1. An Upper Bound
g(n) is an upper bound on f(n).
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EECE 352

2. A Lower Bound
g(n) is a lower bound for f(n)
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EECE 352

3. A Tight Bound
We say that g(n) is an asymptotically tight bound for f(n).
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EECE 352

4. Transitivity The operators are transitive. They are also Reflexive.
and
imply
and
imply
imply
and
The operators are transitive.
They are also Reflexive.
C++ Review
EECE 352

5. Exercises Show that for any constants a and b >0. Is 2n+1 = O(2n) ?
Rewrite in Big-O notation:
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EECE 352

6. Rules to Remember If T1(n) = O(f(n)) and T2(n)=O(g(n)) then
T1(n) + T2(n) = max(O(f(n)), O(g(n)))
T1(n)*T2(n) = O(f(n)*g(n))
If T(n) is a polynomial of degree n, then T(x) = (xn).
logkn = O(n) for any constant k.
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EECE 352

7. Sample Functions
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EECE 352

8. Analyzing Algorithms Model of idealized computer:
The usual set of operators, all take constant time.
No fancy operators.
Infinite memory.
Find the amount of time it takes to finish as a function of the input size.
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EECE 352

9. Loops
A loop of size n that does something that takes time k will take nk total time.
This applies recursively. If the something is also a loop of size n it will take n2k.
And so on...
Recursion
Could be just a for loop, could be worse.
C++ Review
EECE 352