Red $($horizontal line$)$: Constant $1$

Justification:

The line is flat, indicating no growth as n increases.

It's a constant time complexity, O$(1),$ meaning the operation takes the same time regardless of input size.

Blue $($gently rising curve$)$: Logarithmic log n

Justification:

This curve grows very slowly as n increases, typical of logarithmic functions.

Logarithmic complexity, O$($log n$),$ often appears in algorithms that halve the problem size, like binary search.

Green $($diagonal line$)$: Linear n

Justification:

The line grows steadily and is directly proportional to n$.$

This linear growth, O$($n$),$ means the time taken increases directly with the input size, common in simple loops.

Purple $($sharper parabola$)$: Quadratic n$^2$

Justification:

The curve forms a parabola and grows faster than linear.

Quadratic complexity, O$($n$^2),$ often results from nested loops, where the operation count is proportional to the square of the input size.

Orange $($even steeper curve$)$: Cubic n$^3$

Justification:

This curve grows faster than quadratic, indicative of cubic growth.

Cubic complexity, O$($n$^3),$ can occur in algorithms with three nested loops.

Black $($exponentially steep curve$)$: Exponential $2^$n

Justification:

The curve increases rapidly, much faster than polynomial functions.

Exponential growth, O$(2^$n$),$ appears in algorithms that double the work with each additional input, such as certain recursive solutions.

Light Violet $($steepest growth$)$: Factorial n$!$

Justification:

This is the steepest curve, representing factorial growth.

Factorial complexity, O$($n$!),$ occurs in problems where all permutations are considered, like the traveling salesman problem.

If you were to develop an algorithm, which algorithm complexity from above you will target that your algorithm should have and why?