We study the parallel rotor walk process, which works as follows: Consider a graph along with an arbitrary distribution of tokens over its nodes. Every node is equipped with a rotor that points to its neighbours in a fixed circular order. In each round, every node distributes all of its tokens using the rotor. One token is allocated to the neighbour pointed at by the rotor, then the rotor moves to the subsequent neighbour, and so on, until no token remains.The process can be considered as a deterministic analogue of a process in which tokens perform one independent random walk step in each round. We compare the distribution of tokens in the rotor walk process with expected distribution in the random walk model. The similarity between the two processes is measured by their discrepancy, which is the maximum difference between the corresponding distribution entries over all rounds and nodes. We analyze a lazy variation of rotor walks that simulates a random walk with loop probability of 1/2 on each node, and each node sends not all its tokens, but every other token in each round.Viewing the rotor walk as a load balancing process, we prove that the rotor walk falls in the class of bounded-error diffusion processes introduced in [11]. This gives us discrepancy bounds of O(log 3/2 n) and O(1) for hypercube and r-dimensional torus with r = O(1), respectively, which improve over the best existing bounds of O(log 2 n) and O(n 1/r ). Also, as a result of switching to the load balancing view, we observe that the existing load balancing results can be translated to rotor walk discrepancy bounds not previously noticed in the rotor walk literature.We also use the idea of rotor walks to propose and analyze a randomized rounding discrete load balancing process that achieves the same balancing quality as similar protocols [11,3], but uses fewer number of random bits compared to [3], and avoids the negative load problem of [11].

We consider the neighbourhood load balancing problem. Given a network of processors and an arbitrary distribution of tasks over the network, the goal is to balance load by exchanging tasks between neighbours. In the continuous model, tasks can be arbitrarily divided and perfectly balanced state can always be reached. This is not possible in the discrete model where tasks are non-divisible. In this paper we consider the problem in a very general setting, where the tasks can have arbitrary weights and the nodes can have different speeds. Given a continuous load balancing algorithm that balances the load perfectly in T rounds, we convert the algorithm into a discrete version. This new algorithm is deterministic and balances the load in T rounds so that the difference between the average and the maximum load is at most 2d • w max , where d is the maximum degree of the network and w max is the maximum weight of any task. For general graphs, these bounds are asymptotically lower compared to the previous results. The proposed conversion scheme can be applied to a wide class of continuous processes, including first and second order diffusion, dimension exchange, and random matching processes. For the case of identical tasks, we present a randomized version of our algorithm that balances the load up to a discrepancy of O(√ d log n) provided that the initial load on every node is large enough.

Minimum-collisions assignment, in a wireless network, is the distribution of a finite resource set, such that the number of neighbor cells which receive common elements is minimized. In classical operator deployed networks, resources are assigned centrally. Heterogeneous networks contain user deployed cells, therefore centralized assignment is problematic. This paper examines the minimum-collisions assignment problem in the context of physical cell identity (PCI) allocation. Minimumcollisions assignment is NP-complete, therefore a potential-gametheoretic model is proposed as a distributed solution. The players of the game are the cells, actions are the set of PCIs and the utility of a cell is the number of neighbor cells in collision. The price of anarchy and price of stability are derived. Moreover the paper adapts a randomized-distributed-synchronous-update algorithm, for the case, when the number of PCIs is higher than the maximum degree of the neighbor relations graph. It is proven that the algorithm converges to a optimal pure strategy Nash equilibrium in finite time and it is robust to node addition. Simulation results demonstrate that the algorithm is sub-linear in the size of the input graph, thus outperforms best response dynamics.

Given an undirected graph with weights associated with its edges, the Steiner tree problem consists of finding a minimum weight subtree spanning a given subset of (terminal)

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