BELLMAN'S PRINCIPLE OF OPTIMALITY

• Consider the discrete space (grid) shown below:
  
  
  
• Define a path from node (s,t) to (u,v) as an n-tuple:
  
  (s,t), (i₁,j₁), (i₂,j₂), ..., (u,v)
  
• Define a distance in moving from node i_k-1,j_k-1 to i_k,j_k as:
  
  
  d[(i_k,j_k) | (i_k-1,j_k-1)] = d_T[(i_k,j_k) | (i_k-1,j_k-1)] + d_N((i_k,j_k)
  
  
• Define an overall path cost as:
• Bellman's Principle of Optimality:
• This theorem has remarkable consequences: We do not need to exhaustively search for the best path. Instead, we can build the best path by considering a sequence of partial paths, and retaining the best local path:
• The savings in computations are enormous: O(KVL) vs. O(KV^L).
  
  
• For this reason, dynamic programming is one of the most widely used algorithms in computing. It has been applied to many areas of speech recognition including language modeling (search) and scoring (string edits).