Skip to main content

Table 1 For all \(0\le r\le 7\) such that \(m=8l+r\) and \(l \ge 0\), the approximation ratio given by Algorithm 1 is at most \(\frac{11}{8}=1.375\)

From: A new 1.375-approximation algorithm for sorting by transpositions

r

0

1

2

3

4

5

6

7

\(\frac{f(m)+2}{m+2}\)

\(\frac{11l+2}{8l+2}\)

\(\frac{11l+4}{8l+3}\)

\(\frac{11l+5}{8l+4}\)

\(\frac{11l+6}{8l+5}\)

\(\frac{11l+8}{8l+6}\)

\(\frac{11l+9}{8l+7}\)

\(\frac{11l+11}{8l+8}\)

\(\frac{11l+12}{8l+9}\)

\(\frac{f(m)}{m+1}\)

\(\frac{11l}{8l+1}\)

\(\frac{11l+2}{8l+2}\)

\(\frac{11l+3}{8l+3}\)

\(\frac{11l+4}{8l+4}\)

\(\frac{11l+6}{8l+5}\)

\(\frac{11l+7}{8l+6}\)

\(\frac{11l+9}{8l+7}\)

\(\frac{11l+10}{8l+8}\)