Linear-Time Graph Algorithms in GP 2

GP 2 is an experimental programming language based on graph transformation rules which aims to facilitate program analysis and verification. However, implementing graph algorithms efficiently in a rule-based language is challenging because graph pattern matching is expensive. GP 2 mitigates this problem by providing rooted rules which, under mild conditions, can be matched in constant time. In this paper, we present linear-time GP 2 programs for three problems: tree recognition, binary directed acyclic graph (DAG) recognition, and topological sorting. In each case, we show the correctness of the program, prove its linear time complexity, and also give empirical evidence for the linear run time. For DAG recognition and topological sorting, the linear behaviour is achieved by implementing depth-first search strategies based on an encoding of stacks in graphs

Speeding up shortest path algorithms

Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_\psi(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an $\mathcal{O}(m^*n + m \lg n)$ all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.Comment: 10 page

I/O-optimal algorithms on grid graphs

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.Comment: 12 pages' extended abstract plus 12 pages' appendix with details, proofs and calculations. Has not been published in and is currently not under review of any conference or journa

Better to stay apart: asset commonality, bipartite network centrality, and investment strategies

By exploiting a bipartite network representation of the relationships between mutual funds and portfolio holdings, we propose an indicator that we derive from the analysis of the network, labelled the Average Commonality Coefficient (ACC), which measures how frequently the assets in the fund portfolio are present in the portfolios of the other funds of the market. This indicator reflects the investment behavior of funds' managers as a function of the popularity of the assets they held. We show that $ACC$ provides useful information to discriminate between funds investing in niche markets and those investing in more popular assets. More importantly, we find that $ACC$ is able to provide indication on the performance of the funds. In particular, we find that funds investing in less popular assets generally outperform those investing in more popular financial instruments, even when correcting for standard factors. Moreover, funds with a low $ACC$ have been less affected by the 2007-08 global financial crisis, likely because less exposed to fire sales spillovers.Comment: 38 pages, 6 figure

An O(n^{2.75}) algorithm for online topological ordering

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n^{2.75}) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min(m^{3/2} log(n), m^{3/2} + n^2 log(n)) by Katriel and Bodlaender. We also provide an empirical comparison of our algorithm with other algorithms for online topological sorting. Our implementation outperforms them on certain hard instances while it is still competitive on random edge insertion sequences leading to complete DAGs.Comment: 20 pages, long version of SWAT'06 pape

Kernel arquitecture for CAD/CAM in shipbuilding enviroments

The capabilities of complex software products such as CAD/CAM systems are strongly supported by basic information technologies related with data management, visualization, communication, geometry modeling and others related with the development process. These basic information technologies are involved in a continuous evolution process, but over recent years this evolution has been dramatic. The main reason for this has been that new hardware capabilities (including graphic cards) are available at very low cost, but also a contributing factor has been the evolution of the prices of basic software. To take advantage of these new features, the existing CAD/CAM systems must undergo a complete and drastic redesign. This process is complicated but strategic for the future evolution of a system. There are several examples in the market of how a bad decision has lead to a cul-de-sac (both technically and commercially). This paper describes what the authors consider are the basic architectural components of a kernel for a CAD/CAM system oriented to shipbuilding. The proposed solution is a combination of in-house developed frameworks together with commercial products that are accepted as standard components. The proportion of in-house frameworks within this combination of products is a key factor, especially when considering CAD/CAM systems oriented to shipbuilding. General-purpose CAD/CAM systems are mainly oriented to the mechanical CAD market. For this reason several basic products exist devoted to geometry modelling in this context. But these basic products are not well suited to deal with the very specific geometry modelling requirements of a CAD/CAM system oriented to shipbuilding. The complexity of the ship model, the different model requirements through its short and changing life cycle and the many different disciplines involved in the process are reasons for this inadequacy. Apart from these basic frameworks, specific shipbuilding frameworks are also required. This second layer is built over the basic technology components mentioned above. This paper describes in detail the technological frameworks which have been used to develop the latest FORAN version.Postprint (published version

Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes

Mode sorting is an essential function for optical multiplexing systems that exploit the orthogonality of the orbital angular momentum mode space. The familiar log-polar optical transformation provides a simple yet efficient approach whose resolution is, however, restricted by a considerable overlap between adjacent modes resulting from the limited excursion of the phase along a complete circle around the optical vortex axis. We propose and experimentally verify a new optical transformation that maps spirals (instead of concentric circles) to parallel lines. As the phase excursion along a spiral in the wave front of an optical vortex is theoretically unlimited, this new optical transformation can separate orbital angular momentum modes with superior resolution while maintaining unity efficiency

Thick 2D Relations for Document Understanding

We use a propositional language of qualitative rectangle relations to detect the reading order from document images. To this end, we define the notion of a document encoding rule and we analyze possible formalisms to express document encoding rules such as LATEX and SGML. Document encoding rules expressed in the propositional language of rectangles are used to build a reading order detector for document images. In order to achieve robustness and avoid brittleness when applying the system to real life document images, the notion of a thick boundary interpretation for a qualitative relation is introduced. The framework is tested on a collection of heterogeneous document images showing recall rates up to 89%

Improving legibility of natural deduction proofs is not trivial

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page