Specification and complexity - algorithm

1. Algorithm Specification
and Complexity

2. Algorithm Criteria
Every Algorithm must have satisfied following
Criteria:
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3. Algorithm Specification
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4. Algorithm Specification (Cont.)
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5. Algorithm Specification (Cont.)
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6. Algorithm Specification (Cont.)
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7. Algorithm Time Complexity
Asymptotic Notations
Θ, O, Ω
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 Θ (Theta) (Average Case)
 Ο (Big-oh) (Worst Case)(Upper Bound)
 Ω (Omega) (Best case)(Lower Bound)

8. Algorithm Time Complexity
(Cont.)
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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.)
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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.)
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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)
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 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
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