GREEDY ALGORITHMS FOR THE SHORTEST COMMON SUPERSTRING THAT ARE ASYMPTOTICALLY OPTIMAL

There has recently been a resurgence of interest in the shortest commo n superstring problem due to its important applications in molecular b iology (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 beta times (2 less than or equal to bet a less than or equal to 4) worse than optimal. We analyze the problem in a probabilistic framework, and consider the optimal total overlap O -n(opt) and the overlap O-n(gr) produced by various greedy algorithms. These turn out to be asymptotically equivalent. We show that with hig h probability lim(n-->infinity) O-n(opt)/(n log n) = lim(n-->infinity) O-n(gr)/(n log n) = 1/H, where n is the number of original strings an d H is the entropy of the underlying alphabet. Our results hold under a condition that the lengths of all strings are not too short.