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5. Arrange the functionsn , 1000 log n, n log n, 2n!, 2n, 3n, and n2/1,000,000 in a list sothat each function is big-O of the next function.

Question

Arrange the functions in a list such that each function is big-O of the next function:

  • 1000logn1000 \log n
  • n2/1,000,000n^{2}/1,000,000
  • nlognn \log n
  • 2n2n
  • 3n3n
  • 2n!2n!
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Solution

The functions arranged in ascending order of growth rates (each function is big-O of the next function) are:

  1. n
  2. 1000 log n
  3. n log n
  4. n^2/1,000,000
  5. 2^n
  6. 3^n
  7. 2n!

Explanation:

  1. n: This is a linear function and has the lowest growth rate among all the given functions.

  2. 1000 log n: Logarithmic functions grow slower than linear functions for large n.

  3. n log n: This is a linearithmic function and grows faster than both linear and logarithmic functions.

  4. n^2/1,000,000: This is a quadratic function. Despite the large denominator, for sufficiently large n, this function will eventually outgrow the previous ones.

  5. 2^n: This is an exponential function, which grows faster than polynomial functions like n^2.

  6. 3^n: This is also an exponential function, but with a larger base than 2^n, so it grows faster.

  7. 2n!: This is a factorial function, which grows faster than exponential functions.

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