Greedy algorithms for the shortest common superstring that are asymtotically optimal

Abstract

There has recently been a resurgence of interest in the shortest common superstring problem due to its important applications in molecular biology (e.g., recombination of DNA) and data compression. The problem is NP-hard, but it has been known for some time that greedy algorithms work well for this problem. More precisely, it was proved in a recent sequence of papers that in the worst case a greedy algorithm produces a superstring that is at most β times (2≤β≤4) worse than optimal. We analyze the problem in a probabilistic framework,and consider the optimal total overlap O ^opt_n and the overlap O ^gr_n produced by various greedy algorithms. These turn out to be asymptotically equivalent. We show that in several cases, with high probability $\lim _{n \to \infty } \tfrac{{O_n^{opt} }}{{n\log n}} = \lim _{n \to \infty } \tfrac{{O_n^{gr} }}{{n\log n}} = \tfrac{1}{H}$where n is the number of original strings, and H is the entropy of the underlying alphabet. Our results hold under a condition that the lengths of all strings are not too short. Finally, we provide several generalizations and extensions of our basic result.

This work was supported by CCR-9225008.

This research was supported in part by NSF Grants CCR-9201078, NCR-9206315 and NCR-9415491, and in part by NATO Collaborative Grant CGR.950060.