Order of Growth

The ORDER of a running-time function $\Theta$ is the fastest growing term, discarding constant factors.

Insertion Sort:

• Best Case: $an + b \longrightarrow \Theta(n)$
• Worst Case: $an^2 + bn + c \longrightarrow \Theta(n^2)$

XYZ-Sort:

• Worst Case: $\Theta(n^3)$
• Now we can say Insertion-Sort better than XYZ-Sort for large inputs.

Size n
Complexity	10	20	30	40	50	60
n	.00001 s	.00002 s	.00003 s	.00004 s	.00005 s	.00006 s
n2	.0001 s	.0004 s	.0009 s	.0016 s	.0025 s	.0036 s
n3	.001 s	.008 s	.027 s	.064 s	.125 s	.216 s
n5	.1 s	3.2 s	24.3 s	1.7 min	5.2 min	13.0 min
2n	.001 s	1.0 s	17.9 min	12.7 days	35.7 years	366 centuries
3n	.059 s	58 min	6.5 years	3855 centuries	2x108 centuries	1.3 x 1013 centuries