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:
Solution
The functions arranged in ascending order of growth rates (each function is big-O of the next function) are:
- n
- 1000 log n
- n log n
- n^2/1,000,000
- 2^n
- 3^n
- 2n!
Explanation:
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n: This is a linear function and has the lowest growth rate among all the given functions.
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1000 log n: Logarithmic functions grow slower than linear functions for large n.
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n log n: This is a linearithmic function and grows faster than both linear and logarithmic functions.
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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.
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2^n: This is an exponential function, which grows faster than polynomial functions like n^2.
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3^n: This is also an exponential function, but with a larger base than 2^n, so it grows faster.
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2n!: This is a factorial function, which grows faster than exponential functions.
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