Parallel and sequential approximation of shortest superstrings

Abstract

Abstract. Superstrings have many applications in data compression and genetics. However the decision version of the shortest superstring problem is N P-complete. In this paper we examine the complexity of approximating a shortest superstring. There are two basic measures of the approximations: the compression ratio and the approximation ratio. The well known and practical approximation algorithm is the sequential algorithm GREEDY. It approximates the shortest superstring with the compression ratio of 1 2 and with the approximation ratio of 4. Our main results are: 1 An N C algorithm which achieves the compression ratio of 1 4+" .2 The proof that the algorithm GREEDY is not parallelizable, the computation of its output is P-complete.3 An improved sequential algorithm: the approximation ratio is reduced to 2.83. Previously it was reduced by Teng and Yao from 3 to 2.89.4 The design of an RN C algorithm with constant approximation ratio and an N C algorithm with logarithmic approximation ratio.

Lyndon Words and Short Superstrings

“…A lot of effort went into designing approximation algorithms for the problem, Table 1 summarizes these developments. Note that the last two results, by Kaplan et al [10] and Paluch et al [16] Authors Date Factor Li [14] 1990 O(log(n)) Blum, Jiang, Li, Tromp, Yannakakis [4] 1991 3 Teng, Yao [21] 1993 2 8 9 Czumaj, Gasieniec, Piotrów, Rytter [7] 1994 2 5 6 Kosaraju, Park, Stein [12] 1994 2 50…”

Rotation of Periodic Strings and Short Superstrings

“…Their algorithms and analysis rely on the close relation between the shortest superstring problem, that was shown by Turner [22] to be reducible to the traveling salesman problem, and the cycle cover problem. The same relation was exploited in subsequent papers that continued to improve the approximation ratio, by Teng and Yao [21] (≈ 2.89), Czumaj et al [8] (≈ 2.83), Kosaraju et al [13] (≈ 2.79) and Armen and Stein [1,2] (≈ 2.75). Armen and Stein [3] have also recently obtained a 2 2 3 -approximation algorithm, independently of our work 2 .…”

“…Our algorithms are only slightly different from the ones in [1,2,3,4,8,13,21]. We first outline the general approach and then fill in the details of our new constructions and analysis.…”

“…Many researchers have tried to improve the performance of the generic algorithm or its variants by polishing Steps 5 -7. Teng and Yao [21], Czumaj et al [8], and Armen and Stein [1,2] treat the small cycles (i.e. cycles with two or three vertices) in CC with special care.…”

“…cycles with two or three vertices) in CC with special care. Teng and Yao [21] and Czumaj et al [8] do so by finding a short path across the small cycles and Armen and Stein [1,2] by identifying strings that are not much longer than their factors (called short periodic strings in their paper) as the bottleneck, and trying to avoid them in Step 3. Kosaraju et al [13] find a Hamiltonian path with large overlap instead of CC in Steps 5-7.…”