Searching a set/list of items

#### Unsorted items

[draw big vertical array and linked list]

Worst case: unsorted list of unknown items.  Must linearly search each
item and test for equals().  What is order?  O(n).  Why?

#### Sorted

[draw big vertical sorted array; Antelope, Cow, Dog, Hippo, Zebra]

What if you know the list items are sorted (according to some
measure)?  What is the theoretically fastest sort you can do?
O(log_2(n)).  Why?  Because each time you cut the search space in
half.  You may cut n in half exactly log_2(n) times.

#### How can we reduce search time by a multiple factor?

[draw regions in big array]

Divide the whole list into "regions" or "buckets" and then search
linearly in the region.  If there are r regions, your search is how
fast?  O(n/r).  Bigger the r, the faster the search.

How do divide into regions?  See below.

#### What if the items are integers?

[draw a bit array with indices]

Can just turn on bit i if i is in list.  How fast is this?  O(1).
Why?  Can you go faster?

Not very space efficient.  Can use bitsets, but still very sparse.

#### What if you need quasi-O(1) search for non-integers?

Easy: turn the non-integers into integers first and then proceed as
before.  The conversion of non-integers into integers is called
computing a "hash code".  It essentially converts a single big list
into multiple smaller lists (regions or buckets).

[draw smaller array with arrays out the edge]

Fixed arrays for buckets are not very space efficient, so use a linked
list.

[draw smaller array with list emanating]

#### Map vs List

How is a Map different than a list?  List item is key that is mapped
to a value like map(key) = value as in math.

Wrapper: object that holds key/value pair.  Might incorporate the
linked list references into the object to implement hashtable buckets.