Int'l Conf. Foundations of Computer Science | FCS'16 | 1 SESSION GRAPH AND NETWORK BASED ALGORITHMS Chair(s) TBA ISBN: 1-60132-434-0, CSREA Press © 2 Int'l Conf. Foundations of Computer Science | FCS'16 | ISBN: 1-60132-434-0, CSREA Press © Int'l Conf. Foundations of Computer Science | FCS'16 | 3 SELFISH STABILIZATION OF SHORTEST PATH TREE FOR TWO- COLORED GRAPHS A. Anurag Dasgupta1, and B. Anindya Bhattacharya2 1Mathematics and Computer Science, Valdosta State University, Valdosta, GA, USA 2Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA Abstract - In modern day stabilizing distributed systems, each peer networks, mobile computing, topology update, clock process/node or each administrative domain may have selsh synchronization, and many others. motives to optimize its payoff. While maximizing/minimizing own payoffs, the nodes or the domains do not require to give up their stabilization property. Optimizing individual pay offs Selfish stabilization combines the concept of game theory and without sacricing the stabilization property is a relatively stabilization together. There are some strong similarities new trend and this characteristic of the system is termed as between selfish stabilization and game theory, but there are selsh stabilization significant differences too. The players in games are analogous to processes in a stabilizing system, and the The focus of this paper is to investigate the problem of finding equilibrium in games is comparable to the stable configuration a stable shortest path tree for two-colored graphs, where the of stabilizing systems, in as much as both satisfy the colors represent different types of processes or domains. In a convergence and closure properties. However, games usually shortest path tree, for every node, its path along the tree has start from predefined initial configurations, and mostly ignore the minimum possible distance of any path to the root. In this faulty moves or transient state corruptions, which are not paper we study the impact of selfishness on stabilization, necessarily true for stabilizing systems [2]. provide examples to demonstrate the effects of different types of schedulers, and explore how the stabilization time is affected by parameter changes. In traditional stabilizing distributed systems [3], we assume that all processes run some predefined programs or Keywords: Graph theory, stabilization, distributed systems, algorithms. These algorithms are mandated by an external shortest path tree, algorithms, fault tolerance. agency and most often the agency is the owner or the administrator of the entire distributed system. The model is 1 Introduction widely recognized by the stabilization community. This works fine when processes cooperate with one another and share a Stabilization is an important model of fault-tolerance for purely global goal. But in modern times in the Internet, it is distributed computation. The appeal of a stabilizing system possible for the processes to have some private goals besides lies in its robustness and ability to recover from any transient the common global goal. It is quite realistic and fairly fault. A stabilizing distributed system has a subset of desirable common these days to have a distributed system spanning states to which the system converges. These are called the set over multiple administrative domains and therefore processes of legal states. A state not belonging to the set of legal states is having individual goals are not a rare occurrence. On Internet- called an illegal state. A system is stabilizing if and only if it scale distributed systems, each process or each domain may satisfies two properties: a) starting from any state, it is have selfish motives to optimize its own payoff besides the guaranteed that the system will eventually reach a legal state global goal. So the spirit of competition in such cases does not (convergence), and b) given that the system is in a legal state, conflict with the general spirit of cooperation. Optimizing it is guaranteed to stay in a legal state, provided that no fault individual payo s without sacricing the stabilization happens (closure) [1]. The above two properties guarantee that property of the system is termed as selsh stabilization [4]. a stabilizing system will eventually recover from any transient faults that take the system to some arbitrary configuration and The focus of this paper is to finding a selfish-stabilizing this recovery procedure does not require any manual shortest path tree algorithm for two-colored graphs, where the intervention. For the above reasons, stabilizing systems do not colors represent different types of processes or domains. In a need initialization and they can be spontaneously deployed. shortest path tree, for every node, its path along the tree has Because a stabilizing algorithm does not require correct the minimum possible distance of any path to the root. In the initialization and can recover from any transient failures of subsequent sections, we study the impact of selfishness on arbitrary types occurring at any time, stabilization is an stabilization, provide examples to demonstrate the effects of interesting and active research field and it is used in a large different types of schedulers, and explore how the stabilization number of applications, including sensor networks, peer-to- time is affected by changes to a given graph's parameter ISBN: 1-60132-434-0, CSREA Press © 4 Int'l Conf. Foundations of Computer Science | FCS'16 | changes. We also present examples to show how competition (d) is (11, 8). So, different trees yield different costs for blends with cooperation in a stabilizing environment and different colors [4]. provide some experimental results. 2 Background 2.1 Model and Notation Assume a graph G = (V, E). Let V = {1, 2, …, n} denote the set of nodes or processes and E be the set of edges connecting pairs of nodes. Let there are p different subsets or colors of nodes. In our case, p = 2, but in general p could be any value greater than 1. For each subset, we define a separate cost function to map the set of edges to the set of positive integers. Following our selfish stabilization algorithm, starting from any random initial configuration, the different subsets or colors of nodes will cooperate with one another to form a rooted shortest path tree and simultaneously compete against each other to minimize their distance with the root node. We will assume the shared memory model for the communication among the nodes. According to this model, each process can read the states of its 1-distance neighbors and update its own state if required. In each individual step, a process checks a guarded action g A: where g is a Boolean variable. The value of g is a function of the process’s own Fig. 1: Different spanning trees of the graph in part (a) (note state and the states of its immediate neighbors. If g is true, the that not all trees are terminal configurations) process executes action A to perform an update of its own state. If g is false, no action is taken. The global state or configuration of the system consists of the local states of all the processes. Unless stated otherwise, a serial 2.2 Related Work scheduler/daemon schedules the action by randomly choosing a process with an enabled guard to execute its action. Our work is directly related to the paper by Cohen et al. [5] in Let us convert G into a multi-weighted graph by de[ning a which the authors described a selfish stabilization algorithm cost function w of E Np, where N is the set of positive for the minimum spanning tree problem. The algorithm for the minimum spanning tree and the shortest path tree is integers. For every i Ǩ [1. p], the function wi : E N denotes the cost of using edge e (the distance value). Starting essentially the same. In another paper, Dasgupta et al. [6] from any arbitrary initial conguration, the p di erent colors described a probabilistic fault-containment algorithm that of nodes cooperate with one another to form a rooted spanning stabilizes a system from minor failures with a stabilization tree, and at the same time compete against each other to time independent of the network size. In [7], the author minimize their distance value to the root. described a selfish stabilization algorithm for the maximum flow tree problem. Cobb et al. [8] proposed a stabilizing All nodes in the graph have a common global goal in this solution to the stable path problem. Mavronicolas [9] used a problem: starting from an arbitrary initial con[guration, each game theoretic presentation to model security in wireless node collaborate with one another to form a rooted shortest sensor networks where the network security is viewed as a path tree. But in addition to the common goal, the subsets or game between the attackers and the defenders. The last one is colors have their private goals. The private goal of each node only tangentially related to our work. It involves the spirit of is to optimize (in this case, it is a minimization problem) its competition and co-operation simultaneously as in our case, distance value without violating the spanning tree constraints. but stabilization is not an issue. Fig. 1 shows an example of a two-colored graph (a) in which three spanning trees could be obtained at some point of a 3 Algorithm computation (none of these necessarily denotes the terminal configuration). The root is denoted by r and we chose grey In accordance with the shared memory model, each node i can color to indicate the root. For example, the cost of tree (b) is read the states of N(i), the set of its neighbors (excluding i (10, 9), while the cost of tree (c) is (9, 9) and the cost of tree ISBN: 1-60132-434-0, CSREA Press © Int'l Conf.