This document summarizes Huffman code decoding. It takes a sequence to be decoded and a character probability table as input. A binary tree is built from the probabilities, with left branches representing 1 and right 0. The time complexity is O(n^2) where n is the sequence length, as building the tree can resemble an unbalanced tree. Sample run times are provided for sequences of different lengths on 10 possible characters. Pseudocode provides an algorithm to traverse the tree and decode the sequence.

1. Huffman Code Decoding
Rex Yuan
May 13, 2015
Huffman Code Decoding
We take a sequence to be decoded and a table of corresponding characters and
their probabilities as input. Using the given characters and probabilities, this
algorithm build a binary tree by always selecting the two characters with lowest
probabilities and make them a subtree with the sum of their probabilities. When
going from the root, each traversal to the left subtree represents a ”1” and right
a ”0”.
Time Complexity
Suppose we have a total of m possible character to be encoded and a sequence
with length n to be decoded, the time complexity would be linear with regards
to m and n, n2
+ m, i.e.,
O(n2
)
because in worst case we will iterate through all characters and perform a tree-
building operation which in worst case would resemble a completely un-balanced
tree, and thus produce a recursion tree containing n levels, and the operations
required for each level increments from 1 to n − 1, in turn yielding O(n2
) which
dominates over the operations needed for decoding, m.
Run Time Stats
I created the following table using UNIX POSIX time function and round the
mean time of 10 trials to five digits after decimal point to calculate the time
past. All samples have the ten possible characters to be encoded, that is 0 9.
Variable input length implies the length of the sequence to be decoded.
Run Time Stats
Sample Input Length Huffman Decoding
1 35 0.00031
2 37 0.00858
3 368 0.00902
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2. Pseudo Code
Algorithm 1 Huffman Code Decoding Algorithm
procedure DECODE(prob table, seq)
tree = a key-value associating array, with each item’s key the node in that
subtree and value its current sum probability, according to probtable.
code table = an array, with length of the how many characters in
prob table, which maps each code to its corresponding decoded character with
each code initialised as empty strings.
while more than one node in tree do sml sub nodes = key of node with
smallest probability in tree sec sub nodes = key of node with second smallest
probability in tree
for node in sml sub nodes do
code table[node].key = ”1” + code table[node].key
end for
for node in sec sub nodes do
code table[node].key = ”0” + code table[node].key
end for
tree[sml sub nodes + sec sub nodes] = tree[sml sub nodes] +
tree[sec sub nodes]
delete tree[sml sub nodes]
delete tree[sec sub nodes]
end while
temp = ””
result = ””
for code in seq do
temp = temp + code
if temp in code table then
result = result + code table[temp]
temp = ””
end if
end for
return result
end procedure
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