Huffman minimum variance coding

Question

it is well known that Huffman code with minimum variance is preferable. I've digged through entire Polish/English internet and this is what I found: to build Huffman code with minimum variance you need to break ties with one of the following methods (of course probability of node is the most important):

• Select node that represents shortest tree
• Select node that was created earliest (consider leafs as created at start).

the problem is, that I couldn't find any proof of correctness of any of these methods. Can someone proof any of these?

I will gladly clarify anything.

Answer 1

Some systems have an even stronger constraint than "when there's a tie, make the choice that minimizes the maximum depth of a tree" -- they set a hard limit on the maximum depth of the tree ( length-limited, also called minimum variance Huffman coding ):

• "Whether there is a tie or not, build a tree with a maximum depth of at most 16 steps, so the maximum codeword length is 16 bits" (as in the Huffman codes used in JPEG image compression ) ( Jpeg huffman coding procedure )

• "Whether there is a tie or not, build a tree with a maximum depth of at most 15 steps, so the maximum codeword length is 15 bits" (as in the Huffman codes used in DEFLATE and the Huffman codes used in gzip

• "Whether there is a tie or not, build a tree with a maximum depth of at most 12 steps, so the maximum codeword length is 12 bits" ( "Huff0 uses a 12-bit limit." )

My understanding is that people have developed several heuristic algorithms for limiting Huffman codeword lengths that work OK, but the heuristics don't always give exactly the best possible compression .

Several people mention "optimal length-limited Huffman codes", and apparently there exist more than one algorithm to find them:

• Lawrence L. Larmore and Daniel S. Hirschberg. "A Fast Algorithm for Optimal Length-Limited Huffman Codes"
• "Optimal Length-Limited Huffman Codes"
• "A fast algorithm for optimal length-limited Huffman codes"