Question

Draw the frequency array and Huffman tree for the following string: "dogs do not spot hot pots or cats".

Short answer

Count frequencies of each character, build a priority queue, construct the Huffman tree, assign binary codes from the tree, and create a frequency array with these codes.

Step-by-step solution

Count the frequency of each character

Count how many times each character appears in the string 'dogs do not spot hot pots or cats'. Include spaces as separate characters.

List frequencies

List the characters along with their frequencies: d: 2 o: 6 g: 1 s: 5 ' ': 7 n: 1 t: 4 p: 2 h: 1 r: 1 c: 1 a: 1

Create a priority queue of nodes

Each character and its frequency become a node. Insert all nodes into a priority queue (min-heap).

Build Huffman tree

Remove the two nodes with the lowest frequencies from the queue and merge them to create a new node with a frequency equal to the sum of the two nodes. Insert the new node back into the priority queue. Repeat until there is only one node left. This node becomes the root of the Huffman tree.

Draw the Huffman tree

Draw the tree starting from the root. Each internal node has two children, representing the binary codes (0 and 1).

Assign binary codes

Assign a binary code to each character by traversing the Huffman tree from the root to the leaves. Move left (0) or right (1) accordingly.

Create the frequency array

Create an array listing each character along with its frequency and corresponding Huffman binary code.

Key concepts

Frequency Array

A frequency array helps to count the frequencies of characters in a string. It's the first step in Huffman encoding. Start by tallying how many times each character appears. For instance, in the string 'dogs do not spot hot pots or cats', we count each character: 'd' appears 2 times, 'o' appears 6 times, and so on. This is crucial because it sets the foundation for building the Huffman tree.
By organizing these counts into an array, we can proceed to the next steps efficiently. It essentially looks like this:

• d: 2
• o: 6
• g: 1
• s: 5
• ' ': 7
• n: 1
• t: 4
• p: 2
• h: 1
• r: 1
• c: 1
• a: 1

After creating this array, you’ll have a clear picture of the character frequency, allowing you to build an efficient Huffman tree next.

Huffman Tree

The Huffman tree is where the magic of optimal character encoding happens. It’s a binary tree where each character from your frequency array becomes a leaf node. Start by creating nodes for each character and insert them into a priority queue. Merge the two nodes with the smallest frequencies and insert the new node back into the queue. Repeat this until you have only one node left, which will be the root of your Huffman tree.
For instance, with our array, you merge nodes with lower frequencies first, progressively building internal nodes until the tree is complete. This structure is crucial because it forms the basis for assigning efficient binary codes to characters.

Priority Queue

In Huffman encoding, a priority queue (often implemented as a min-heap) is used to build the tree efficiently. It ensures that nodes with the smallest frequencies are processed first.
You start by inserting all character nodes into this queue. Each time you remove the two nodes with the lowest frequencies to merge them. The sum of their frequencies becomes the frequency of the new parent node, which is inserted back into the queue.
This process continues until there's a single node left. This node will serve as the root of the Huffman tree. Using a priority queue ensures that the process is efficient and follows the greedy algorithm essential for Huffman encoding.

Binary Codes

Binary codes are what make Huffman encoding useful for data compression. After you’ve built the Huffman tree, you assign binary codes to each character based on the path from the root to the leaf of that character.
In the tree, moving left is encoded as 0 and moving right as 1. This means that characters with lower frequencies (and thus longer paths in the tree) have longer binary codes, while frequent characters get shorter codes.
For example, if 'o' has a frequency of 6, its binary code might be short, like '10', while 'g' with a frequency of 1 might have a longer code like '1100'. This optimizes the overall space required for encoding the original string, leading to efficient data compression.