Optimal Prefix-Free Codes for Unequal Letter Costs: Dynamic Programming with the Monge Property

In this paper we discuss the problem of finding optimal prefix-free codes for unequal letter costs, a variation of the classical Huffman coding problem. Our problem consists of finding a minimal cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, with lengths α and β. The most efficient algorithm known previously requires O(n^{2 + max(α, β)}) time to construct such a minimal-cost set of n codewords, provided α and β are integers. In this paper we provide an O(n^max(α, β)) time algorithm. Our improvement comes from the use of a more sophisticated modeling of the problem, combined with the observation that the problem possesses a “Monge property” and that the SMAWK algorithm on monotone matrices can therefore be applied.     

‘Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems’ [Numer. Linear Algebra Appl. 14 (2007) 217–235]

In this note, some errors in the article (Numer. Linear Algebra Appl. 2007; 14 :217–235) are pointed out and some correct results are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Continuity Properties of Walras Equilibrium Points

We explore convergence notions for bivariate functions that yield convergence and stability results for their max/inf points. The results are then applied to obtain continuity results for Walras equilibrium points under perturbations of the utility functions of the agents.