Efficient algorithms for consensus string problems minimizing both distance sum and radius

The consensus (string) problem is finding a representative string, called a consensus, of a given set S of strings. In this paper we deal with consensus problems considering both distance sum and radius, where the distance sum is the sum of (Hamming) distances from the strings in S to the consensus and the radius is the longest (Hamming) distance from the strings in S to the consensus. Although there have been results considering either distance sum or radius, there have been no results considering both, to the best of our knowledge. We present the first algorithms for two consensus problems considering both distance sum and radius for three strings: one problem is to find an optimal consensus minimizing both distance sum and radius. The other problem is to find a bounded consensus such that the distance sum is at most s and the radius is at most r for given constants s and r . Our algorithms are based on characterization of the lower bounds of distance sum and radius, and thus they solve the problems efficiently. Both algorithms run in linear time.