quartz/content/notes/set.md

number headings	title	aliases	tags
auto, first-level 1, max 6, 1.1	set	sets, Set, Sets	cosc201 datastructure

links: java docs for set interface

A collection of items with no repetition allowed

How do we want to be able to use them?

• We want to be able to add to them
• And Remove from them
• And check if it contains something

This gives us the methods:

• Add(x)
• Remove(x)
• Contains(x)

Binary search trees are data types that supprts this data type when there is an ordering on the underlying elements A binary search tree can be used to implement a set when there is no order. hash sets and hash maps can be used when there is order.

1 Implementations

1.1 Basic set

Simplest way to implement a set is using an array.

Code for basic set

• Contains: Simple linear search
• Add: Check if it is present, if not add it to the end
• Remove: Delete the element and replace it with the last element
  • This leaves empty elements at the end.
  • So we keep track of the size

All three operations are O(n) as they must all iterate over the entire set

1.2 Ordered set

A set with some underlying "natural" order. E.g., dictionary order for string objects.

We would also like to be able to do an in order traversal of an ordered set

If the set is static

• store using sorted array
• use binary search to find elements --> O(lg\ n)
• traverse by incementing a counter --> O(lg\ n) to init then O(1)

But then:

• Add(x) Insert x if its not already present, so we start a search which is fine, but then insertion is O(n)
• Remove(x) find x if present, then move eyerthing beyond it back over the top --> O(n)

This is fine if we dont expect to use the dynamic operations a lot.

Another approach might be to maintain a main array and a subsidary add and remove arrays and only periodically do the updates to the main array. but this gets complicated very quickly

A better way of doing this is using a binary search tree