CS 46B - Lecture 23

Pre-class reading

• Worked Example 17.1

Huffman coding

Huffman coding

• Leaves contain symbols that we want to encode
• To encode a symbol, walk from root to symbol
• Left turn = 0, right turn = 1
• Example: H = 0001, A = 10
• Interior nodes are empty
• No code is a prefix of another
• Goal: Most frequent codes have shortest lengths

Lecture 23 Clicker Question 1

What is the Encoding of ALOHA?

1. 100000011000110
2. 100000001000110
3. 10000000100010
4. Something else

Lecture 23 Clicker Question 2

Which of these is NOT a Huffman code?

1. 7-bit ASCII
2. UTF-8
3. Morse code
4. None of them are

Building a Good Huffman Tree

• Example: child painting
• Most of the pixels are white (50%), there are lots of orange (20%) and pink (20%) pixels, and small amounts of yellow (5%), blue (3%), and green (2%)

Building a Good Huffman Tree

• Want short code for white, long code for green
• Huffman algorithm builds optimal tree
• Invented when he was a graduate student

Huffman Algorithm

Add all nodes to a priority queue
While there are two nodes left
	 Remove the two nodes with the smallest frequencies
	 Make them children of a parent whose frequency is the sum of the child frequencies
	 Add the parent to the priority queue

Huffman Algorithm

Huffman Algorithm

• After the tree has been constructed, the frequencies are no longer needed

• White   0
  Pink    100
  Yellow  1010
  Blue    10110
  Green   10111
  Orange   11

• Not a code for encrypting; it's public
• Note prefix property. White is 0; no other codeword starts with 0.

Huffman Tree Implementation

class Node implements Comparable<Node>
{
   public char character;
   public int frequency;
   public Node left;
   public Node right;
   public int compareTo(Node other) { ... }
}

• Implement Comparable so we can put the nodes in a priority queue
• This implementation uses counts, not percentages for frequencies
• Tree constructor receives a Map<Character, Integer> with the frequencies

Huffman Tree Implementation

• Make a node for each character
• Add nodes to priority queue

PriorityQueue<Node> nodes = new PriorityQueue<Node>();
for (char ch : frequencies.keySet())
{
   Node newNode = new Node();
   newNode.character = ch;
   newNode.frequency = frequencies.get(ch);
   nodes.add(newNode);
}

Huffman Tree Implementation

• Combine the nodes with the lowest frequencies

while (nodes.size() > 1)
{
   Node smallest = nodes.remove();
   Node nextSmallest = nodes.remove();
   Node newNode = new Node();
   newNode.frequency = smallest.frequency + nextSmallest.frequency;
   newNode.left = smallest;
   newNode.right = nextSmallest;
   nodes.add(newNode);
}
root = nodes.remove();

Lecture 23 Clicker Question 3

Which line needs to be filled in in the definition of the compareTo method of the Node class to make the preceding code work?

public int compareTo(Node other) { return ___; }

1. character - other.character
2. other.character - character
3. frequency - other.frequency
4. other.frequency - frequency

Decoding

public String decode(String input)
{
   String result = "";
   Node n = root;
   for (int i = 0; i < input.length(); i++)
   {
      char ch = input.charAt(i);
      if (ch == '0') { n = n.left; }
      else { n = n.right; }
   }
   if (n.left == null) // n is a leaf
   {
      result = result + n.character;
      n = root;
   }
   return result;
}

Encoding

• Don't want to search through the leaves each time we find a character
• Make a map from each character to its encoding
• Recursively populate with this helper method

class Node implements Comparable<Node>
{
   . . .
   public void fillEncodingMap(Map<Character, String> map, String prefix)
   {
      if (left == null) // It’s a leaf
      {
         map.put(character, prefix);
      }
      else
      {
         left.fillEncodingMap(map, prefix + "0");
         right.fillEncodingMap(map, prefix + "1");
      }
   }
}

Lecture 23 Clicker Question 4

Another way of doing the encoding would be to make a map from each character to the node that represents it.

Map<Character, Node> encodingMap;

When doing this, what else do you need to do?

1. Give each node a link to the parent
2. Add an instance variable to each node that contains the encoding, and set it when the node is constructed.
3. Either of these will work
4. Neither of these will work