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

Algorithm to merge two sorted arrays with third array, Q. Write down an alg...

Q. Write down an algorithm to merge the two sorted arrays into the third array. Do  not perform the sort function in the third array.                           Ans: void m

Define the term counting - pseudocode, Define the term counting - Pseudocod...

Define the term counting - Pseudocode Counting in 1s is quite simple; use of statement count = count + 1 would enable counting to be done (for example in controlling a repeat

Determine the effect of ruby in implementation of string, Determine the eff...

Determine the effect of Ruby in implementation of string Ruby has a String class whose instances are mutable sequences of Unicode characters. Symbol class instances are charact

Explain the rgb model, RGB Model The RGB model is based on the assumpti...

RGB Model The RGB model is based on the assumption that any desired shade of colour can be obtained by mixing the correct amounts of red, green, and blue light. The exact hues

Write algorithm for post-order traversal, P os t - o r d e r T ...

P os t - o r d e r T r av er sal :  This can be done by both iteratively and recursively. The iterative solution would require a modification or alteration of the in-

Arrays, This unit discussed about data structure called Arrays. The easiest...

This unit discussed about data structure called Arrays. The easiest form of array is a one-dimensional array which may be described as a finite ordered set of homogeneous elements

Enumerate about the data structure, Enumerate about the Data structure ...

Enumerate about the Data structure An arrangement of data in memory locations to signify values of the carrier set of an abstract data type. Realizing computational mechanis

What is a spanning tree of a graph, What is a Spanning tree of a graph? ...

What is a Spanning tree of a graph? A Spanning Tree is any tree having of vertices of graph tree and some edges of graph is known as a spanning tree.

Algorithm, Write an algorithm for compound interest.

Write an algorithm for compound interest.

Applications of linear and binary search, The searching method are applicab...

The searching method are applicable to a number of places in current's world, may it be Internet, search engines, text pattern matching, on line enquiry, finding a record from data

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