By Vijay V. Vazirani

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Covering the elemental options utilized in the most recent learn paintings, the writer consolidates development made to this point, together with a few very contemporary and promising effects, and conveys the sweetness and pleasure of labor within the box. He provides transparent, lucid factors of key effects and concepts, with intuitive proofs, and gives severe examples and various illustrations to assist elucidate the algorithms. a few of the effects offered were simplified and new insights supplied. Of curiosity to theoretical computing device scientists, operations researchers, and discrete mathematicians.

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**Sample text**

To see that the approximation factor of this algorithm is no better than 2, consider an input consisting of 3 strings: abk, bkc, and bk+l. If the first two strings are selected in the first iteration, the greedy algorithm produces the string abkcbk+l. This is almost twice as long as the shortest superstring, abk+ 1 c. We will obtain a 2Hn factor approximation algorithm, using the greedy set cover algorithm. The set cover instance, denoted by S, is constructed as follows. For si, Sj E S and k > 0, if the last k symbols of si are the same as the first k symbols of sj, let O'ijk be the string obtained by overlapping these k positions of si and Sj: k CTijk Let M be the set that consists of the strings O'ijk, for all valid choices of i, j, k.

If the first two strings are selected in the first iteration, the greedy algorithm produces the string abkcbk+l. This is almost twice as long as the shortest superstring, abk+ 1 c. We will obtain a 2Hn factor approximation algorithm, using the greedy set cover algorithm. The set cover instance, denoted by S, is constructed as follows. For si, Sj E S and k > 0, if the last k symbols of si are the same as the first k symbols of sj, let O'ijk be the string obtained by overlapping these k positions of si and Sj: k CTijk Let M be the set that consists of the strings O'ijk, for all valid choices of i, j, k.

Renumber the n strings in the order in which their leftmost occurrences start. Again, since no string is a substring of another, this is also the order in which they end. s II Se 2 S& 3 I I! I Sea : 11'1 ! ! 11'3 We will partition the ordered list of strings s 1, ... , sn in groups as described below. Each group will consist of a contiguous set of strings from this 22 2 Set Cover list. Let bi and ei denote the index of the first and last string in the ith group (bi = ei is allowed). Thus, b1 = 1.