An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.

Abstract — The problem of route address lookup has received much attention recently and several algorithms and data structures for performing address lookups at high speeds have been proposed. In this paper we consider one such data structure – a binary search tree built on the intervals created by the routing table prefixes. We wish to exploit the difference in the probabilities with which the various leaves of the tree (where the intervals are stored) are accessed by incoming packets in order to speedup the lookup process. More precisely, we seek an answer to the question “How can the search tree be drawn so as to minimize the average packet lookup time while keeping the worst-case lookup time within a fixed bound? ” We use ideas from information theory to derive efficient algorithms for computing near-optimal routing lookup trees. Finally, we consider the practicality of our algorithms through analysis and simulation.

: 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...

Abstract. The minimum-redundancy prefix code problem is to determine a list of integer codeword lengths I = [li l i E {1... n}], given a list of n symbol weightsp = [pili C {1.n}], such that ~' ~ 2-l ' < 1, 9 &quot; i = l--n and ~i=1 lipi is minimised. An extension is the minimum-redundancy length-limited prefix code problem, in which the further constraint li < L is imposed, for all i C {1...n} and some integer L> [log 2 hi. The package-merge algorithm of Larmore and Hirschberg generates lengthlimited codes in O(nL) time using O(n) words of auxiliary space. Here we show how the size of the work space can be reduced to O(L2). This represents a useful improvement, since for practical purposes L is O(log n). 1

Most data that is inherently discrete needs to be compressed in such a way that it can be recovered exactly, without any loss. Examples include text of all kinds, experimental results, and statistical databases. Other forms of data may need to be stored exactly, such as images---particularly bilevel ones, or ones arising in medical and remotesensing applications, or ones that may be required to be certified true for legal reasons. Moreover, during the process of lossy compression, many occasions for lossless compression of coefficients or other information arise. This paper surveys techniques for lossless compression. The process of compression can be broken down into modeling and coding. We provide an extensive discussion of coding techniques, and then introduce methods of modeling that are appropriate for text and images. Standard methods used in popular utilities (in the case of text) and international standards (in the case of images) are described. Keywords Text compression, ima...

. 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...

Suppose that $S$ is a string of length $m$ drawn from an alphabet of $n$ characters, $d$ of which occur in $S$. Let $P$ be the relative frequency distribution of characters in $S$. We present a new algorithm for dynamic coding that uses at most \(\lceil \lg n \rceil 1\) bits to encode each character in $S$

Huffman coding finds a prefix code that minimizes mean codeword length for a given probability distribution over a finite number of items. Campbell generalized the Huffman problem to a family of problems in which the goal is to minimize not mean codeword length � i pili but rather a generalized mean of the form ϕ−1 ( � i piϕ(li)), where li denotes the length of the ith codeword, pi denotes the corresponding probability, and ϕ is a monotonically increasing cost function. Such generalized means — also known as quasiarithmetic or quasilinear means — have a number of diverse applications, including applications in queueing. Several quasiarithmetic-mean problems have novel simple redundancy bounds in terms of a generalized entropy. A related property involves the existence of optimal codes: For “well-behaved ” cost functions, optimal codes always exist for (possibly infinite-alphabet) sources having finite generalized entropy. Solving finite instances of such problems is done by generalizing an algorithm for finding length-limited binary codes to a new algorithm for finding optimal binary codes for any quasiarithmetic mean with a convex cost function. This algorithm can be performed using quadratic time and linear space, and can be extended to other penalty functions, some of which are solvable with similar space and time complexity, and others of which are solvable with slightly greater complexity. This reduces the computational complexity of a problem involving minimum delay in a queue, allows combinations of previously considered problems to be optimized, and greatly expands the space of problems solvable in quadratic time and linear space. The algorithm can be extended for purposes such as breaking ties among possibly different optimal codes, as with bottom-merge Huffman coding.

We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Monge-array results for two shortest path problems to a general algebraic setting, with values in an ordered commutative semigroup, if the semigroup operator is strictly compatible with the order relation. We show how our algorithms can be modified to solve bottleneck shortest path problems, even though strict compatibility does not hold in that case. For example, we give a linear time algorithm for the unrestricted shortest path bottleneck problem on n nodes, also O(kn) and O(n 3/2 log 5/2 n) time algorithms for the k-shortest path bottleneck problem.