Lemma 17.2

Huffman exhibits the greedy choice property.

Proof:

• Assume x and y have lowest frequencies.
• If x and y have lowest frequencies, then there exists an optimal code in which x and y are at the maximum depth (greedy choice).
• Greedy choice would put them where b and c are in T.
• Assume f(x) $\leq$ f(y) and f(b) $\leq$ f(c). We know f(x) $\leq$ f(b) and f(y) $\leq$ f(c).
• B(T) - B(T') =
  
  $$=\; f[x]d_T(x) \;+\; f[b]d_T(b) \;-\; f[x] d_{T'}(x) - f[b]d_{T'}(b)$$
  
  
  
  
  $$=\; f[x]d_T(x) \;+\; f[b]d_T(b) - f[x]d_T(b) - f[b]d_T(x)$$
  
  
  
  
  $$=\; (f[b] \;-\; f[x])(d_T(b) \;-\; d_T(x))$$
  
  
  
  
  $$\geq \; 0$$
  
  

• Thus, moving x to the bottom (similarly, y to the bottom) yields a better (optimal) solution.