Question

Function Growth i Organize the following functions into six columns. Items in the same column should have the same asymptotic growth rates (big-O). If a column is to the left of another column, all its growth rates should be slower than those of the column to its right. na, n!, log, n, n log2 n, 3n, 5n2 + 3,2", 10000, n log3 n, 100n, 3log3n

Answer 1

na | log | n log2 n | n | 3n | 100n | n log3 n | 3log3n | 5n^2 + 3 | 2^n | 10000 |

The columns are organized based on the dominant term in the asymptotic growth rate (Big-O) of each function as n approaches infinity:

Column 1 (na): Constant functions like "na" have a growth rate of O(1), meaning they remain bounded by a constant value as n increases.

Column 2 (log, log2 n): Logarithmic functions like "log" and "log2 n" grow slowly, their Big-O being O(log n).

Column 3 (n log2 n): Functions like "n log2 n" have a slightly faster growth than pure logarithms, being O(n log n).

Column 4 (n): Linear functions like "n" have a Big-O of O(n), meaning they grow proportionally to n.

Column 5 (3n): Functions like "3n" have a slightly faster growth than pure linear functions, being O(n).

Column 6 (100n, n log3 n, 3log3n): Functions with factors like n multiplied by a constant or logarithm grow faster than linear functions, having a Big-O of O(n).

Column 7 (5n^2 + 3): Quadratic functions like "5n^2 + 3" grow faster than linear functions, with a Big-O of O(n^2).

Column 8 (2^n): Exponential functions like "2^n" have the fastest growth rate in this list, being O(2^n).

Column 9 (10000): Constant values like "10000" remain bounded by a constant value, similar to "na".