The theta-notation, Data Structure & Algorithms

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).

Posted Date: 4/4/2013 5:42:48 AM | Location : United States







Related Discussions:- The theta-notation, Assignment Help, Ask Question on The theta-notation, Get Answer, Expert's Help, The theta-notation Discussions

Write discussion on The theta-notation
Your posts are moderated
Related Questions

It is a naturally occurring sorting method exemplified through a card player arranging the cards dealt to him. He picks up the cards like they are dealt & added them into the neede

Q. Describe the basic concept of binary search technique? Is it more efficient than the sequential search?         Ans : The bin ary search technique:- This tec

Given are the definitions of some important terms: 1) Field: This is an elementary data item characterized by its size, length and type. For instance, Name

Merging two sequence using CREW merge

Q. Prove the hypothesis that "A tree having 'm' nodes has exactly (m-1) branches".      Ans: A tree having m number of nodes has exactly (m-1) branches Proof: A root

Explain State Space Tree If it is convenient to execute backtracking by constructing a tree of choices being made, the tree is known as a state space tree. Its root indicates a

What is String Carrier set of the String ADT is the set of all finite sequences of characters from some alphabet, including empty sequence (the empty string). Operations on s

Demonstrate that Dijkstra's algorithm does not necessarily work if some of the costs are negative by finding a digraph with negative costs (but no negative cost dicircuits) for whi

The minimum cost spanning tree has broad applications in distinct fields. It represents several complicated real world problems such as: 1. Minimum distance for travelling all o