
Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
contains information and examples of Big omega notation
Typology: Exercises
1 / 1
This page cannot be seen from the preview
Don't miss anything!
Example 1: Prove that running time T( n ) = n^3 + 20 n + 1 is O( n^3 ) Proof: by the Big-Oh definition, T( n ) is O( n^3 ) if T( n ) ≤ c · n^3 for some n ≥ n 0. Let us check this condition: if n^3 + 20 n + 1 ≤ c · n^3 then 1 + (^) n^20 2 + n^13 ≤ c. Therefore, the Big-Oh condition holds for n ≥ n 0 = 1 and c ≥ 22 (= 1 + 20 + 1). Larger values of n 0 result in smaller factors c (e.g., for n 0 = 10 c ≥ 1.201 and so on) but in any case the above statement is valid.
Example 2: Prove that running time T( n ) = n^3 + 20 n + 1 is not O( n^2 ) Proof: by the Big-Oh definition, T( n ) is O( n^2 ) if T( n ) ≤ c · n^2 for some n ≥ n 0. Let us check this condition: if n^3 + 20 n + 1 ≤ c · n^2 then n + (^20) n^ + n^12 ≤ c. Therefore, the Big-Oh condition cannot hold (the left side of the latter inequality is growing infinitely, so that there is no such constant factor c ).
Example 3: Prove that running time T( n ) = n^3 + 20 n + 1 is O( n^4 ) Proof: by the Big-Oh definition, T( n ) is O( n^4 ) if T( n ) ≤ c · n^4 for some n ≥ n 0. Let us check this condition: if n^3 + 20 n + 1 ≤ c · n^4 then (^) n^1^ + (^) n^203 + n^14 ≤ c. Therefore, the Big-Oh condition holds for n ≥ n 0 = 1 and c ≥ 22 (= 1 + 20 + 1). Larger values of n 0 result in smaller factors c (e.g., for n 0 = 10 c ≥ 0.10201 and so on) but in any case the above statement is valid.
Example 4: Prove that running time T( n ) = n^3 + 20 n is Ω( n^2 ) Proof: by the Big-Omega definition, T( n ) is Ω( n^2 ) if T( n ) ≥ c · n^2 for some n ≥ n 0. Let us check this condition: if n^3 + 20 n ≥ c · n^2 then n + (^20) n^ ≥ c. The left side of this inequality has the minimum value of 8.94 for n = 20 ≅ 4. 47 Therefore, the Big-Omega condition holds for n ≥ n 0 = 5 and c ≤ 9. Larger values of n 0 result in larger factors c (e.g., for n 0 = 10 c ≤ 12.01) but in any case the above statement is valid.