[algorithm] Polynomial time and exponential time

Could someone explain the difference between polynomial-time, non-polynomial-time, and exponential-time algorithms?

For example, if an algorithm takes O(n^2) time, then which category is it in?

Polynomial time.

A polynomial is a sum of terms that look like Constant * x^k Exponential means something like Constant * k^x

(in both cases, k is a constant and x is a variable).

The execution time of exponential algorithms grows much faster than that of polynomial ones.

o(n sequre) is polynimal time complexity while o(2^n) is exponential time complexity if p=np when best case , in the worst case p=np not equal becasue when input size n grow so long or input sizer increase so longer its going to worst case and handling so complexity growth rate increase and depend on n size of input when input is small it is polynimal when input size large and large so p=np not equal it means growth rate depend on size of input "N". optimization, sat, clique, and independ set also met in exponential to polynimal.

Below are some common Big-O functions while analyzing algorithms.

• O(1) - constant time
• O(log(n)) - logarithmic time
• O((log(n))^c) - polylogarithmic time
• O(n) - linear time
• O(n²) - quadratic time
• O(n^c) - polynomial time
• O(cⁿ) - exponential time
• O(n!) - factorial time

(n = size of input, c = some constant)

Here is the model graph representing Big-O complexity of some functions

cheers :-)

graph credits http://bigocheatsheet.com/

O(n^2) is polynomial time. The polynomial is f(n) = n^2. On the other hand, O(2^n) is exponential time, where the exponential function implied is f(n) = 2^n. The difference is whether the function of n places n in the base of an exponentiation, or in the exponent itself.

Any exponential growth function will grow significantly faster (long term) than any polynomial function, so the distinction is relevant to the efficiency of an algorithm, especially for large values of n.

Exponential (You have an exponential function if MINIMAL ONE EXPONENT is dependent on a parameter):

• E.g. f(x) = constant ^ x

Polynomial (You have a polynomial function if NO EXPONENT is dependent on some function parameters):

• E.g. f(x) = x ^ constant

polynomial time O(n)^k means Number of operations are proportional to power k of the size of input

exponential time O(k)^n means Number of operations are proportional to the exponent of the size of input