Locating information with uncertainty in fully interconnected networks: The case of nondistributed memory

We consider the problem of searching for a piece of information in a fully interconnected computer network (also called a complete network or clique) by exploiting advice about its location from the network nodes. Each node contains a database that “knows” what kind of documents or information is stored in other nodes (e.g., a node could be a Web server that answers queries about documents stored on the Web). The databases in each node, when queried, provide a pointer that leads to the node that contains the information. However, this information is up-to-date (or correct) with some bounded probability. While, in principle, one may always locate the information by simply visiting the network nodes in some prescribed ordering, this requires a time complexity in the order of the number of nodes of the network. In this paper, we provide algorithms for locating an information node in the complete communication network, which take advantage of advice given from network nodes. The nodes may either give correct advice, by pointing directly to the information node, or give wrong advice, by pointing elsewhere. On the lower-bounds' side, we show that no fixed-memory (i.e., with memory independent of the network size) deterministic algorithm may locate the information node in a constant (independent of the network size) expected number of steps. Moreover, if p=ω(1/n) is the probability that a node of an n-node clique gives correct advice, we show that no algorithm may locate the information node in an expected number of steps less than 1/p–o(1). To study how the expected number of steps is affected by the amount of memory allowed to the algorithms, we give a memoryless randomized algorithm with expected number of steps 4/p+o(1/p)+o(1) and a 1-bit randomized algorithm requiring on the average at most 2/p+o(1) steps. In addition, in the memoryless case, we also prove a 4/p lower bound for the expected number of steps in the case where the nodes giving faulty advice may decide on the content of this advice in any possible way and not merely at random (adversarial fault model). Finally, for the case where faulty nodes behave randomly, we given an optimal, unlimited memory deterministic algorithm with expected number of steps bounded from above by 1/p+o(1/p)+1.