Models , Tractability , and Complexity

There are many examples for the fact that the growth of a system is limited if this system relies on important central instances for supply or control. In a growing system, the central instances turn into bottlenecks, thus making the system inefficient. Examples for this phenomenon can be found in computer science, in history, and even in the behavior of dinosaurs. Distributed systems have great potential to bypass such bottlenecks, but due to ”friction” issues such as incomplete knowledge, we may not be able to exploit their potential. In this thesis we show that some of these issues can be handled quite well in weak and realistic models. More specifically, we present and discuss three examples: 1. Distributing Rare Resources: Roman Domination and Win–Win Given a graph, servers (resources) should be placed on nodes to service a pair of requests that can occur at nodes. Since resources are rare, the number of used resources should be minimized. What is the computational complexity of this problems, and how much control, communication, or interaction is needed to organize the service? We give detailed characterization of the computational complexity of these problems. Especially, we show that these problems are NP–hard; and for Planar Roman Domination, a Polynomial Time Approximation Scheme (PTAS) is presented. 2. Distributed Data Structure Given a distributed system of processors/computers/servers with weak assumptions, is it possible to build and maintain a distributed data structure in this system? We propose a distributed dictionary that supports insert and search operations and that tolerates arbitrary single server crashes. In contrast to other proposals of distributed fault tolerant search structures, our solution works in an asynchronous and weak environmental setting. 3. Distributed Coordination: Point Formation There are n robots in the Euclidean plane that should move to one destination point. What is a minimal set of abilities that the robots have to have in order to complete this task? We define the concept of contraction functions and show that with this concept, the point formation problem can be solved in an asynchronous model. The results show that these problems can be solved in weak and realistic settings. We will discuss these problems individually, but we will discuss the connections between them as well. Interestingly, we will find connections between the point formation problem and facility location as well.