1. Analysis of Algorithms II
22-Feb-19

2. Basics
Before we attempt to analyze an algorithm, we need to define two things:
How we measure the size of the input
How we measure the time (or space) requirements
Once we have done this, we find an equation that describes the time (or space) requirements in terms of the size of the input
We simplify the equation by discarding constants and discarding all but the fastest-growing term

3. Size of the input
Usually it’s quite easy to define the size of the input
If we are sorting an array, it’s the size of the array
If we are computing n!, the number n is the “size” of the problem
Sometimes more than one number is required
If we are trying to pack objects into boxes, the results might depend on both the number of objects and the number of boxes
Sometimes it’s very hard to define “size of the input”
Consider: f(n) = if n is 1, then 1; else if n is even, then f(n/2); else f(3*n + 1)
The obvious measure of size, n, is not actually a very good measure
To see this, compute f(7) and f(8)

4. Measuring requirements
If we want to know how much time or space an algorithm takes, we can do empirical tests—run the algorithm over different sizes of input, and measure the results
This is not analysis
However, empirical testing is useful as a check on analysis
Analysis means figuring out the time or space requirements
Measuring space is usually straightforward
Look at the sizes of the data structures
Measuring time is usually done by counting characteristic operations
Characteristic operation is a difficult term to define
In any algorithm, there is some code that is executed the most times
This is in an innermost loop, or a deepest recursion
This code requires “constant time” (time bounded by a constant)
Example: Counting the comparisons needed in an array search

5. Big-O and friends
Informal definitions:
Given a complexity function f(n),
(f(n)) is the set of complexity functions that are lower bounds on f(n)
O(f(n)) is the set of complexity functions that are upper bounds on f(n)
(f(n)) is the set of complexity functions that, given the correct constants, correctly describes f(n)
Example: If f(n) = 17x3 + 4x – 12, then
(f(n)) contains 1, x, x2, log x, x log x, etc.
O(f(n)) contains x4, x5, 2x, etc.
(f(n)) contains x3

6. Formal definition of Big-O*
A function f(n) is O(g(n)) if there exist positive constants c and N such that, for all n > N, < f(n) < cg(n)
That is, if n is big enough (larger than N—we don’t care about small problems), then cg(n) will be bigger than f(n)
Example: 5x is O(n3) because < 5n2 + 6 < 2n3 whenever n > 3 (c = 2, N = 3)
We could just as well use c = 1, N = 6, or c = 50, N = 50
Of course, 5x2 + 6 is also O(n4), O(2n), and even O(n2)

7. Formal definition of Big-*
A function f(n) is (g(n)) if there exist positive constants c and N such that, for all n > N, < cg(n) < f(n)
That is, if n is big enough (larger than N—we don’t care about small problems), then cg(n) will be smaller than f(n)
Example: 5x is (n) because < 20n < 5n2 + 6 whenever n > 4 (c=20, N=4)
We could just as well use c = 50, N = 50
Of course, 5x2 + 6 is also O(log n), O(n), and even O(n2)

8. Formal definition of Big-*
A function f(n) is (g(n)) if there exist positive constants c1 and c2 and N such that, for all n > N, < c1g(n) < f(n) < c2g(n)
That is, if n is big enough (larger than N), then c1g(n) will be smaller than f(n) and c2g(n) will be larger than f(n)
In a sense,  is the “best” complexity of f(n)
Example: 5x is (n2) because n2 < 5n2 + 6 < 6n2 whenever n > (c1 = 1, c2 = 6)

9. Graphs Points to notice: f(n) cg(n) f(n) is O(g(n)) N f(n) cg(n)
f(n) is (g(n)) N
f(n)
c2g(n)
c1g(n)
f(n) is (g(n)) N
Points to notice:
What happens near the beginning (n < N) is not important
cg(n) always passes through 0, but f(n) might not (why?)
In the third diagram, c1g(n) and c2g(n) have the same “shape” (why?)

10. Informal review For any function f(n), and large enough values of n,
f(n) = O(g(n)) if cg(n) is greater than f(n),
f(n) = theta(g(n)) if c1g(n) is greater than f(n) and c2g(n) is less than f(n),
f(n) = omega(g(n)) if cg(n) is less than f(n),
...for suitably chosen values of c, c1, and c2

11. The End
The formal definitions were taken, with some slight modifications, from Introduction to Algorithms, by Thomas H. Cormen, Charles E. Leiserson, Donald L. Rivest, and Clifford Stein