Asymptotic notation, Data Structure & Algorithms

Assignment Help:

Asymptotic notation

Let us describe a few functions in terms of above asymptotic notation.

Example: f(n) = 3n3 + 2n2 + 4n + 3

= 3n3 + 2n2 + O (n), as 4n + 3 is of O (n)

= 3n3+ O (n2), as 2n2 + O (n)   is O (n2)

= O (n3)

Example: f(n) = n² + 3n + 4 is O(n²), since n² + 3n + 4 < 2n² for all n > 10.

Through definition of big-O, 3n + 4 is also O(n²), too, although as a convention, we employ the tighter bound to the function, i.e., O(n).

Here are some rules regarding big-O notation:

1. f(n) = O(f(n)) for any function f. In other terms, every function is bounded by itself.

2. aknk + ak-1 n k-1 + • • • + a1n + a0 = O(nk) for every k ≥ 0 & for all a0, a1, . . . , ak ∈ R.

In other terms, every polynomial of degree k may be bounded through the function nk. Smaller order terms can be avoided in big-O notation.

3. Basis of Logarithm can be avoided in big-O notation that means logan = O(logb n) for any bases a, b. Generally we write O(log n) to specify a logarithm n to any base.

4. Any logarithmic function may be bounded through a polynomial that means. logb n = O(nc) for any b (base of logarithm) & any positive exponent c > 0.

5. Any polynomial function may be limited by an exponential function i.e. nk = O (bn.).

6.Any exponential function may be limited by the factorial function. For instance, an = O(n!) for any base a.


Related Discussions:- Asymptotic notation

Algorithm for insertion into the circular queue, Algorithm for insertion of...

Algorithm for insertion of any element into the circular queue: Step-1: If "rear" of the queue is pointing at the last position then go to step-2 or else Step-3 Step-2: make

Er diagram, Ask queConsider the following functional dependencies: Applican...

Ask queConsider the following functional dependencies: Applicant_ID -> Applicant_Name Applicant_ID -> Applicant_Address Position_ID -> Positoin_Title Position_ID -> Date_Position_O

Algorithm for the selection sort, Q. Give the algorithm for the selection s...

Q. Give the algorithm for the selection sort. Describe the behaviours of selection sort when the input given is already sorted.

Stack and array, how to implement multiple stack using single dimension arr...

how to implement multiple stack using single dimension array in c

Explain worst fit method, Worst Fit method:- In this method the system alw...

Worst Fit method:- In this method the system always allocate a portion of the largest free block in memory. The philosophy behind this method is that by using small number of a ve

The quick sort algorithm exploit design technique, The quick sort algorithm...

The quick sort algorithm exploit design technique Divide and Conquer

Functions for inserting and deleting at either of the end, Q. Develop a rep...

Q. Develop a representation for a list where insertions and deletions can be done at either end. Such a structure is known as a Deque (Double ended queue). Write functions for inse

Explain np-complete decision problem, a. Determine the result of inserting ...

a. Determine the result of inserting the keys 4,19, 17, 11, 3, 12, 8, 20, 22, 23, 13, 18, 14, 16, 1, 2, 24, 25, 26, 5 in order to an empty B-Tree of degree 3. Only draw the configu

B-tree, Unlike a binary-tree, each node of a B-tree may have a number of ke...

Unlike a binary-tree, each node of a B-tree may have a number of keys and children. The keys are stored or saved in non-decreasing order. Each key has an related child that is the

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd