3104 words - 13 pages

COMPUTATIONAL COMPLEXITY

OPERATIONS RESEARCH

Pooja Punjabi

Manash Hazarika

INDIAN INSTITUTE OF MANAGEMENT, KOZHIKODE

COMPUTATIONAL COMPLEXITY

Computational Complexity is a measure of the computational time taken by a particular algorithm. In a scenario where there are multiple algorithms available for a particular problem, the effectiveness of any particular algorithm is gauged on the basis of the time constraint. This is done by breaking the algorithm into its basic steps and then taking a count of each of them. Hence greater is the number of steps, greater is the complexity. Now for example, if we take two 5 bit binary numbers and XOR them, the number of steps taken is 5 and if the ...view middle of the document...

This factor becomes important because different machines may take different times `while processing the same ‘b’ basic steps. Another important point to consider is that for any algorithm with computational complexity defined as O(b), if we are to multiply the size of the problem by X, the computational time will have the same effect of being multiplied by X as well.

Big O notation is used to describe the asymptotic behavior of the functions in complexity theory. It tells you at what rate will the function grow. Rate at which a function grows is called order, hence letter O is used as its notation. Let f(x) and g(x) be two functions defined in real numbers. Then the big O notation would be represented as:

f(x) = O(g(x)) (for x belonging to real numbers) if and only if there exist certain constants N and C such that

|f(x)| <= C|g(x)| for all x>N

Basically it means that our function cannot grow faster than the function defining its time dependence complexity.

Commonly used functions as g(x) to describe time dependence are as follows written in increasing order of their growth rate (c is some constant)

O(1) constant

O(log(n)) logarithmic

O((log(n))c) polylogarithmic

O(n) linear

O(n2) quadratic

O(nc) polynomial

O(cn) exponential

As explained above, when it comes to addition, the complexity of adding two binary numbers is given as O(b). However if we are to multiply those numbers, the complexity changes to O(b2). We can take a simple example to illustrate this change. Let’s say we are to multiply two numbers 34 and 56. This will involve multiplication in 4 basic steps and then making carry over adjustments accordingly to get the final result. Hence the complexity for this case becomes O(22). Similarly, if we double the bit size of the numbers to be multiplied, the time required changes to O[(2b)2] = O(4b2).

So complexity theory basically classifies the problems on the basis of the difficulty faced in solving them.

This brings us to the concept of P and NP problems:

1. Deterministic Polynomial-Time (P) Type

A problem is basically classified as that belonging to the P (polynomial time) class if there is at least one algorithm which can solve that problem; the number of steps employed by the algorithm being restricted by a polynomial in ‘n’, where n gives the length of the input used.

An algorithm, existing or newly created is said to abide by polynomial time if, for a given input, the number of steps essential to complete the algorithm is given by O(nk) for some nonnegative integer k, where k gives the complexity of the input provided. Algorithms using polynomial-time are considered to be fast, feasible and efficient. A large number of day to day mathematical applications such as addition, subtraction, multiplication, and division in addition to square roots, logarithms and powers can be performed using polynomial time.

A specific term which comes into being in the deterministic approach is the time complexity of...

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