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Article Dans Une Revue Electronic Notes in Discrete Mathematics Année : 2010

On the Convex Hull of Huffman Trees

Jean-François Maurras
  • Fonction : Auteur
Thanh Hai Nguyen
  • Fonction : Auteur
Viet Hung Nguyen

Résumé

A well-known kind of binary tree in information theory is Huffman tree. Given any linear objective function f, Huffman has given an algorithm allowing to find an optimal Huffman tree minimizing f. In this paper, we associate to each Huffman point of n nodes, a point in QnQn called Huffman point whose components are the length of the path from the root of the tree to each leaf. In this paper, we study the Huffmanhedron, PHnPHn, which is the convex hull of the Huffman points in QnQn. In particular, we present a family of facet-defining inequalities for PHnPHn whose coefficients form a Fibonacci sequence. We describe several lifting and composition methods which allow to derive new facet-defining inequalities from existing ones. Finally, we show that these methods together with the Fibonacci family of facet-defining inequalities characterize all facets of non-negative coefficients for PHnPHn containing a deepest Huffman point, i.e. a permutation of the Huffman point (n−1,n−1,n−2,…,3,2,1)(n−1,n−1,n−2,…,3,2,1).

Dates et versions

hal-01170335 , version 1 (01-07-2015)

Identifiants

Citer

Jean-François Maurras, Thanh Hai Nguyen, Viet Hung Nguyen. On the Convex Hull of Huffman Trees. Electronic Notes in Discrete Mathematics, 2010, 36, pp.1009-1016. ⟨10.1016/j.endm.2010.05.128⟩. ⟨hal-01170335⟩
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