: Given an alphabet fa 1 ; : : : ; ang with corresponding set of weights fw 1 ; : : : ; wng, and a number L dlog ne, we introduce an O(n log n+n log w) algorithm for constructing a suboptimal prefix code with restricted maximal length L, where w is the highest presented weight. The number of additional bits per symbol generated by our code is not greater than 1=/ L\Gammadlog(n+dlog ne\GammaL)e\Gamma2 when L ? dlog ne + 1, where / is the golden ratio 1:618. An important feature of the proposed algorithm is its implementation simplicity. The algorithm is basically a selected sequence of Huffman trees construction for modified weights. Keywords: Prefix codes, Huffman Trees, Lagragean Duality Resumo: Dado um alfabeto fa 1 ; : : : ; ang com pesos correspondentes fw 1 ; : : : ; wng e um n'umero L dlog ne, n'os apresentamoso um algoritmo de de complexidade O(n log n + n log w)para construit c'odigos de prefixo sub'otimos com restric~ao de comprimento L, onde w 'e o maior peso do dado co...

: Consider an alphabet \Sigma = fa 1 ; : : : ; ang with corresponding symbol probabilities p 1 ; : : : ; pn . The L\Gammarestricted prefix code is a prefix code where all the code lengths are not greater than L. The value L is a given integer such that L dlog ne. Define the average code length difference by ffl = P n i=1 p i :l i \Gamma P n i=1 p i :l i , where l 1 ; : : : ; l n are the code lengths of the optimal L-restricted prefix code for \Sigma and l 1 ; : : : ; l n are the code lengths of the optimal prefix code for \Sigma. Let / be the golden ratio 1,618. In this paper, we show that ffl ! 1=/ L\Gammadlog(n+dlog ne\GammaL)e\Gamma1 when L ? dlog ne. We also prove the sharp bound ffl ! dlog ne \Gamma 1, when L = dlog ne. By showing the lower bound 1 / L\Gammadlog ne+2+dlog n n\GammaL e \Gamma1 on the maximum value of ffl, we guarantee that our bound is asymptotically tight in the range dlog ne ! L n=2. Furthermore, we present an O(n) time and space 1=/ L\Gammadlo...

. Given an alphabet \Sigma = fa1 ; : : : ; ang with a corresponding list of positive weights fw1 ; : : : ; wng and a length restriction L, the length-restricted prefix code problem is to find, a prefix code that minimizes P n i=1 w i l i , where l i , the length of the codeword assigned to a i , cannot be greater than L, for i = 1; : : : ; n. In this paper, we present an efficient implementation of the WARM-UP algorithm, an approximative method for this problem. The worst-case time complexity of WARM-UP is O(n log n +n log wn ), where wn is the greatest weight. However, some experiments with a previous implementation of WARM-UP show that it runs in linear time for several practical cases, if the input weights are already sorted. In addition, it often produces optimal codes. The proposed implementation combines two new enhancements to reduce the space usage of WARM-UP and to improve its execution time. As a result, it is about ten times faster than the previous implementat...

Huffman codes, combined with word-based models, are considered efficient compression schemes for fulltext retrieval systems. The decoding rate for these schemes can be substantially improved if the maximum length of the codewords is not greater then the machine word size L. However, if the vocabulary is large, simple methods for generating optimal length-restricted codes are either too slow or require a significantly large amount of memory. In this paper we present an inplace, simple and fast implementation for the BRCI (Initials of Build, Remove, Condense and Insert) algorithm, an approximative method for length-restricted coding. It overwrites a sorted input list of n weights with the corresponding codeword lengths in O(n) time. In addition, the worst-case compression loss introduced by BRCI codes with respect to unrestricted Huffman codes is proved to be negligible for all practical values of both L and n. 1 Introduction Zobel and Moffat [18] have proposed an innovative compressio...