Abstract. String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing or compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string A is a sequence P = (P1, P2, . . . , Pm) of strings, called the blocks, whose concatenation is equal to A. Given a partition P of a string A and a partition Q of a string B, we say that the pair P, Q is a common partition of A and B if Q is a permutation of P. The minimum common string partition problem (MCSP) is to find a common partition of two strings A and B with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most k times in each input string, is denoted by k-MCSP. In this paper, we show that 2-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a 1.1037-approximation for 2-MCSP and a linear time 4-approximation algorithm for 3-MCSP. We are not aware of any better approximations.
Fire departments have replaced traditional uniforms with modern, more thermal protective gear. Although the new uniforms afford superior burn protection, they may reduce work time. Our purpose was to determine if exercise time was (1) reduced by wearing the modern versus traditional uniform, and (2) increased by a design change to a modified modern uniform (T-shirt and short pants rather than a shirt and long pants under the outer uniform). Male firefighters (n = 23; age 27 to 59) performed a maximum exercise test in gym clothes (maximal oxygen consumption = 46 +/- 9 ml/kg/min) and then returned on separate days to exercise using a moderately high intensity, constant work rate treadmill protocol while wearing fire fighting breathing apparatus and each of three uniforms. Firefighters exceeded anaerobic threshold by 1 minute and eventually reached or exceeded maximum heart rate and maximal oxygen consumption. Exercise time in modern (15 +/- 3 min) was significantly less than in traditional (18 +/- 5 min) uniform. Exercise time in modified modern (17 +/- 5 min) was significantly greater than in modern and not significantly different than in traditional uniforms. The rate of change in oxygen consumption and water loss were significantly affected by uniform type, with faster rates in modern compared with modified modern or traditional uniforms. These findings show the impact that design changes have on energy demands and exercise duration.
String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing and compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string $A$ is a sequence ${\cal P} = (P_1,P_2,\dots,P_m)$ of strings, called the blocks, whose concatenation is equal to $A$. Given a partition ${\cal P}$ of a string $A$ and a partition ${\cal Q}$ of a string $B$, we say that the pair $\langle{{\cal P},{\cal Q}}\rangle$ is a common partition of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$. The minimum common string partition problem (MCSP) is to find a common partition of two strings $A$ and $B$ with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most $k$ times in each input string, is denoted by $k$-MCSP. In this paper, we show that $2$-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a $1.1037$-approximation for $2$-MCSP and a linear time $4$-approximation algorithm for $3$-MCSP. We are not aware of any better approximations.
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