3. What is an Algorithm?

4. Algorithm Specification

7. Asymptotic Notations
Q, O, W

8. Asymptotic Notation Q(Theta)(Average Case)
O(Big-oh)(Worst Case)(Upper Bound)
W(Best case)(Lower Bound)

9. -notation
For function g(n), we define (g(n)), big-Theta of n, as the set:
(g(n)) = {f(n) :  positive constants c1, c2, and n0, such that n  n0,
we have 0  c1g(n)  f(n)  c2g(n) }
f(n) and g(n) are nonnegative, for large n.

10. O-notation
For function g(n), we define O(g(n)), big-O of n, as the set:
O(g(n)) = {f(n) :  positive constants c and n0, such that n  n0,
we have 0  f(n)  cg(n) }
g(n) is an asymptotic upper bound for f(n).

11.  -notation
For function g(n), we define (g(n)), big-Omega of n, as the set:
(g(n)) = {f(n) :  positive constants c and n0, such that n  n0,
we have 0  cg(n)  f(n)}
g(n) is an asymptotic lower bound for f(n).

12. Example 3n2 + 17 (1), (n), (n2)  lower bounds
O(n2), O(n3), ...  upper bounds
(n2)  exact bound

13. Example for Practice Examples 3n+2=O(n) /* 3n+24n for n2 */
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 */

14. Relations Between Q, O, W

15. Relations Between Q, W, O I.e., (g(n)) = O(g(n)) Ç W(g(n))
Theorem : For any two functions g(n) and f(n), f(n) = (g(n)) iff
f(n) = O(g(n)) and f(n) = (g(n)).
I.e., (g(n)) = O(g(n)) Ç W(g(n))
In practice, asymptotically tight bounds are obtained from asymptotic upper and lower bounds.

16. o-notation For a given function g(n), the set little-o:
o(g(n)) = {f(n):  c > 0,  n0 > 0 such that  n  n0, we have 0  f(n) < cg(n)}.
f(n) becomes insignificant relative to g(n) as n approaches infinity:
lim [f(n) / g(n)] = 0
n
g(n) is an upper bound for f(n) that is not asymptotically tight.

17. ω-notation For a given function g(n), the set little-: ω
ω(g(n)) = {f(n):  c > 0,  n0 > 0 such that  n  n0, we have 0  cg(n) <f(n)}.
f(n) becomes insignificant relative to g(n) as n approaches infinity:
lim [f(n) / g(n)] = 0
n
g(n) is an lower bound for f(n) that is not asymptotically tight.

18. Algorithm
Definition An algorithm is a finite set of instructions that accomplishes a particular task.
Criteria
input
output
definiteness: clear and unambiguous
finiteness: terminate after a finite number of steps
effectiveness: instruction is basic enough to be carried out

19. Measurements Criteria Performance Analysis (machine independent)
Is it correct?
Is it readable? …
Performance Analysis (machine independent)
space complexity: storage requirement
time complexity: computing time
Performance Measurement (machine dependent)

20. Space Complexity S(P)=C+SP(I)
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

21. O(1): constant
O(n): linear
O(n2): quadratic
O(n3): cubic
O(2n): exponential
O(logn)
O(nlogn)

22. Function values

23. Plot of function values
nlogn n
logn

24. Times on a 1 billion instruction per second computer