The trend towards very large DNA sequencing projects, such as those being undertaken as part of the human genome initiative, necessitates the development of efficient and precise algorithms for assembling a long DNA sequence from the fragments obtained by shotgun sequencing or other methods. The sequence reconstruction problem that we take as our formulation of DNA sequence assembly is a variation of the shortest common superstring problem, complicated by the presence of sequencing errors and reverse complements of fragments. Since the simpler superstring problem is NP-hard, any efficient reconstruction procedure must resort to heuristics. In this paper, however, a four phase approach based on rigorous design criteria is presented, and has been found to be very accurate in practice. Our method is robust in the sense that it can accommodate high sequencing error rates and list a series of alternate solutions in the event that several appear equally good. Moreover it uses a limited form ...

Contents 1 Introduction 1 2 Notation and preliminaries 4 2.1 Notation and basic definitions : : : : : : : : : : : : : : : : : : 4 2.2 Longest common subsequences : : : : : : : : : : : : : : : : : : 7 2.3 Computing longest common subsequences : : : : : : : : : : : 10 2.4 Expected length of longest common subsequences : : : : : : : 14 3 Lower Bounds 20 3.1 Css machines : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 3.2 Analysis of css machines : : : : : : : : : : : : : : : : : : : : : 26 3.3 Design of css machines : : : : : : : : : : : : : : : : : : : : : : 31 3.4 Labeled css machines : : : : : : : : : : : : : : : : : : : : : : : 38 4 Upper bounds 45 4.1 Collations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45 4.2 Previous upper bounds : : : : : : : : : : : : : : : : : : : : : : 51 4.3 Simple upper bound (binary alphabet) : : : : : : : : : : : : : 55 4.4 Simple upper bound (alphabet size 3) : : : : : : : : : : : : : : 59 4.5 Upper bounds for binary alphabet : :

Let f(n) be the expected length of a longest common subsequence of two random sequences over a fixed alphabet of size k. It is known that f(n) ! ck n for some constant ck . We define a collation as a pair of sequences with marked matches. A dominated collation is a collation that is not matched optimally. Upper bounds for ck can be derived from upper bounds for the number of nondominated collations. Using local properties of matches we can eliminate many nondominated collations and improve upper bounds for ck . 1 Introduction The problem of finding longest common subsequences arises in various situations. As typical we can mention approximate string matching and text comparisons (e.g. the diff function in UNIX) [1, 11]. Another important area where the longest common subsequence problem appears is molecular biology. The longest common subsequence problem is a special case of the more general sequence alignment problem. A survey on the longest common subsequence problem can be found in...