamp ort and Lynch [LL90] mention that: \there is one im- the goal of self-stabilizati on is for the system to reach \nor- p ortant prop erty that is harder to achieve in co ordination mal" op eration, starting from an arbitrary initial state. Of problems than in contention { namely self-stabilizatio n". course, while making the scheme fault-tolerant, we should The random walk solution can b e extended to achieve not sacri ce eciency during \normal" op eration. That is, round-robin prop erty in the following fashion: all tokens the scheme should still b e ecient in b oth the sp eed and travel only in one direction say clo ckwise, since space requirements. In particular, the token, if not needed self-stabilizin g ring-orientation in constant space is p ossible at a no de, must b e able to go through the switching hardware [IJ90b]; at each step, each token ips a coin and either stays i.e. through nite state machine fast. That is, the hard- for a single clo ck-tick or advances. When two tokens meet ware should not rely on a software clock or other software they play elimination tournament. Again, eventually and mechanisms and we should deal with faults while maintain- wepay in stabilizatio n time only one token will remain. No- ing high-sp eed message-passing communication, esp ecially tice, however, that this solution is also totally unacceptable, of control information. as we need a steady state in which tokens can propagates in The inherent problem in any message-driven system is hardware without waiting for a clo ck or other delaywe call communication deadlo ck i.e. all messages are lost, see for such a solution fair. example [DIM91 ]. Also, any shared memory system of even In this pap er, we solve the problem of fair round-robin probabilis tic nite state automata for token-management token managementscheme by actually considering and solv- can p otentially deadlo ck in a state without tokens. Israeli ing a strictly stronger problem in fact, equivalent to leader- and Jalfon [IJ90a] show that deadlo ck-freeness implies that election of automata-based bidirectional round-robin token logn bits are needed to represent a token where n is the management scheme: tokens have \p ositive" or \negative" size of the ring for any deadlo ck-free solution. signs, and dep ending on the sign they travel in opp osite di- In practice, in normal op eration the high-sp eed switch rections on the ring. In the steady state, there are exactly can work while the software monitors it; which implies a two cycling tokens, one in each direction. twolayer solution. An ecient solution should minimize the use of the monitoring software layer and let the fast switch 1.4 The Price of Self-Stabilizing Leader Election do all the work. The slow software layer of a no de has a clo ck and large memory and thus a no de can intro duce a In Dijkstra's original pap er in addition to the imp ossibi li ty new token into a system after a timeout which corresp onds of deterministic round-robin token managementscheme, he to a maximal round-trip delay of a token i.e., when it detects presented a deterministic automata- based token- manage- that no token passed through a hardware switch the ab ove ment algorithm in a presence of a leader.