Consensus Optimizing Both Distance Sum and Radius

The consensus string problem is finding a representative string (consensus) of a given set S of strings. In this paper we deal with the consensus string problems optimizing 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 as far as we know. We present two algorithms to solve the consensus string problems optimizing both distance sum and radius for three strings. The first algorithm finds the optimal consensus string that minimizes both distance sum and radius, and the second algorithm finds the bounded consensus string such that, given constants s and r, the distance sum is at most s and the radius is at most r. Both algorithms take linear time.