What is an easy way for finding C and N when proving the Big-Oh of an Algorithm?

Question

I\'m starting to learn about Big-Oh notation.

What is an easy way for finding C and N₀ for a given function?

Say, for example:

(n+1)⁵

Answer 1

You can pick a constant c by adding the coefficients of each term in your polynomial. Since

| n⁵ + 5n⁴ + 0n³ + 10n² + 5n¹ + 1n⁰ | <= | n⁵ + 5n⁵ + 0n⁵ + 10n⁵ + 5n⁵ + 1n⁵ |

and you can simplify both sides to get

| n⁵ + 5n⁴ + 10n² + 5n + 1 | <= | 22n⁵ |

So c = 22, and this will always hold true for any n >= 1.

It's almost always possible to find a lower c by raising N₀, but this method works, and you can do it in your head.

(The absolute value operations around the polynomials are to account for negative coefficients.)