SMAWK algorithm

Abstract: In this paper we discuss a variation of the classical Huffman coding problem: finding optimal prefix-free codes for unequal letter costs. 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 required O(n2+max(αβ)) time to construct such a minimal-cost set of n codewords. In this paper we provide an O(nmax(αβ)) 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.

Abstract: This study examines quantization for outputs of binary-input discrete memoryless channels (B-DMCs) by concatenating its output with another DMC, so-called a quantizer. As an objective function of channel quantization, we employ the α-mutual information of a B-DMC, which connects to more powerful coding theorem than the ordinary mutual information. Showing a Monge property of the α-mutual information, we propose an optimal quantizer design algorithm for given B-DMC in polynomial time complexity with respect to the output alphabet size and the quantized level. Since the proposed method employs the SMAWK algorithm due to the Monge property, our algorithm is faster than a naive dynamic programming.

Abstract: In this paper we study algorithms for the max-plus product of Monge matrices These algorithms use the underlying regularities of the matrices to be faster than the general multiplication algorithm, hence saving time A nonnaive solution is to iterate the SMAWK algorithm For specific classes there are more efficient algorithms We present a new multiplication algorithm (MMT), that is efficient for general Monge matrices and also for specific classes The theoretical and empirical analysis shows that MMT operates in near optimal space and time Hence we give further insight into an open problem proposed by Landau The resulting algorithms are relevant for bio-informatics, namely because Monge matrices occur in string alignment problems

Abstract: This paper shows that finding the row minima (maxima) in an n×n totally monotone matrix in the worst case requires any algorithm to make 3n−5 comparisons or 4n−5 matrix accesses. Where the, so called, SMAWK algorithm of Aggarwal et al. finds the row minima in no more than 5n − 2 lg n − 6 comparisons.