Specification and complexity - algorithm
Download as PPT, PDF4 likes3,266 views
This document discusses algorithm specification and complexity. It defines the criteria an algorithm must satisfy and introduces asymptotic notations for analyzing time complexity, including Θ (average case), O (worst case upper bound), and Ω (best case lower bound). Examples are provided to illustrate analyzing algorithms using these notations. The relations between Θ, Ω, and O are explained, stating that Θ represents the exact bound. The document also covers algorithm space complexity, distinguishing between fixed space requirements and variable space requirements dependent on input characteristics.
Specification and complexity - algorithm
- 2. Algorithm Criteria Every Algorithm must have satisfied following Criteria: 01/04/16 2
- 7. Algorithm Time Complexity Asymptotic Notations Θ, O, Ω 01/04/16 7 Θ (Theta) (Average Case) Ο (Big-oh) (Worst Case)(Upper Bound) Ω (Omega) (Best case)(Lower Bound)
- 8. Algorithm Time Complexity (Cont.) 01/04/16 8 Example: f(n) = 3n2 + 17 Ω(1), Ω(n), Ω(n2 ) lower bounds O(n2 ), O(n3 ), ... upper bounds Θ(n2 ) exact bound
- 9. Algorithm Time Complexity (Cont.) 01/04/16 9 Examples (For Practices): 3n+2=O(n) /* 3n+2≤4n for n≥2 */ 3n+3=O(n) /* 3n+3≤4n for n≥3 */ 100n+6=O(n) /* 100n+6≤101n for n≥10 */ 10n2 +4n+2=O(n2 ) /* 10n2 +4n+2≤11n2 for n≥5 */ 6*2n +n2 =O(2n ) /* 6*2n +n2 ≤7*2n for n≥4 */
- 10. Algorithm Time Complexity (Cont.) 01/04/16 10 Relations Between Θ, Ω, O: Theorem : For any two functions g(n) and f(n), f(n) = Θ(g(n)) if f(n) = O(g(n)) and f(n) = Ω(g(n)). Theorem : For any two functions g(n) and f(n), f(n) = Θ(g(n)) if f(n) = O(g(n)) and f(n) = Ω(g(n)). I.e., Θ(g(n)) = O(g(n)) ∩ Ω(g(n)) In practice, asymptotically tight bounds are obtained from asymptotic upper and lower bounds.
- 11. Algorithm Space Complexity S(P)=C+SP(I) 01/04/16 11 Fixed Space Requirements (C) Independent of the characteristics of the inputs and outputs instruction space space for simple variables, fixed-size structured variable, constants Variable Space Requirements (SP(I)) depend on the instance characteristic I number, size, values of inputs and outputs associated with I recursive stack space, formal parameters, local variables, return address
- 12. Any Questions????? Thanks To Everybody 01/04/16 12