Avrim Blum
Ming Li
Abstract
We consider the following problem: given a collection of strings s1,…, sm, find the shortest string s such that each si appears as a substring (a consecutive block) of s. Although this problem is known to be NP-hard, a simple greedy procedure appears to do quite well and is routinely used in DNA sequencing and data compression practice, namely: repeatedly merge the pair of (distinct) strings with maximum overlap until only one string remains. Let n denote the length of the optimal superstring. A common conjecture states that the above greedy procedure produces a superstring of length O(n) (in fact, 2n), yet the only previous nontrivial bound known for any polynomial-time algorithm is a recent O(n log n) result.
We show that the greedy algorithm does in fact achieve a constant factor approximation, proving an upper bound of 4n. Furthermore, we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3n. We also show the superstring problem to be MAXSNP-hard, which implies that a polynomial-time approximation scheme for this problem is unlikely.
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Index Terms
- Linear approximation of shortest superstrings
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Reviews
Ralph Walter Wilkerson
A greedy algorithm to solve the following problem is studied: for a finite set of strings, find the shortest common superstring S such that every string in the set is a substring of S . This problem occurs in data compression and most recently has received considerable attention due to its relevance in the DNA sequencing problem. While the problem is known to be NP-hard, the paper shows that the greedy algorithm produces a superstring of length O n , where n is the length of the optimal superstring. In fact, an upper bound of 4n is achieved. A modified version of this algorithm can achieve 3n as an upper bound. The paper includes all basic notation and detailed proofs concerning the correctness of the algorithms.
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