For each group of functions, sort the functions in increasing order of asymptotic (big-O) complexity: (a) [5 points] Group 1: \[ \begin{array}{l} f_{1}(n)=n^{0.999999} \log n \\ f_{2}(n)=10000000 n \\ f_{3}(n)=1.000001^{n} \\ f_{4}(n)=n^{2} \end{array} \] (b) [5 points] Group 2: \[ \begin{aligned} f_{1}(n) & =2^{2^{1000000}} \\ f_{2}(n) & =2^{100000 n} \\ f_{3}(n) & =\binom{n}{2} \\ f_{4}(n) & =n \sqrt{n} \end{aligned} \] (c) [5 points] Group 3: \[ \begin{array}{l} f_{1}(n)=n^{\sqrt{n}} \\ f_{2}(n)=2^{n} \\ f_{3}(n)=n^{10} \cdot 2^{n / 2} \\ f_{4}(n)=\sum_{i=1}^{n}(i+1) \end{array} \]

To sort the functions in each group by their asymptotic (big-O) complexity, we will analyze the growth rates of each function as \( n \) approaches infinity. ### Group 1: \[ \begin{array}{l} f_{1}(n)=n^{0.999999} \log n \\ f_{2}(n)=10000000 n \\ f_{3}(n)=1.000001^{n} \\ f_{4}(n)=n^{2} \end{array} \] 1. **\( f_{1}(n) = n^{0.999999} \log n \)**: This function grows slightly slower than linear due to the \( n^{0.999999} \) term, but the logarithmic factor increases it slightly. 2. **\( f_{2}(n) = 10000000 n \)**: This is a linear function, which grows faster than \( f_{1}(n) \) as \( n \) becomes large. 3. **\( f_{3}(n) = 1.000001^{n} \)**: This is an exponential function, which grows much faster than any polynomial function as \( n \) increases. 4. **\( f_{4}(n) = n^{2} \)**: This is a polynomial function of degree 2, which grows faster than linear but slower than exponential. **Sorted Order for Group 1**: 1. \( f_{1}(n) \) 2. \( f_{2}(n) \) 3. \( f_{4}(n) \) 4. \( f_{3}(n) \) ### Group 2: \[ \begin{aligned} f_{1}(n) & =2^{2^{1000000}} \\ f_{2}(n) & =2^{100000 n} \\ f_{3}(n) & =\binom{n}{2} \\ f_{4}(n) & =n \sqrt{n} \end{aligned} \] 1. **\( f_{1}(n) = 2^{2^{1000000}} \)**: This is a constant function since it does not depend on \( n \). 2. **\( f_{2}(n) = 2^{100000 n} \)**: This is an exponential function, which grows very fast as \( n \) increases. 3. **\( f_{3}(n) = \binom{n}{2} = \frac{n(n-1)}{2} \)**: This is a quadratic function, which grows slower than exponential functions. 4. **\( f_{4}(n) = n \sqrt{n} = n^{1.5} \)**: This is a polynomial function of degree 1.5, which grows faster than quadratic but slower than exponential. **Sorted Order for Group 2**: 1. \( f_{1}(n) \) 2. \( f_{3}(n) \) 3. \( f_{4}(n) \) 4. \( f_{2}(n) \) ### Group 3: \[ \begin{array}{l} f_{1}(n)=n^{\sqrt{n}} \\ f_{2}(n)=2^{n} \\ f_{3}(n)=n^{10} \cdot 2^{n / 2} \\ f_{4}(n)=\sum_{i=1}^{n}(i+1) = \frac{n(n+3)}{2} \end{array} \] 1. **\( f_{1}(n) = n^{\sqrt{n}} \)**: This function grows faster than any polynomial but slower than exponential functions. 2. **\( f_{2}(n) = 2^{n} \)**: This is an exponential function, which grows faster than \( n^{\sqrt{n}} \). 3. **\( f_{3}(n) = n^{10} \cdot 2^{n / 2} \)**: This is a combination of polynomial and exponential growth, and it grows faster than \( 2^{n} \). 4. **\( f_{4}(n) = \frac{n(n+3)}{2} \)**: This is a quadratic function, which grows slower than \( n^{10} \cdot 2^{n / 2} \). **Sorted Order for Group 3**: 1. \( f_{4}(n) \) 2. \( f_{1}(n) \) 3. \( f_{2}(n) \) 4. \( f_{3}(n) \) ### Summary of Sorted Orders: - **Group 1**: \( f_{1}(n), f_{2}(n), f_{4}(n), f_{3}(n) \) - **Group 2**: \( f_{1}(n), f_{3}(n), f_{4}(n), f_{2}(n) \) - **Group 3**: \( f_{4}(n), f_{1}(n), f_{2}(n), f_{3}(n) \)