We study the memory requirements of self-stabilizing leader election (SSLE) protocols. We are mainly interested in two types of systems: anonymous systems and id-based systems. We consider two classes of protocols: deterministic ones and randomized ones.We prove that a non-constant lower bound on the memory space is required by a SSLE protocol on unidirectional, anonymous rings (even if the protocol is randomized).We show that, if there is a deterministic protocol solving a problem on id-based systems where the processor memory space is constant and the id-values are not bounded then there is a deterministic protocol on anonymous systems using constant memory space that solves the same problem. Thus impossibility results on anonymous rings (i.e. one may design a deterministic SSLE protocol, only on prime size rings, under a centralized daemon) can be extended to those kinds of id-based rings. Nevertheless, it is possible to design a silent and deterministic SSLE protocol requiring constant memory space on unidirectional, id-based rings where the id-values are bounded. We present such a protocol.We also present a randomized SSLE protocol and a token circulation protocol under an unfair, distributed daemon on anonymous and unidirectional rings of any size. We give a lower bound on memory space requirement proving that these protocols are space optimal. The memory space required is constant on average.

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We present a randomized self-stabilizing leader election protocol and a randomized self-stabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols are space optimal. We also give a formal and complete proof of these protocols. To this end, we develop a complete model for probabilistic self-stabilizing distributed systems which clearly separates the non deterministic behavior of the scheduler from the randomized behavior of the protocol. This framework includes all the necessary tools for proving the selfstabilization of a randomized distributed system: definition of a probabilistic space and definition of the self-stabilization of a randomized protocol. We also propose a new technique of scheduler management through a self-stabilizing protocol composition (cross-over composition). Roughly speaking, we force all computations to have a fairness property under any scheduler, even under an unfair one.

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The spanning tree construction is a fundamental task in communication networks.Improving the efficiency of the underlying spanning tree algorithm usually also corresponds to the improvement of the efficiency of the entire system. One of the important performance issues of selfstabilizing algorithms is the memory requirement per processor. The self-stabilizing spanning-tree algorithms to-date need a distance variable, which keeps track of the current level of the processor in the BFS tree. Thus, these BFS spanning tree construction algorithms have the space complexity of at least O(log N) bits per processor, where IV is the number of processors. Some authors have proposed specific data structures to store the distance variable in a distributed manner, thus reducing the memory requirement. We present a self-stabilizing BFS spanning tree construction algorithm which requires only O(1) bits of memory per link. The algorithm uses neither the distance variable nor any special data structure to achieve the memory requirement. One of the desirable features of the protocols written in large distributed systems is that the cost does not depend on the globaf properties, such as network size, which can change over time. Our afgorithm has this feature: when the network size changes, the algorithm does not need to be modified. The code at a processor needs to be modified only when the degree of the processor changes (a locally checkable property). It is known that no deterministic algorithm cart construct a spanning tree in an anonymous (uniform) network. The best that can be proposed is a semi-uniform deterministic algorithm as ours, in which all processors except one execute the same code. We call the distinguished processor the legal root (also denoted r-) which eventually becomes the root of the BFS trees. Each processor i maintains the following variables: (i) TS and P: Pointers to one of its neighbors (called i's parent) or to NULL; (ii) C: Color of i E {O, 1}; (iii) S: Status of i E {Idle, Working, Power, Erroneous}; (iv) ph: Phase of i C {a, b}.Our algorithm is a non-terminating algorithm.The legal root alternately builds O-colored and l-colored BFS spanq A large version ie in the Proc. of the Third Workshop on SelfStabilizing Sy.steme, Permission to make digitnlflmrd copies of all or ptwl of this m:ltcri:llfor pwsonrd or clmsroom use is grimted without fee provided that the copies are not nmde or distributed for profit or commerxinl advanlagc. Ihe copyright notice. the title of the publication mld iis date ;Ippcar.md nolicc is given that copyright is hy pemliesion oflhe ACM. Inc. "1'o copy otherwise. to republish, to post on servers or to redislribulc to Iisls. tcquircs spocdic permission end/or fee 1 997POD('97 Sm7(rI lkv+ora ( '.1 ( !S~i Copyrighl 1997 AChl 0-89791 -952 -1/97/8 ..$3.50ning trees (the obtained trees may differ from a construction to another one). We use ralor to denote the color of the current tree. The color is used to distinguish the processors that are from those that are not part of the current t...

We present a deterministic distributed depth-first token passing protocol on a rooted network. This protocol uses neither the processor identifiers nor the size of the network, but assumes the existence of a distinguished processor, called the root of the network. The protocol is self-stabilizing, meaning that starting from an arbitrary state (in response to an arbitrary perturbation modifying the memory state), it is guaranteed to reach a state with no more than one token in the network. Our protocol implements a strictly fair token circulation scheme. The proposed protocol has extremely small state requirement -only 3(∆ + 1) states per processor, i.e., O(log ∆) bits per processor, where ∆ is the degree of the network. The protocol can be used to implement a strictly fair distributed mutual exclusion in any rooted network. This protocol can also be used to construct a DFS spanning tree.