The theta-notation, Data Structure & Algorithms

Assignment Help:

This notation bounds a function to in constant factors. We say f(n) = Θ(g(n)) if there presents positive constants n0, c1 and c2 such that to the right of n0 the value of f(n) always lies between c1g(n) and c2g(n), both inclusive. The given Figure gives an idea about function f(n) and g(n) where f(n) = Θ(g(n)) . We will say that the function g(n) is asymptotically tight bound for f(n).

1562_The Theta-Notation.png

Figure: Plot of  f(n) = Θ(g(n))

 

For instance, let us show that the function f(n) =  (1/3) n2   - 4n = Θ(n2).

Now, we must determined three positive constants, c 1, c 2 and no as

c1n2 ≤1/3 n2 - 4n c2 n2 for all n no

c1 ≤ 1/3- 4/n  ≤ c2

By choosing no = 1 and c2 ≥ 1/3 the right hand inequality holds true.

 

Likewise, by choosing no = 13, c1  ≤  1/39, the right hand inequality holds true. Thus, for c1 = 1/39 , c2 = 1/3 and no  ≥ 13, it follows that 1/3 n2 - 4n = Θ (n2).

Surely, there are other alternative for c1, c2 and no. Now we might illustrates that the function f(n) = 6n3    ≠  Θ (n2).

In order to prove this, let us suppose that c3 and no exist such that 6n3   ≤ c3n2  for n ≥ no, But this fails for adequately large n. Therefore 6n3    ≠  Θ (n2).


Related Discussions:- The theta-notation

Create a function to show data structure, Given a number that is represente...

Given a number that is represented in your data structure, you will need a function that prints it out in base 215 in such a way that its contents can be checked for correctness. Y

Data structures, 1.  You are required to hand in both a hard copy and an el...

1.  You are required to hand in both a hard copy and an electronic copy of the written report on the project described in A, including all the diagrams you have drawn.  2.  You

Algorithm, Describe different methods of developing algorithms with example...

Describe different methods of developing algorithms with examples.

Implementing abstract data types, Implementing abstract data types A co...

Implementing abstract data types A course in data structures and algorithms is hence a course in implementing abstract data types. It may seem that we are paying a lot of atten

Binary search tree, write an algorithm to delete an element x from binary...

write an algorithm to delete an element x from binary search with time complex

Illustrate the wire frame representation, RENDERING, SHADING AND COLOURING ...

RENDERING, SHADING AND COLOURING By introducing hidden line removal we have already taken one step away from wire-frame drawings towards being able to realistically model and d

What is an algorithm, What is an algorithm?  What are the characteristics o...

What is an algorithm?  What are the characteristics of a good algorithm? An algorithm is "a step-by-step process for accomplishing some task'' An algorithm can be given in many

Sorting, compare and contrast the bubble sort,quick sort,merge sort and rad...

compare and contrast the bubble sort,quick sort,merge sort and radix sort

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