Tight Bounds on the Redundancy of Huffman Codes
نویسندگان
چکیده
Consider a discrete finite source with N symbols, and with the probability distribution p := (u1, u2, . . . , uN). It is well-known that the Huffman encoding algorithm [1] provides an optimal prefix code for this source. A D-ary Huffman code is usually represented using a D-ary tree T , whose leaves correspond to the source symbols; The D edges emanating from each intermediate node of T are labeled with the D letters of the alphabet, and the codeword corresponding to a symbol is the string of labels on the path from the root to the corresponding leaf. Huffman’s algorithm is a recursive bottom-up construction of T , where at each time the smallest D probabilities are merged into a new unit, and henceforth represented by an intermediate node in the tree. Throughout this paper, unless D is explicitly specified, we talk about the binary Huffman codes. Denote by l(u) the length of the path from the root to a node u on the Huffman tree T . Then the expected length of the Huffman code is defined as
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