关于Herman算法的稳定性
On Stabilization in Herman's Algorithm
摘要：
Herman算法是一种同步随机协议，用于在由N个进程组成的令牌环中实现自稳定。令牌的交互使得协议的动态性非常难以分析。在本文中，我们研究的期望时间稳定的初始配置。简单地说，该算法几乎肯定会从任何初始配置实现稳定，并且已知最坏情况下的预期稳定时间（相对于初始配置）为Theta（N^2）。我们的第一个贡献是给出了期望稳定时间的上界为0.64 N^2，改进了以前的上界，并缩小了与现有最佳下界的差距。我们还介绍了一个异步版本的协议，在这种情况下显示了类似的O（N^2）收敛界。假设错误产生的腐败的一些数量k位，其中k是固定的独立的环的大小，我们表明，预期的时间稳定是O（N）。这揭示了赫尔曼算法的一个迄今为止未知的和非常理想的属性：它从有界误差中快速恢复。我们还表明，如果初始配置产生的重置每个位独立和均匀随机，然后稳定是显着快于在最坏的情况下。
英文摘要：
Herman's algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration. It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worst-case expected time to stabilization (with respect to the initial configuration) is Theta(N^2). Our first contribution is to give an upper bound of 0.64 N^2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N^2) convergence bound in this case. Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman's algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case.
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