Locating information with uncertainty in fully interconnected networks

We consider the problem of searching for a piece of information in a fully interconnected computer 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 are 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, that 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. We show that, on the average, if the probability p that a node provides correct advice is asymptotically larger than 1/n, where it is the number of the computer nodes, then the average, time complexity for locating the information node is, asymptotically, 1/p or 2/p depending on the available memory. The probability p may, in general, be a function of the number of network nodes n. On the lower bounds side, we prove that no fixed memory deterministic algorithm can locate the information node in finite expected number of steps. We also prove a lower bound of 1/p for the expected number or steps of any algorithm that locates the information node in the complete network.