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On the Huffman and Alphabetic Tree Problem with General Cost Functions

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Abstract

We address generalized versions of the Huffman and Alphabetic Tree Problem where the cost caused by each individual leaf i, instead of being linear, depends on its depth in the tree by an arbitrary function. The objective is to minimize either the total cost or the maximum cost among all leaves. We review and extend the known results in this direction and devise a number of new algorithms and hardness proofs.

It turns out that the Dynamic Programming approach for the Alphabetic Tree Problem can be extended to arbitrary cost functions, resulting in a time O(n 4) optimal algorithm using space O(n 3). We identify classes of cost functions where the well-known trick to reduce the runtime by a factor of n via a “monotonicity” property can be applied.

For the generalized Huffman Tree Problem we show that even the k-ary version can be solved by a generalized version of the Coin Collector Algorithm of Larmore and Hirschberg (in Proc. SODA’90, pp. 310–318, 1990) when the cost functions are nondecreasing and convex. Furthermore, we give an O(n 2logn) algorithm for the worst case minimization variants of both the Huffman and Alphabetic Tree Problem with nondecreasing cost functions.

Investigating the limits of computational tractability, we show that the Huffman Tree Problem in its full generality is inapproximable unless P = NP, no matter if the objective function is the sum of leaf costs or their maximum. The alphabetic version becomes NP-hard when the leaf costs are interdependent.

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Acknowledgements

We wish to thank Sebastian Kamprath for a helpful comment regarding the presentation of Sect. 2.1.

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Authors and Affiliations

  1. Department of Information and Computer Sciences, Toyohashi University of Technology, 1-1 Tenpaku-cho, Toyohashi, 441-8580, Japan

    Hiroshi Fujiwara

  2. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Tobias Jacobs

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  1. Hiroshi Fujiwara
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  2. Tobias Jacobs
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Correspondence to Hiroshi Fujiwara.

Additional information

The work was supported by KAKENHI (19700015, 23700014, and 23500014) and a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).

A preliminary version of this paper has appeared in the proceedings of the 18th Annual European Symposium on Algorithms, 2010.

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Fujiwara, H., Jacobs, T. On the Huffman and Alphabetic Tree Problem with General Cost Functions. Algorithmica 69, 582–604 (2014). https://doi.org/10.1007/s00453-013-9755-6

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