Asymptotic notations ada
- 1. Prepared by- Shaishav Shah(170120116094) Guided by – Prof. Bhumi Shah Gandhinagar Institute of Technology SUBJECT - ADA(2150703 ) Asymptotic Notations
- 3. Asymptotic Notations • Execution time of an algorithm depends on the instruction set, processor speed, disk I/O speed, etc. • Hence, we estimate the efficiency of an algorithm asymptotically. • Time function of an algorithm is represented by T(n), where n is the input size.
- 4. Types of asymptotic notations used to represent the complexity of an algorithm • Following asymptotic notations are used to calculate the running time complexity of an algorithm. – O − Big Oh – Ω − Big omega – θ − Big theta – o − Little Oh – ω − Little omega
- 5. O: Asymptotic Upper Bound • ‘O’ (Big Oh) is the most commonly used notation. • A function f(n) can be represented is the order of g(n) that is O(g(n)), if there exists a value of positive integer n as n0 and a positive constant c such that − f(n) ⩽ c.g(n) for n > n0 in all case • Hence, function g(n) is an upper bound for function f(n), as g(n) grows faster than f(n).
- 6. Example Let us consider a given function, f(n)=4.n3+10.n2+5.n+1f(n)=4.n3+10.n2+5.n+1 Considering g(n)=n3, f(n)⩽5.g(n) for all the values of n>2 Hence, the complexity of f(n) can be represented as O(g(n)), i.e. O(n3).
- 7. Ω: Asymptotic Lower Bound • We say that f(n)=Ω(g(n)) when there exists constant c that f(n)⩾c.g(n)for all sufficiently large value of n. • Here n is a positive integer. • It means function g is a lower bound for function f; after a certain value of n, f will never go below g.
- 8. Example Let us consider a given function, f(n)=4.n3+10.n2+5.n+1. Considering g(n)=n3g(n)=n3, f(n)⩾4.g(n) for all the values of n>0. Hence, the complexity of f(n) can be represented as Ω(g(n)), i.e. Ω(n3).
- 9. θ: Asymptotic Tight Bound • We say that f(n)=θ(g(n)) when there exist constants c1 and c2 that c1.g(n)⩽f(n)⩽c2.g(n) for all sufficiently large value of n. • Here n is a positive integer. • This means function g is a tight bound for function f.
- 10. Example Let us consider a given function, f(n)=4.n3+10.n2+5.n+1 Considering g(n)=n3, 4.g(n)⩽f(n)⩽5.g(n) for all the large values of n. Hence, the complexity of f(n) can be represented as θ(g(n)), i.e. θ(n3).
- 11. O - Notation • The asymptotic upper bound provided by O- notation may or may not be asymptotically tight. • The bound 2.n2=O(n2) is asymptotically tight, but the bound 2.n=O(n2) is not. • We use o-notation to denote an upper bound that is not asymptotically tight. • We formally define o(g(n)) (little-oh of g of n) as the set f(n) = o(g(n)) for any positive constant c>0 and there exists a value n0>0, such that 0⩽f(n)⩽c.g(n). • Intuitively, in the o-notation, the function f(n) becomes insignificant relative to g(n) as n approaches infinity; that is, limn→∞(f(n)/g(n))=0
- 12. Example • Let us consider the same function, f(n)=4.n3+10.n2+5.n+1 • Considering g(n)=n4, limn→∞(4.n3+10.n2+5.n+1/n4)=0 • Hence, the complexity of f(n) can be represented as o(g(n)), i.e. o(n4).
- 13. ω – Notation • We use ω-notation to denote a lower bound that is not asymptotically tight. • Formally, however, we define ω(g(n)) (little-omega of g of n) as the set f(n) = ω(g(n)) for any positive constant C > 0 and there exists a value n0>0, such that 0⩽c.g(n)<f(n). • For example, n2/2=ω(n), but n2/2≠ω(n2). The relation f(n)=ω(g(n))implies that the following limit exists limn→∞(f(n)/g(n))=∞ • That is, f(n) becomes arbitrarily large relative to g(n) as n approaches infinity.
- 14. Example Let us consider same function, f(n)=4.n3+10.n2+5.n+1 Considering g(n)=n2, limn→∞(4.n3+10.n2+5.n+1/n2)=∞ Hence, the complexity of f(n) can be represented as o(g(n)), i.e. ω(n2).
- 15. Thank You