霍夫曼编码树的最大高度

Question

What is the maximum height of a tree made with the huffman coding algorithm assuming that all bytes are accepted into it.

I'm curious since I somehow managed to get a path of 9 bits when I tried to compress I file which I generated randomly. Which means that I essentially inflate the size of the file. Though there might be a problem somewhere in the program that I don't know about.

Answer 1

If by "all bytes" you mean all 256 possible byte values as your set of symbols, the answer is that the maximum depth, and so the length of the longest code, is 255.

To get this though requires very large numbers for the frequencies of the symbols. The sequence that does this with the smallest total count are the Lucas numbers, with the zeroth Lucas number of 2 broken into two 1's. So:

1, 1, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ...

where the terms starting with 4 are the sum of the previous two terms, just like the Fibonacci sequence. The last term in that sequence for 256 symbols is:

121020968315000050139390193037122554865361969834971243

About 10 ⁵³ .

As for your random data of all possible bytes, in general you can't compress it. However if you got one code of 9 bits in length, then that means the code is at least as good as assigning 8 bits to all of the symbols. So you did not inflate the file, but rather left it the same size or reduced it by a bit or more. However that's not taking into account the fact that you have to also send the code itself so that the decoder on the other end can decode. That will more than kill any tiny bit of compression you got.