期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2003
卷号：2003
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：We study private computations in information-theoretical settings on networks that are not 2-connected. Non-2-connected networks are ``non-private'' in the sense that most functions cannot privately be computed on such networks. We relax the notion of privacy by introducing lossy private protocols, which generalize private protocols. We measure the information each player gains during the computation. Good protocols should minimize the amount of information it loses to the players. The loss of a protocol to a player is the logarithm of the number of different probability distributions on the communication strings a player can observe. For optimal protocols, this is justified by the following result: For a particular content of any player's random tape, the distributions the player observes have pairwise fidelity zero. Thus the player can easily distinguish the distributions. The simplest non-2-connected networks consists of two blocks that share one bridge node. We prove that on such networks, communication complexity and the loss of a private protocol are closely related: Up to constant factors, they are the same. Then we study 1-phase protocols, an analogue of 1-round communication protocols. In such a protocol each bridge node may communicate with each block only once. We investigate in which order a bridge node should communicate with the blocks to minimize the loss of information. In particular, for symmetric functions it is optimal to sort the components by increasing size. Then we design a 1-phase protocol that for symmetric functions simultaneously minimizes the loss at all nodes, where the minimum is taken over all 1-phase protocols. Finally, we prove a phase hierarchy. For any k there is a function such that every (k-1)-phase protocol for this function has an information loss that is exponentially greater than that of the best k-phase protocol.
