On the Huffman and Alphabetic Tree Problem with General Cost Functions

We address generalized versions of the Huffman and Alphabetic Tree Problem where the cost caused by each individual leaf i, instead of being linear, depends on its depth in the tree by an arbitrary function. The objective is to minimize either the total cost or the maximum cost among all leaves. We review and extend the known results in this direction and devise a number of new algorithms and hardness proofs. It turns out that the Dynamic Programming approach for the Alphabetic Tree Problem can be extended to arbitrary cost functions, resulting in a time O(n (4)) optimal algorithm using space O(n (3)). We identify classes of cost functions where the well-known trick to reduce the runtime by a factor of n via a "monotonicity" property can be applied. For the generalized Huffman Tree Problem we show that even the k-ary version can be solved by a generalized version of the Coin Collector Algorithm of Larmore and Hirschberg (in Proc. SODA'90, pp. 310-318, 1990) when the cost functions are nondecreasing and convex. Furthermore, we give an O(n (2)logn) algorithm for the worst case minimization variants of both the Huffman and Alphabetic Tree Problem with nondecreasing cost functions. Investigating the limits of computational tractability, we show that the Huffman Tree Problem in its full generality is inapproximable unless P = NP, no matter if the objective function is the sum of leaf costs or their maximum. The alphabetic version becomes NP-hard when the leaf costs are interdependent.ArticleALGORITHMICA. 69(3): 582-604 (2014)journal articl

Minimax Trees in Linear Time with Applications

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves\u27 depths, it minimizes the maximum of any leaf\u27s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, $O (n log n)$-time algorithm for building them. Drmota and Szpankowski (2002) gave another $O (n log n)$-time algorithm, which takes linear time when the weights are already sorted by their fractional parts. In this paper we give the first linear-time algorithm for building minimax trees for unsorted real weights

Hu-Tucker alogorithm for building optimal alphabetic binary search trees

The purpose of this thesis is to study the behavior of the Hu- Tucker algorithm for building Optimal Alphabetic Binary Search Trees (OABST), to design an efficient implementation, and to evaluate the performance of the algorithm, and the implementation. The three phases of the algorithm are described and their time complexities evaluated. Two separate implementations for the most expensive phase, Combination, are presented achieving 0(n2) and O(nlogn) time and 0(n) space complexity. The break even point between them is experimentally established and the complexities of the implementations are compared against their theoretical time complexities. The electronic version of The Complete Works of William Shakespeare is compressed using the Hu- Tucker algorithm and other popular compression algorithms to compare the performance of the different techniques. The experiments justified the price that has to be paid to implement the Hu- Tucker algorithm. It is shown that an efficient implementation can process extremely large data sets relatively fast and can achieve optimality close to the Optimal Binary Tree, built using the Huffman algorithm, however the OABST can be used in both encoding and decoding processes, unlike the OBT where an additional mapping mechanism is needed for the decoding phase

Generalizations of Length Limited Huffman Coding for Hierarchical Memory Settings

In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the soft-length limited Huffman coding problem. In this version, there is a penalty associated with each of the n characters of the alphabet whose encodings exceed a specified bound D(? n) where the penalty increases linearly with the length of the encoding beyond D. The goal of the problem is to find a prefix-free encoding having minimum average code length and total penalty within a pre-specified bound P. This generalizes the LLHC problem. We present an algorithm to solve this problem that runs in time O(nD). We study a further generalization in which the penalty function and the objective function can both be arbitrary monotonically non-decreasing functions of the codeword length. We provide dynamic programming based exact and PTAS algorithms for this setting

