Kazuyuki Amano , Yuta Kojima , Toshiya Kurabayashi , Keita Kurihara , Masahiro Nakamura , Ayaka Omi , Toshiyuki Tanaka and Koichi Yamazaki
Article: How to Solve the Torus Puzzle
Algorithms 2012, 5(1), 18-29; doi:10.3390/a5010018 - published online 13 January 2012
Show/Hide AbstractAbstract: In this paper, we consider the following sliding puzzle called torus puzzle. In an m by n board, there are mn pieces numbered from 1 to mn. Initially, the pieces are placed in ascending order. Then they are scrambled by rotating the rows and columns without the player’s knowledge. The objective of the torus puzzle is to rearrange the pieces in ascending order by rotating the rows and columns. We provide a solution to this puzzle. In addition, we provide lower and upper bounds on the number of steps for solving the puzzle. Moreover, we consider a variant of the torus puzzle in which each piece is colored either black or white, and we present a hardness result for solving it.

Antonella Di Lillo , Ajay Daptardar , Kevin Thomas , James A. Storer and Giovanni Motta
Article: Compression-Based Tools for Navigation with an Image Database
Algorithms 2012, 5(1), 1-17; doi:10.3390/a5010001 - published online 10 January 2012
Show/Hide AbstractAbstract: We present tools that can be used within a larger system referred to as a passive assistant. The system receives information from a mobile device, as well as information from an image database such as Google Street View, and employs image processing to provide useful information about a local urban environment to a user who is visually impaired. The first stage acquires and computes accurate location information, the second stage performs texture and color analysis of a scene, and the third stage provides specific object recognition and navigation information. These second and third stages rely on compression-based tools (dimensionality reduction, vector quantization, and coding) that are enhanced by knowledge of (approximate) location of objects.

John Kieffer
Article: A Catalog of Self-Affine Hierarchical Entropy Functions
Algorithms 2011, 4(4), 307-333; doi:10.3390/a4040307 - published online 1 November 2011
Show/Hide AbstractAbstract: For fixed k ≥ 2 and fixed data alphabet of cardinality m, the hierarchical type class of a data string of length n = k^j for some j ≥ 1 is formed by permuting the string in all possible ways under permutations arising from the isomorphisms of the unique finite rooted tree of depth j which has n leaves and k children for each non-leaf vertex. Suppose the data strings in a hierarchical type class are losslessly encoded via binary codewords of minimal length. A hierarchical entropy function is a function on the set of m-dimensional probability distributions which describes the asymptotic compression rate performance of this lossless encoding scheme as the data length n is allowed to grow without bound. We determine infinitely many hierarchical entropy functions which are each self-affine. For each such function, an explicit iterated function system is found such that the graph of the function is the attractor of the system.

Tobias Marschall and Sven Rahmann
Article: An Algorithm to Compute the Character Access Count Distribution for Pattern Matching Algorithms
Algorithms 2011, 4(4), 285-306; doi:10.3390/a4040285 - published online 31 October 2011
Show/Hide AbstractAbstract: We propose a framework for the exact probabilistic analysis of window-based pattern matching algorithms, such as Boyer–Moore, Horspool, Backward DAWG Matching, Backward Oracle Matching, and more. In particular, we develop an algorithm that efficiently computes the distribution of a pattern matching algorithm’s running time cost (such as the number of text character accesses) for any given pattern in a random text model. Text models range from simple uniform models to higher-order Markov models or hidden Markov models (HMMs). Furthermore, we provide an algorithm to compute the exact distribution of differences in running time cost of two pattern matching algorithms. Methodologically, we use extensions of finite automata which we call deterministic arithmetic automata (DAAs) and probabilistic arithmetic automata (PAAs) [1]. Given an algorithm, a pattern, and a text model, a PAA is constructed from which the sought distributions can be derived using dynamic programming. To our knowledge, this is the first time that substring- or suffix-based pattern matching algorithms are analyzed exactly by computing the whole distribution of running time cost. Experimentally, we compare Horspool’s algorithm, Backward DAWG Matching, and Backward Oracle Matching on prototypical patterns of short length and provide statistics on the size of minimal DAAs for these computations.

Rafael Carrascosa , François Coste , Matthias Gallé and Gabriel Infante-Lopez
Article: The Smallest Grammar Problem as Constituents Choice and Minimal Grammar Parsing
Algorithms 2011, 4(4), 262-284; doi:10.3390/a4040262 - published online 26 October 2011
Show/Hide AbstractAbstract: The smallest grammar problem—namely, finding a smallest context-free grammar that generates exactly one sequence—is of practical and theoretical importance in fields such as Kolmogorov complexity, data compression and pattern discovery. We propose a new perspective on this problem by splitting it into two tasks: (1) choosing which words will be the constituents of the grammar and (2) searching for the smallest grammar given this set of constituents. We show how to solve the second task in polynomial time parsing longer constituent with smaller ones. We propose new algorithms based on classical practical algorithms that use this optimization to find small grammars. Our algorithms consistently find smaller grammars on a classical benchmark reducing the size in 10% in some cases. Moreover, our formulation allows us to define interesting bounds on the number of small grammars and to empirically compare different grammars of small size.

Jose Miguel Tarongi and Adriano Camps
Article: Radio Frequency Interference Detection and Mitigation Algorithms Based on Spectrogram Analysis
Algorithms 2011, 4(4), 239-261; doi:10.3390/a4040239 - published online 25 October 2011
Show/Hide AbstractAbstract: Radio Frequency Interference (RFI) detection and mitigation algorithms based on a signal’s spectrogram (frequency and time domain representation) are presented. The radiometric signal’s spectrogram is treated as an image, and therefore image processing techniques are applied to detect and mitigate RFI by two-dimensional filtering. A series of Monte-Carlo simulations have been performed to evaluate the performance of a simple thresholding algorithm and a modified two-dimensional Wiener filter.