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61 lines
2.2 KiB
Markdown
61 lines
2.2 KiB
Markdown
---
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number headings: auto, first-level 1, max 6, 1.1
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title: "set"
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aliases: sets, Set, Sets
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tags:
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- cosc201
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- datastructure
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---
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links: [java docs](https://docs.oracle.com/javase/7/docs/api/java/util/Set.html) for set interface
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> A collection of items with no repetition allowed
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How do we want to be able to use them?
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- We want to be able to add to them
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- And Remove from them
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- And check if it contains something
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This gives us the methods:
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- `Add(x)`
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- `Remove(x)`
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- `Contains(x)`
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Binary search trees are data types that supprts this data type when there is an ordering on the underlying elements
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A [binary search tree](notes/binary-search-tree.md) can be used to implement a set when there is no order. [hash sets and hash maps](notes/hash-map.md) can be used when there is order.
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# 1 Implementations
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## 1.1 Basic set
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Simplest way to implement a set is using an array.
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[Code for basic set](https://blackboard.otago.ac.nz/bbcswebdav/pid-2890167-dt-content-rid-18354837_1/courses/COSC201_S1DNIE_2022/BasicSet.java)
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- Contains: Simple linear search
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- Add: Check if it is present, if not add it to the end
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- Remove: Delete the element and replace it with the last element
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- This leaves empty elements at the end.
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- So we keep track of the size
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All three operations are $O(n)$ as they must all iterate over the entire set
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## 1.2 Ordered set
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A set with some underlying "natural" order. E.g., dictionary order for `string` objects.
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We would also like to be able to do an in order traversal of an ordered set
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If the set is static
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- store using sorted array
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- use binary search to find elements --> $O(lg\ n)$
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- traverse by incementing a counter --> $O(lg\ n)$ to init then $O(1)$
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But then:
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- `Add(x)` Insert `x` if its not already present, so we start a search which is fine, but then insertion is $O(n)$
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- `Remove(x)` find `x` if present, then move eyerthing beyond it back over the top --> $O(n)$
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This is fine if we dont expect to use the dynamic operations a lot.
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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
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A better way of doing this is using a [binary search tree](notes/binary-search-tree.md) |