1. Complexity
2. Searching
   1. Linear search
   2. Binary search

Complexity

Ignore differences which are constant - e.g., treat n and n/2 as same order of magnitude.

Definition: We say that g(n) is O(f(n)) if there exist two constants C and k such that |g(n)| <= C |f(n)| for all n > k.

Most common are

O(1) - for any constant

O(log n), O(n), O(n log n), O(n²), ..., O(2ⁿ)

Usually use these to measure time and space complexity of algorithms.

Make table of values to show difference.

Suppose have operations with time complexity O(log n), O(n), O(n log n), O(n²), and O(2ⁿ).

And suppose all work on problem of size n in time t. How much time to do problem 10, 100, or 1000 times larger?

size	10 n	100 n	1000 n
O(log n)	3+t	7+t	10+t
O(n)	10t	100t	1,000t
O(n log n)	>10t	>100t	>1,000t
O(n2)	100t	10,000t	1,000,000t
O(2n)	~t10	~t100	~t1000

TIME TO SOLVE PROBLEM

*Note that the last line depends on the fact that the constant is 1, otherwise the times are somewhat different.

Suppose get new machine that allows certain speed-up. How much larger problems can be solved? If original machine allowed solution of problem of size k in time t, then

speed-up	1x	10x	100x	1000x
O(log n)	k	k10	k100	k1000
O(n)	k	10k	100k	1,000k
O(n log n)	k	<10k	<100k	<1,000k
O(n2)	k	3k+	10k	30k+
O(2n)	k	k+3	k+7	k+10

SIZE OF PROBLEM

We will use big Oh notation to help us measure complexity of algorithms.

Searching

Code for all searches is on-line in Sort program example SortingDemo.java.

Linear search is O(n), while binary search i O(log n) comparisons.

Concrete comparison of worst cases: # of comparisons:

Search\# elts	10	100	1000	1,000,000
linear	10	100	1000	1,000,000
binary	8	14	20	40