环形网格上的多色发电机
Multicolored Dynamos on Toroidal Meshes
摘要：
在一个图上检测出能够“激活”图中指定数量的顶点的最小数量的节点（目标集）的存在被称为目标集选择问题（TSS），该问题由Kempe、Kleinberg和Tardos提出。在TSS的设置中，节点有两种可能的状态（活动或非活动），触发节点激活的阈值由其活动邻居的数量给出。在基于多数的系统中处理容错，两种可能的状态被用来表示故障或非故障节点，并且阈值由大多数邻居的状态给出。在这里，主要的工作是确定导致整个系统故障行为的初始故障的分布。这种激活模式，也被称为动态垄断（dynamic monopoly，简称dynamo），是由Peleg在1996年提出的。在本文中，我们扩展了TSS问题的设置表示节点的状态与“多色”集。该问题的扩展版本可以描述如下：设G是一个简单连通图，其中每个节点被分配来自有限有序集C = {1，. . ., k}的颜色。在每个本地时间步，每个节点可以根据本地配置重新为自己着色，颜色由其大多数邻居保持。给定G，我们研究的颜色的初始分布导致系统的一个k单色配置在环形网格，专注于最小数量的初始k色节点。我们发现上界和下界的大小的发电机，然后特殊类的发电机，概述了通过一种新的方法的基础上，包围模式，其特征在于。
英文摘要：
Detecting on a graph the presence of the minimum number of nodes (target set) that will be able to "activate" a prescribed number of vertices in the graph is called the target set selection problem (TSS) proposed by Kempe, Kleinberg, and Tardos. In TSS's settings, nodes have two possible states (active or non-active) and the threshold triggering the activation of a node is given by the number of its active neighbors. Dealing with fault tolerance in a majority based system the two possible states are used to denote faulty or non-faulty nodes, and the threshold is given by the state of the majority of neighbors. Here, the major effort was in determining the distribution of initial faults leading the entire system to a faulty behavior. Such an activation pattern, also known as dynamic monopoly (or shortly dynamo), was introduced by Peleg in 1996. In this paper we extend the TSS problem's settings by representing nodes' states with a "multicolored" set. The extended version of the problem can be described as follows: let G be a simple connected graph where every node is assigned a color from a finite ordered set C = {1, . . ., k} of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. Given G, we study the initial distributions of colors leading the system to a k monochromatic configuration in toroidal meshes, focusing on the minimum number of initial k-colored nodes. We find upper and lower bounds to the size of a dynamo, and then special classes of dynamos, outlined by means of a new approach based on recoloring patterns, are characterized.
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