Consider n nodes connected to a single coordinator. Each node receives an individual online data stream of numbers and, at any point in time, the coordinator has to know the k nodes currently observing the largest values, for a given k between 1 and n. We design and analyze an algorithm that solves this problem while bounding the amount of messages exchanged between the nodes and the coordinator. Our algorithm employs the idea of using filters which, intuitively speaking, leads to few messages to be sent, if the new input is "similar" to the previous ones. The algorithm uses a number of messages that is on expectation by a factor of O((log + k) log n) larger than that of an offline algorithm that sets filters in an optimal way, where is upper bounded by the largest value observed by any node.
In their work @cite_0 , Yi and Zhang were the first to study streaming algorithms with respect to their competitiveness. In their model there is one node and one coordinator and the goal is to keep the coordinator informed about the current value of a multivalued function @math that is observed by the node and changes its value over time, while minimizing the number of messages. Among others, for @math , Yi and Zhang present an algorithm that is @math -competitive if the last value received by the coordinator might deviate by @math from the current value of @math . For arbitrary @math , a competitiveness of @math is shown.
@cite_0
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This paper presents two efficient flooding algorithms based on 1-hop neighbor information. In the first part of the paper, we consider sender-based flooding algorithms, specifically the algorithm proposed by In their paper, propose a sender-based flooding algorithm that can achieve local optimality by selecting the minimum number of forwarding nodes in the lowest computational time complexity O(n logn), where n is the number of neighbors. We show that this optimality only holds for a subclass of sender-based algorithms. We propose an efficient sender-based flooding algorithm based on 1-hop neighbor information that reduces the time complexity of computing forwarding nodes to O(n). In Liu's algorithm, n nodes are selected to forward the message in the worst case, whereas in our proposed algorithm, the number of forwarding nodes in the worst case is 11. In the second part of the paper we propose a simple and highly efficient receiver-based flooding algorithm. When nodes are uniformly distributed, we prove that the probability of two neighbor nodes broadcasting the same messageneighbor nodes broadcasting the same message exponentially decreases when the distance between them decreases or when the node density increases. The analytical results are confirmed using simulation.