Huffman Codes
Radu Trˆımbit¸a¸s
November 11, 2012
Huffman invented a greedy algorithm that constructs an optimal prefix code
called a Huffman code. In the pseudocode that follows (Algorithm 1), we assume
that Cis a set of n characters and that each character c∈Cis an object
with an attribute c.freq giving its frequency. The algorithm builds the tree T
corresponding to the optimal code in a bottom-up manner. It begins with a set
of |C|leaves and performs a sequence of |C| − 1 merging operations to create
the final tree. The algorithm uses a min-priority queue Q, keyed on the freq
attribute, to identify the two least-frequent objects to merge together. When
we merge two objects, the result is a new object whose frequency is the sum of
the frequencies of the two objects that were merged.
Algorithm 1 Huffman(C)
1: n:= |C|;
2: Q:= C;
3: for i:= 1 to n−1do
4: allocate a new node z
5: z.lef t := x:= Extract-Min(Q);
6: z.right := y:= Extract-Min(Q);
7: z.f req := x.f req +y.fr eq;
8: Insert(Q,z );
9: end for
10: return Extract-Min(Q); {return the root of the tree}
a b c d e f
Frequency (in thousands) 45 13 12 16 9 5
Fixed-length codeword 000 001 010 011 100 101
Variable-length codeword 0 101 100 111 1101 1100
Table 1: A character-coding problem. A data file of 100,000 characters contains
only the characters af, with the frequencies indicated. If we assign each character
a 3-bit codeword, we can encode the file in 300,000 bits. Using the variable-
length code shown, we can encode the file in only 224,000 bits.
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