Big-O Notation

For functions f(n) and g(n), we say that "f(n) is Big-O of g(n)" if

There exists a constant c > 0 there exists a constant n0 such that
for all n ≥ n0, 0 ≤ f(n) ≤ c * g(n)

We write f(n) = O(g(n)), but the "=" is not the usual meaning.

The intention is to allow us to say

Technically, 101,000,000*n is O(n), but programs with that running time is still very slow.

Arrays vs Linked Lists

Asymptotic running times for n items:

  Insert Search Delete*
Arrays O(1) O(n) O(n) or O(1)
Sorted Arrays O(n) O(log n) O(n)
Linked Lists O(1) O(n) O(n) or O(1)
Sorted Linked Lists O(n) O(n) O(n) or O(1)

*Running time for an unsorted array is O(1) if you are given the index of the item to be deleted (replace with last item). Running time for deletion from a linked list is O(1) if you already have a pointer for the node to be deleted. Otherwise, deletion is O(n).

Arrays vs Linked Lists

Suppose we have n items inserted and do n searches:

If we have all n items before hand, we can sort them all at the same time in O(n log n), plus n * O(log n) for n searches for O(n log n) time total.

If not, we need binary search trees.

A Binary Search Tree

For each node in a binary search tree, its left child (if any) holds a smaller number and its right child (if any) holds a larger number.

Balanced Binary Search Trees

There are several schemes for maintaining a O(log n) height for a binary search tree n nodes: AVL trees, Red-Black trees, B-trees, 2-3 trees, ...

For such trees:

Hash Tables

Hash Tables

Using a "good" hash function:

In the worst case, all items have the same hash index and a closed hash table degenerates into an unsorted array.

There are provably good hash functions.