Approximately Optimum Search Trees in External Memory Models

We examine optimal and near optimal solutions to the classic binary search tree problem of Knuth. We are given a set of n keys (originally known as words), B_1, B_2, ..., B_n and 2n+1 frequencies. {p_1, p_2, ..., p_n} represent the probabilities of searching for each given key, and {q_0, q_1, ..., q_n} represent the probabilities of searching in the gaps between and outside of these keys. We have that Σ_{i=0}^n q_i + Σ_{i=1}^n p_i = 1. We also assume without loss of generality that q_{i-1}+p_i+q_i != 0 for any i ϵ {1,...,n}. The keys must make up the internal nodes of the tree while the gaps make up the leaves. Our goal is to construct a binary search tree such that expected cost of search is minimized. First, we re-examine an approximate solution of Guttler, Mehlhorn and Schneider which was shown to have a worst case bound of c * H + 2 where c >= 1/(H(1/3,2/3)) ~ 1.08, and H = Σ_{i=1}^{n} p_i * log_2(1/p_i) + Σ_{j=0}^{n} q_i * log_2(1/q_j) is the entropy of the distribution. We give an improved worst case bound on the heuristic of H+4. Next, we examine the optimum binary search tree problem under a model of external memory. We use the Hierarchical Memory Model of Aggarwal et al. The model has an unlimited number of registers, R_1, R_2, ... each with its own location in memory (a positive integer). We have a set of memory sizes m_1, m_2, ..., m_l which are monotonically increasing. Each memory level has a finite size except m_l which we assume has infinite size. Each memory level has an associated cost of access c_1, c_2, ..., c_l. We assume that c_1 < c_2 < ... < c_l. We propose two approximate solutions which run in O(n) time where n is the number of words in our data set. Using these methods, we improve upon a bound given in Thite's 2001 thesis under the related HMM_2 model in the approximate setting. We also examine the related problem of binary trees on multisets of probabilities where keys are unordered and we do not differentiate between which probabilities must be leaves, and which must be internal nodes. We provide a simple O(n log_2(n)) algorithm that is within an additive (n+1)(2n) of optimal on a multiset of n keys

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Function-specific schemes for verifiable computation

An integral component of modern computing is the ability to outsource data and computation to powerful remote servers, for instance, in the context of cloud computing or remote file storage. While participants can benefit from this interaction, a fundamental security issue that arises is that of integrity of computation: How can the end-user be certain that the result of a computation over the outsourced data has not been tampered with (not even by a compromised or adversarial server)? Cryptographic schemes for verifiable computation address this problem by accompanying each result with a proof that can be used to check the correctness of the performed computation. Recent advances in the field have led to the first implementations of schemes that can verify arbitrary computations. However, in practice the overhead of these general-purpose constructions remains prohibitive for most applications, with proof computation times (at the server) in the order of minutes or even hours for real-world problem instances. A different approach for designing such schemes targets specific types of computation and builds custom-made protocols, sacrificing generality for efficiency. An important representative of this function-specific approach is an authenticated data structure (ADS), where a specialized protocol is designed that supports query types associated with a particular outsourced dataset. This thesis presents three novel ADS constructions for the important query types of set operations, multi-dimensional range search, and pattern matching, and proves their security under cryptographic assumptions over bilinear groups. The scheme for set operations can support nested queries (e.g., two unions followed by an intersection of the results), extending previous works that only accommodate a single operation. The range search ADS provides an exponential (in the number of attributes in the dataset) asymptotic improvement from previous schemes for storage and computation costs. Finally, the pattern matching ADS supports text pattern and XML path queries with minimal cost, e.g., the overhead at the server is less than 4% compared to simply computing the result, for all our tested settings. The experimental evaluation of all three constructions shows significant improvements in proof-computation time over general-purpose schemes

Logic-based machine learning using a bounded hypothesis space: the lattice structure, refinement operators and a genetic algorithm approach

Rich representation inherited from computational logic makes logic-based machine learning a competent method for application domains involving relational background knowledge and structured data. There is however a trade-off between the expressive power of the representation and the computational costs. Inductive Logic Programming (ILP) systems employ different kind of biases and heuristics to cope with the complexity of the search, which otherwise is intractable. Searching the hypothesis space bounded below by a bottom clause is the basis of several state-of-the-art ILP systems (e.g. Progol and Aleph). However, the structure of the search space and the properties of the refinement operators for theses systems have not been previously characterised. The contributions of this thesis can be summarised as follows: (i) characterising the properties, structure and morphisms of bounded subsumption lattice (ii) analysis of bounded refinement operators and stochastic refinement and (iii) implementation and empirical evaluation of stochastic search algorithms and in particular a Genetic Algorithm (GA) approach for bounded subsumption. In this thesis we introduce the concept of bounded subsumption and study the lattice and cover structure of bounded subsumption. We show the morphisms between the lattice of bounded subsumption, an atomic lattice and the lattice of partitions. We also show that ideal refinement operators exist for bounded subsumption and that, by contrast with general subsumption, efficient least and minimal generalisation operators can be designed for bounded subsumption. In this thesis we also show how refinement operators can be adapted for a stochastic search and give an analysis of refinement operators within the framework of stochastic refinement search. We also discuss genetic search for learning first-order clauses and describe a framework for genetic and stochastic refinement search for bounded subsumption. on. Finally, ILP algorithms and implementations which are based on this framework are described and evaluated.Open Acces