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[\<- 05/15](05-15.md)
---
# An Application of Binary Trees: Huffman Coding
## Encoding
- Examples
- Text
- Video
- Image
- Audio
- Computer only understands low layer bits (0s and 1s)
- Encoding is turning those examples into the low layer bits
### An Application - Encoding
- An example - using only 1 and 0s to represent following cases
- true or false - only need 1 bit
- 1 is true
- 0 is false
- 4 directions: south, north, east, and west - need 2 bits
- 00 is south
- 01 is north
- 10 is east
- 11 is west
- ASCII is a fixed width encoding. Every character requires 8 bits to encode
- Motiviation: Can we save some bits in encoding?
- Thoughts:
- some characters (like e and n) occur frequently. whereas characters (like q and z) occur very infrequently
- Do you still remember probability search (move to front heuristic)?
### One Example
- Ex. "the fat cat sat on the mat"
- If we use fixed-length binary code to represent the above sentence, what will that be? 26\*8
- Can we save some bits by using fewer bits for more frequent letters?
- Frequency of letters:
- a:4, c:1, e:2, f:1, h:2, m:1, n:1, o:1, s:1, t:6, " ":6
- Bits of letters
- t & " ", frequently occur - 1 bit
- a, h, e, less frequently - 2bits
- etc.
## Huffman Coding
- Huffman Coding: **variable width encoding**
- We are likely to **save on the total number of bits** required to encode a file/document if we represented frequently occurring character with fewer bits and infrequently occurring characters with more than 8 bits
- Question: How do we know, given a variable length encoding, where characters stop and start?
- ex: if e=11, z=1111... so what's 111111?
- Answer: We use **unique prefixes**. Each character will have its own unique prefix -> how to implement?
- We build a binary tree. By having only leaf nodes to represent characters, we are able to guarantee unique prefixes
## Huffman Tree
- Use binary trees (i.e. Huffman tree). e.g. 'ABCA'
- B: 00
- C: 01
- A: 1
- Question: How do we determine the optimal encoding given the frequencies of the characters?
- Answer: Use the leaf nodes that are farther away from the root to represent characters appear less frequently
- so now you have the "Huffman tree"
- label the left and right sub-branches of every node as 0 and 1 respectively
## Huffman Coding
- Ex. "the fat cat sat on the mat"
- a:4, c:1, e:2, f:1, h:2, m:1, n:1, o:1, s:1, t:6, " ":6
- create a node for each letter and sort according to their values
- remove two minimum nodes, combine with a new node, and insert this node with a new weight - the combined value of its two child nodes
- Is it better than the fixed length encoding?
- This is our Huffman Tree
- Our letters translate to:
- a -> 011
- c -> 00000
- e -> 0100
- f -> 00001
- h -> 0101
- m -> 00010
- n -> 00011
- o -> 0010
- s -> 0011
- t -> 10
- " " -> 11
- This encoding only requires 80 bits (which significantly saves the number of bits needed for encoding)
---
### Code Structure
- A struct node includes:
- `struct node *parent`
- `int count`
1. Open file
- count frequency of each character
- need a while loop, need to use `getc()`
- put characters in an array, assigning them an index reflecting their ascii code
- the last slot on the array will be the EOF
- the length of the array will be 257 (2^8 + 1)
2. `NODE *mknode(int data, NODE *left, NODE *right);`
- We can use this function to generate the leaf nodes as well
- also need node comparison
```
NODE *mknode(int data, NODE *left, NODE *right){
NODE *np = malloc(...);
assert(...);
np->parent = NULL;
np->count = data;
//connect
}
```
3. `pq = createqueue()`
- call addEntry: `addEntry(pq, nodes[i]);`
- `l = removeEntry(pq);`
- `r = removeEntry(pq);`
- `new = mknode(l->count + r->conut, l, r)`
- `addEntry(pq, np);`
4. print
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