树中延迟引起的指数记忆间隙
Delays Induce an Exponential Memory Gap for Rendezvous in Trees
摘要：
图中的会合是指两个移动Agent在未知的匿名连通图的某个节点上相遇。在本文中，我们专注于会合在树上，类似的努力，已经取得了解决探索问题与紧凑的自动机，我们研究的内存大小的移动代理，允许确定性地解决会合问题。我们假设代理是相同的，并在同步轮移动。我们首先表明，如果代理的开始时间之间的延迟是任意的，那么集合所需的内存下限是Ω（log n）位，即使是长度为n的线。这个下限满足一个先前已知的上限为O（log n）位的任意图的大小最多为n的交会。我们的主要结果是一个证明，同时开始会合所需的内存量基本上取决于树的叶子的数量L，并指数影响较少的节点数n。事实上，我们提出了两个相同的代理O（日志L + loglog n）位的内存，解决了在所有树的会合问题，最多n个节点，最多L叶。因此，对于一类树与polymerically许多叶子，有一个指数差距的最小内存大小之间的情况下，任意延迟和延迟为零的情况下会合。此外，我们表明，我们的上限是最佳的证明，欧米茄（log L + loglog n）$位的内存所需的交会，即使在类树的程度有界的3。
英文摘要：
The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this paper, we focus on rendezvous in trees, and, analogously to the efforts that have been made for solving the exploration problem with compact automata, we study the size of memory of mobile agents that permits to solve the rendezvous problem deterministically. We assume that the agents are identical, and move in synchronous rounds. We first show that if the delay between the starting times of the agents is arbitrary, then the lower bound on memory required for rendezvous is Omega(log n) bits, even for the line of length n. This lower bound meets a previously known upper bound of O(log n) bits for rendezvous in arbitrary graphs of size at most n. Our main result is a proof that the amount of memory needed for rendezvous with simultaneous start depends essentially on the number L of leaves of the tree, and is exponentially less impacted by the number n of nodes. Indeed, we present two identical agents with O(log L + loglog n) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most L leaves. Hence, for the class of trees with polylogarithmically many leaves, there is an exponential gap in minimum memory size needed for rendezvous between the scenario with arbitrary delay and the scenario with delay zero. Moreover, we show that our upper bound is optimal by proving that Omega(log L + loglog n)$ bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.
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