Memory efficient, self-stabilizing algorithm to construct BFS spanning trees

Abstract

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...

Light enabling snap-stabilization of fundamental protocols

“…The protocol provided in Cournier et al [2005] In arbitrary rooted networks, a Breath-First Spanning Tree Construction, noted BFSTC, consists in the computation of a rooted tree Tree(r) containing all processes and such that the distance of any process along the tree to the root is the smallest one. There exists many self-stabilizing BFSTC protocols [Huang and Chen 1992;Johnen 1997]. Delaët et al [2005]; propose a general scheme allowing in particular to implement a self-stabilizing BFSTC.…”

“…Delaët et al [2005]; propose a general scheme allowing in particular to implement a self-stabilizing BFSTC. The protocols proposed in Johnen [1997] and Delaët et al [2005] work under a distributed unfair daemon. However, except Johnen [1997], all these protocols are silent, that is, using such protocols, the system converges to a fix point.…”

“…Also, the simplicity of their codes will show that in practice as in theory BreakingIn is easier to obtain than stabilization. Hence, in contrast with the specific (self-or snap-) stabilizing solutions that usually implement complex mechanism [Huang and Chen 1993;Johnen 1997;Cournier et al 2005;Cournier et al 2006], our transformer can be seen as a powerful tool to simplify the design of snap-stabilizing protocols. Finally, we will show that, thanks to a counting property of our transformer, our depth-first token circulation can be used to solve in few lines the mutual exclusion in a snap-stabilizing manner.…”

Distributed Maintenance of a Spanning Tree Using Labeled Tree Encoding

“…Algorithms have also been proposed for other types of trees -like DFS tree [CD94] and minimum spanning tree [AS97]. There have been other papers which try to optimize on the memory used and stabilization time [Joh97], [AK93].…”

Self-stabilizing neighborhood synchronizer in tree networks

“…The self-stabilizing spanning tree construction algorithms have been proposed in [1], [3], [5], [10], [12], and [15]. Any of these algorithms can be combined with our synchronizer to design a synchronizer for a general network.…”