
The Huffman Tree Problem with Unit Step FunctionsThe Huffman Tree Problem with Unit Step Functions 
"/Fujiwara, Hiroshi/"Fujiwara, Hiroshi ,
"/Nakamura, Takuya/"Nakamura, Takuya ,
"/Fujito, Toshihiro/"Fujito, Toshihiro
E98A
(
6
)
, pp.1189

1196 , 201506 , IEICEINST ELECTRONICS INFORMATION COMMUNICATIONS ENG
Description
A binary tree is regarded as a prefixfree binary code, in which the weighted sum of the lengths of rootleaf paths is equal to the expected codeword length. Huffman's algorithm computes an optimal tree in O(n log n) time, where n is the number of leaves. The problem was later generalized by allowing each leaf to have its own function of its depth and setting the sum of the function values as the objective function. The generalized problem was proved to be NPhard. In this paper we study the case where every function is a unit step function, that is, a function that takes a lower constant value if the depth does not exceed a threshold, and a higher constant value otherwise. We show that for this case, the problem can be solved in O(n log n) time, by reducing it to the Coin Collector's problem.
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