Searching with lies under error cost constraints

Authors:
Rudolf Ahlswede;Ferdinando Cicalese;Christian Deppe
Affiliations:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany;Department of Computer Science and Applications, University of Salerno, I-84081 Baronissi (SA), Italy and AG Genominformatik, Technische Fakultät, Universität Bielefeld, Germany;Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

The Renyi-Berlekamp-Ulam game is a classical model for the problem of determining the minimum number of queries to find an unknown member in a finite set when up to a finite number of the answers may be erroneous. In the variant considered in this paper, questions with q many possible answers are allowed, further lies are constrained by a bipartite graph with edges weighted by 0,1,2,... (the ''channel''). The channel @C is an arbitrary assignment stipulating the cost of the different possible lies, i.e., of each answer ji when the correct answer is i by @C(i,j). It is also assumed that a maximum cost e (sum of the cost of all wrong answers) can be afforded by the responder during the whole game. We provide tight asymptotic bounds for the number of questions needed to solve this problem. The appropriate searching strategies are actually provided. We also show that adaptiveness can be dramatically reduced when the channel satisfies certain symmetry constraints.