Specification and complexity - algorithm
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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