## Abstract

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.

Original language | English |
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Pages (from-to) | 5239-5246 |

Number of pages | 8 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 39 |

DOIs | |

State | Published - 9 Sep 2011 |

### Bibliographical note

Funding Information:The first author was partially supported by NSF grant CCR-09-04581, ISF grant 347/09, the Israel–Korea Scientific Research Cooperation, and BSF grant 2008217. The second author was partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grants 35/05 and 347/09, the Israel–Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States–Israel Binational Science Foundation (BSF) and DFG. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0007860). The fourth author was partially supported by the Korea Foundation for International Cooperation of Science Technology (KICOS) through a grant provided by the Korean Ministry of Education, Science & Technology (MEST) in 2009 (No. K20717000007-09B0100-00710) and Frontier Functional Proteomics Project from Korean Ministry of Education, Science and Technology (FPR08-A1-020). The fifth author was supported by NAP of Korea Research Council of Fundamental Science & Technology. The sixth author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0090441) and by Inha University Research Grant.

## Keywords

- Consensus strings
- Distance sum
- Multiple alignments
- Radius

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science