Time- and space-efficient self-stabilizing algorithms

Abstract

In a distributed system error handling is inherently more difficult than in conventional systems that have a central control unit. To recover from an erroneous state the nodes have to cooperate and coordinate their actions based on local information only. Self-stabilization is a general approach to make a distributed system tolerate arbitrary transient faults by design. A self-stabilizing algorithm reaches a legitimate configuration in a finite number of steps by itself without any external intervention, regardless of the initial configuration. Furthermore, once having reached legitimacy this property is preserved. An important characteristic of an algorithm is its worst-case runtime and its memory requirements. This thesis presents new timeand space-efficient self-stabilizing algorithms for well-known problems in algorithmic graph theory and provides new complexity analyses for existing algorithms. The main focus is on proof techniques used in the complexity analyses and the design of the algorithms. All algorithms presented in this thesis assume the most general concept with respect to concurrency. The maximum weight matching problem is a fundamental problem in graph theory with a variety of applications. In 2007, Manne and Mjelde presented the first selfstabilizing algorithm to compute a 2-approximation for this problem. They proved an exponential upper bound on the time complexity until stabilization is reached for both the sequential and the concurrent setting. This thesis presents a new proof technique based on graph reduction to analyze the complexity of self-stabilizing algorithms. It is used to show that the algorithm of Manne and Mjelde in fact stabilizes within polynomial time assuming sequential execution and that a modified version of the algorithm also stabilizes within polynomial time in a concurrent setting. Connected dominating sets are a vital structure for many applications. By relaxing the connectivity requirement the number of nodes can be reduced significantly. The first self-stabilizing algorithm for the weakly connected minimal dominating set problem was presented by Srimani and Xu in 2007. For the worst-case runtime they proved an exponential upper bound. It remained an open problem whether this limit is sharp. This thesis provides an example that shows that their algorithm indeed has an exponential time complexity. Furthermore, a new self-stabilizing algorithm is presented that stabilizes within polynomial time. Another classical problem in graph theory is the computation of a minimum vertex cover. Currently, all self-stabilizing algorithms for this problem assume symmetrybreaking mechanisms, such as restricted concurrency, unique identifiers, or randomization. This thesis presents a deterministic self-stabilizing algorithm to compute a (3− 2 ∆+1)-approximation of a minimum vertex cover in anonymous networks. It reaches stabilization within polynomial runtime and requires O(log n) storage per node. For trees the algorithm computes a 2-approximation of a minimum vertex cover. In 2008, Dong et al. introduced the edge-monitoring problem and provided a distributed algorithm to solve it. In this thesis the first self-stabilizing algorithm for this problem is developed. Several versions of the edge-monitoring problem are considered. The proposed algorithms have polynomial time complexity.