具有非齐次步长的随机游动覆盖时的紧界
Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths
摘要：
在生物世界的所有尺度上都观察到了不同长度的随机定向步骤的搜索模式，从微观到生态，包括蛋白质马达，细菌，T细胞，蜜蜂，海洋捕食者等等。通过不同的模型，它已被证明，采用不同的步长的大小可以大大提高搜索效率。然而，搜索效率和步长的数量之间的精确连接的曲目的搜索引擎尚未确定。受一维地形中生物实例的启发，最近的一篇论文研究了n节点循环上的最佳覆盖时间，该时间可以通过使用k步长的随机行走过程来实现。通过调整的长度和相应的概率，作者在其中表明，最佳的覆盖时间约为n 1+ $Θ$（1/k）。虽然这个界限对于k值很大很有用，但对于生物学中感兴趣的小k值几乎没有信息。在本文中，我们提供了一个紧界的覆盖时间这样的行走，对任何整数k > 1。具体地说，直到低阶多对数因子，覆盖时间的上限是n的指数为1+ 1/（2k-1）的多项式。对于k = 2、3、4和5，指数因此分别为4/3、6/5、8/7和10/9。非正式地，我们的结果意味着，只要步长k的数量不是太大，将额外的步长的过程的剧目，使提高覆盖时间的多项式因子，但改善的程度逐渐减少与k。
英文摘要：
Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from the microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n 1+$Θ$(1/k). While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k > 1. Specifically, up to lower order polylogarithmic factors, the upper bound on the cover time is a polynomial in n of exponent 1+ 1/(2k--1). For k = 2, 3, 4 and 5 the exponent is thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k.
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