Objectives

Notes

• Huffman code is constructed by
  • Making all symbols unique trees
  • Selecting the two symbols with the lowest frequency and combining those into a new tree, with the symbols as children.
  • Continue this process until only a single tree remains.
• Since traversing a tree from the root to a leaf indicates the encoding for a symbol
  • The lowest frequency symbols will have the longest path
  • And hence the longest encoding
  • And the opposite for the most frequently used symbols.
• The greedy choice is to select the lest frequently used trees and unite them,
• This is prefix-free as the symbols will always be leaves.
• The algorithm

  HUFFMAN(Σ)
      Input: Σ An alphabet with associated symbols and frequency
      Output: A prefix-free tree with minimal encoding costs.
  
   n ← |Σ|
   Q ← Σ    // note Q is a min-heap
   for i ← 1 to n-1 do
      allocate a new node z
      z.left = x = Q.ExtractMin()
      z.right = y = Q.ExtractMin()
      z.frequency = x.frequency + y.frequency
      Q.Insert(z)
   return Q.ExtractMin()
   

• This is from CLRS
• What is the performance?
  • Line 2 is O(n) where n = |Σ|
  • The loop in 3 is also O(n) but is the body O(1)?
  • Remember, with a heap, ExtractMin, and Insert are both O(log₂n)
  • So the loop in 3-8 is O(n log₂ n)