1. Asymptotes: Why? How to describe an algorithm’s running time?
(or space, …)
How does the running time depend on the input?
T(x) = running time for instance x
Problem: Impractical to use, e.g.,
“15 steps to sort [ ], 13 steps to sort [ ], …”
Need to abstract away from the individual instances.

2. Asymptotes: Why? Standard solution: Abstract based on size of input.
How does the running time depend on the input?
T(n) = running time for instances of size n
Problem: Time also depends on other factors.
E.g., on sortedness of array.

3. Solution: Provide a bound over these instances.
Asymptotes: Why?
Solution: Provide a bound over these instances.
Most common. Default.
Worst case T(n) = max{T(x) | x is an instance of size n}
Best case T(n) = min{T(x) | x is an instance of size n}
Average case T(n) = |x|=n Pr{x}  T(x)
Determining the input probability distribution can be difficult.

4. Asymptotes: Why? What’s confusing about this notation?
Worst case T(n) = max{T(x) | x is an instance of size n}
Best case T(n) = min{T(x) | x is an instance of size n}
Average case T(n) = |x|=n Pr{x}  T(x)
Two different kinds of functions:
T(instance) T(size of instance)
Won’t use T(instance) notation again, so can ignore.

5. Asymptotes: Why? Problem: T(n) = 3n2 + 14n + 27
Too much detail: constants may reflect implementation details & lower terms are insignificant. n
3n2
14n+17 1 3 31 10
300
157
100
30,000
1,417
1000
3,000,000
14,017
10000
300,000,000
140,017
Solution: Ignore the constants & low-order terms. (Omitted details still important pragmatically.)
3n2 > 14n+17
 “large enough” n

6. Upper Bounds
Creating an algorithm proves we can solve the problem within a given bound.
But another algorithm might be faster.
E.g., sorting an array.
Insertion sort  O(n2)
What are example algorithms for
O(1), O(log n), O(n), O(n log n), O(n2), O(n3), O(2n)?

7. # comparisons needed in worst case  (n log n)
Lower Bounds
Sometimes can prove that we cannot compute something without a sufficient amount of time.
That doesn't necessarily mean we know how to compute it in this lower bound.
E.g., sorting an array.
# comparisons needed in worst case  (n log n)
Shown in COMP 482.

8. Definitions: O,  T(n)  O(g(n))   constants C,k > 0
such that
 nk, T(n)  Cg(n)
Cg(n) k
T(n)  (g’(n)) 
 constants C’,k’ > 0
such that
 nk’, T(n)  C’g’(n)
C’g’(n) k’
T(n)

9. Also, O(n2), O(5n), … Can always weaken the bound.
Examples: O, 
2n+13  O( ? )
O(n)
Also, O(n2), O(5n), … Can always weaken the bound.
2n+13  ( ? )
(n), also (log n), (1), …
2n  O(n) ? (n) ?
Given a C, 2n  Cn, for all but small n.
(n), not O(n).
nlog n  O(n5) ?
No. Given a C, log n  C5, for all large enough n. Thus, (n5).

10. Definitions:  T(n)  (g(n))  T(n)  O(g(n)) and T(n)  (g(n))
Ideally, find algorithms that are asymptotically as good as possible.

11. Notation O(), (), () are sets of functions.
But common to abuse notation, writing
T(n) = O(…)
instead of T(n)  O(…)
as well as T(n) = f(n) + O(…)