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Over the last decade, PageRank has gained importance in a wide range of applications and domains, ever since it first proved to be effective in determining node importance in large graphs (and was a pioneering idea behind Google's search engine). In distributed computing alone, PageRank vector, or more generally random walk based quantities have been used for several different applications ranging from determining important nodes, load balancing, search, and identifying connectivity structures. Surprisingly, however, there has been little work towards designing provably efficient fully-distributed algorithms for computing PageRank. The difficulty is that traditional matrix-vector multiplication style iterative methods may not always adapt well to the distributed setting owing to communication bandwidth restrictions and convergence rates.In this paper, we present fast random walk-based distributed algorithms for computing PageRanks in general graphs and prove strong bounds on the round complexity. We first present a distributed algorithm that takes O log n/ rounds with high probability on any graph (directed or undirected), where n is the network size and is the reset probability used in the PageRank computation (typically is a fixed constant). We then present a faster algorithm that takes O √ log n/ rounds in undirected graphs. Both of the above algorithms are scalable, as each node sends only small (polylog n) number of bits over each edge per round. To the best of our knowledge, these are the first fully distributed algorithms for computing PageRank vector with provably efficient running time.

We study robust and efficient distributed algorithms for searching, storing, and maintaining data in dynamic Peer-to-Peer (P2P) networks. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to guarantee, despite high node churn rate, that a large number of nodes in the network can store, retrieve, and maintain a large number of data items. Our main contributions are fast randomized distributed algorithms that guarantee the above with high probability even under high adversarial churn. In particular, we present the following main results:1. A randomized distributed search algorithm that with high probability guarantees that searches from as many as n − o(n) nodes (n is the stable network size) succeed in O(log n)-rounds despite O(n/ log 1+δ n) churn, for any small constant δ > 0, per round. We assume that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm).2. A storage and maintenance algorithm that guarantees, with high probability, data items can be efficiently stored (with only Θ(log n) copies of each data item) and maintained in a dynamic P2P network with churn rate up to O(n/ log 1+δ n) per round. Our search algorithm together with our storage and maintenance algorithm guarantees that as many as n − o(n) nodes can efficiently store, maintain, and search even under O(n/ log 1+δ n) churn per round. Our algorithms require only polylogarithmic in n bits to be processed and sent (per round) by each node.To the best of our knowledge, our algorithms are the first-known, fully-distributed storage and search algorithms that provably work under highly dynamic settings (i.e., high churn rates per step). Furthermore, they are localized (i.e., do not require any global topological knowledge) and scalable. A technical contribution of this paper, which may be of independent interest, is

The mobile robot dispersion problem on graphs asks k ≤ n robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a configuration in which each robot is on a distinct node of the graph. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we provide two novel deterministic algorithms for dispersion, one for arbitrary graphs and another for grid graphs, in a synchronous setting where all robots perform their actions in every time step. Our algorithm for arbitrary graphs has O(min(m, k∆) · log k) steps runtime using O(log n) bits of memory at each robot, where m is the number of edges and ∆ is the maximum degree of the graph. This is an exponential improvement over the O(mk) steps best previously known algorithm. In particular, the runtime of our algorithm is optimal (up to a O(log k) factor) in constant-degree arbitrary graphs. Our algorithm for grid graphs has O(min(k, √ n)) steps runtime using Θ(log k) bits at each robot. This is the first algorithm for dispersion in grid graphs. Moreover, this algorithm is optimal for both memory and time when k = Ω(n). problem). This problem has many practical applications, for example, in relocating selfdriven electric cars (robots) to recharge stations (nodes), assuming that the cars have smart devices to communicate with each other to find a free/empty charging station [1,16]. This problem is also important due to its relationship to many other well-studied autonomous robot coordination problems, such as exploration, scattering, load balancing, covering, and self-deployment [1,16]. One of the key aspects of mobile-robot research is to understand how to use the resource-limited robots to accomplish some large task in a distributed manner [10,11]. In this paper, we study the trade-off between memory requirement of robots and the time to solve Dispersion on graphs.Augustine and Moses Jr.[1] studied Dispersion assuming k = n. They proved a memory lower bound of Ω(log n) bits at each robot and a time lower bound of Ω(D) (Ω(n) in arbitrary graphs) for any deterministic algorithm in any graph, where D is the diameter of the graph. They then provided deterministic algorithms using O(log n) bits at each robot to solve Dispersion on lines, rings, and trees in O(n) time. For arbitrary graphs, they provided two algorithms, one using O(log n) bits at each robot with O(mn) time and another using O(n log n) bits at each robot with O(m) time, where m is the number of edges in the graph. Recently, Kshemkalyani and Ali [16] provided an Ω(k) time lower bound for arbitrary graphs for k ≤ n. They then provided three deterministic algorithms for Dispersion in arbitrary graphs: (i) The first algorithm using O(k log ∆) bits at each robot with O(m) time, (ii) The second algorithm using O(D log ∆) bits at each robot with O(∆ D ) time, and ...

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