The Time Complexity of Hsu and Huang's Self-Stabilizing Maximal Matching Algorithm

The idea of self-stabilization was introduced by Edsger W. Dijkstra in 1974 [1]. Self-stabilizing algorithms enable systems to be started in an arbitrary state and still converge to a desired behavior. In this letter, we discuss the time complexity of the self-stabilizing algorithm proposed by Hsu and Huang in [2], which finds a maximal matching in a network. This algorithm is the first self-stabilizing maximal matching algorithm and has been regularly cited in the literature. Because of its technical importance, the time complexity of this particular algorithm has been well studied. In [2], Hsu and Huang show that it is bounded by O(n3), where n is the number of nodes. In [3], Tel provides an almost tight upper bound, which is 2n 2+2n+1 if n is even and 2n 2+n− 2 if n is odd. In [4] Tel gives a more concise proof for the O(n2) bound than [3]. In [5] Hedetniemi, Jacobs and Srimani provide an upper bound of 2m + n, where m is the number of edges. This gives a better bound than the one in [3] only if the network is sparse. In this letter, we provide the exact time complexity of the Hsu-Huan algorithm.