The Designs and Analyses of Self-Stabilizing Algorithm for Maximal Weight Matching Problem on Complete Graphs

[[abstract]]In 1974, Dijsktra defined a self-stabilizing system as a system which is guaranteed to arrive at a legitimate state in a finite number of steps regardless of its initial state. Since his introduction, self-stabilizing algorithms gained wide-spread research interest. The objectives of this research are to design and analyze self-stabilizing algorithms for maximal weight matching problem. Firstly, Hsu and Huang proved that the time complexity of their self-stabilizing algorithm for finding a maximal matching in distributed networks is O(n3), where n is the number of nodes in the graph. In 1994, Tel introduced a variant function to show that the time complexity of Hsu-Huang's algorithm is O(n2). In this paper, we design a self-stabilizing algorithm for maximal weight matching of the complete graph and prove its correctness. The maximal weight matching problem is defined not only to find the maximal matching of the complete graph, but also to let the total weight of the matching edges be maximal. We combine Hsu-Huang's maximal matching alogrithm and new swapping rules. This system possesses the properties of fault tolerance and self-stabilization and has a time complexity O(n2+nk), where k is the largest weight over all edges in the graph.

[[alternative]]The Designs and Analyses of Self-Stabilizing Algorithms for Maximal Matching and Stable-Marriage Problems

[[abstract]]In 1974, Dijsktra defined a self-stabilizing system as a system which is guaranteed to arrive at a legitimate state in a finite number of steps regardless of its initial state. Since his introduction, self-stabilizing algorithms gained wide-spread research interest. The objectives of this research are to design and analyze self-stabilizing algorithms for maximal matching and stable-marriage problems. Firstly, we discuss Hsu-Huang's self-stabilizing algorithm for finding a maximal matching in distributed networks. Hsu and Huang proved that the time complexity of their algorithm is O(n3) , where n is the number of nodes in the graph. In 1994, Tel introduced a variant function to show that the time complexity of Hsu-Huang's algorithm is O(n2) . In this research, we present a new method to expose that this algorithm has a time complexity of Theta(n2) . Secondly, we design a self-stabilizing algorithm for maximal weight matching problem for complete graphs and prove its correctness. The maximal weight matching problem is defined not only to find the maximal matching of the complete graph, but also to let the total weight of the matching edges be maximal. We combine Hsu-Huang's maximal matching algorithm and new swapping rules. This system possesses the properties of fault tolerance and self-stabilization and has a time complexity O(n2+nk) , where k is the largest weight over all edges in the graph. Thirdly, the stable-marriage problem is that of matching n men and n women, each of them has ranked the members of the opposite sex in order of preference, so that no unmatched couple both prefer each other to their partners under the matching. In the research, we also use the swapping rule according to the preference to make the matching couple stable. Our self-stabilizing algorithm guarantees that the system will reach a legitimate stable state in O(n3) steps to find one stable matching from any initial state.