Linear Compressed Pattern Matching for Polynomial Rewriting (Extended Abstract)

This paper is an extended abstract of an analysis of term rewriting where the terms in the rewrite rules as well as the term to be rewritten are compressed by a singleton tree grammar (STG). This form of compression is more general than node sharing or representing terms as dags since also partial trees (contexts) can be shared in the compression. In the first part efficient but complex algorithms for detecting applicability of a rewrite rule under STG-compression are constructed and analyzed. The second part applies these results to term rewriting sequences. The main result for submatching is that finding a redex of a left-linear rule can be performed in polynomial time under STG-compression. The main implications for rewriting and (single-position or parallel) rewriting steps are: (i) under STG-compression, n rewriting steps can be performed in nondeterministic polynomial time. (ii) under STG-compression and for left-linear rewrite rules a sequence of n rewriting steps can be performed in polynomial time, and (iii) for compressed rewrite rules where the left hand sides are either DAG-compressed or ground and STG-compressed, and an STG-compressed target term, n rewriting steps can be performed in polynomial time.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599

Activity recognition from videos with parallel hypergraph matching on GPUs

In this paper, we propose a method for activity recognition from videos based on sparse local features and hypergraph matching. We benefit from special properties of the temporal domain in the data to derive a sequential and fast graph matching algorithm for GPUs. Traditionally, graphs and hypergraphs are frequently used to recognize complex and often non-rigid patterns in computer vision, either through graph matching or point-set matching with graphs. Most formulations resort to the minimization of a difficult discrete energy function mixing geometric or structural terms with data attached terms involving appearance features. Traditional methods solve this minimization problem approximately, for instance with spectral techniques. In this work, instead of solving the problem approximatively, the exact solution for the optimal assignment is calculated in parallel on GPUs. The graphical structure is simplified and regularized, which allows to derive an efficient recursive minimization algorithm. The algorithm distributes subproblems over the calculation units of a GPU, which solves them in parallel, allowing the system to run faster than real-time on medium-end GPUs

Pattern matching of compressed terms and contexts and polynomial rewriting

A generalization of the compressed string pattern match that applies to terms with variables is investigated: Given terms s and t compressed by singleton tree grammars, the task is to find an instance of s that occurs as a subterm in t. We show that this problem is in NP and that the task can be performed in time O(ncjVar(s)j), including the construction of the compressed substitution, and a representation of all occurrences. We show that the special case where s is uncompressed can be performed in polynomial time. As a nice application we show that for an equational deduction of t to t0 by an equality axiom l = r (a rewrite) a single step can be performed in polynomial time in the size of compression of t and l; r if the number of variables is fixed in l. We also show that n rewriting steps can be performed in polynomial time, if the equational axioms are compressed and assumed to be constant for the rewriting sequence. Another potential application are querying mechanisms on compressed XML-data bases

A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments

Computationally efficient induction of classification rules with the PMCRI and J-PMCRI frameworks

In order to gain knowledge from large databases, scalable data mining technologies are needed. Data are captured on a large scale and thus databases are increasing at a fast pace. This leads to the utilisation of parallel computing technologies in order to cope with large amounts of data. In the area of classification rule induction, parallelisation of classification rules has focused on the divide and conquer approach, also known as the Top Down Induction of Decision Trees (TDIDT). An alternative approach to classification rule induction is separate and conquer which has only recently been in the focus of parallelisation. This work introduces and evaluates empirically a framework for the parallel induction of classification rules, generated by members of the Prism family of algorithms. All members of the Prism family of algorithms follow the separate and conquer approach.are increasing at a fast pace. This leads to the utilisation of parallel computing technologies in order to cope with large amounts of data. In the area of classification rule induction, parallelisation of classification rules has focused on the divide and conquer approach, also known as the Top Down Induction of Decision Trees (TDIDT). An alternative approach to classification rule induction is separate and conquer which has only recently been in the focus of parallelisation. This work introduces and evaluates empirically a framework for the parallel induction of classification rules, generated by members of the Prism family of algorithms. All members of the Prism family of algorithms follow the separate and conquer approach

Efficient Pattern Matching in Python

Pattern matching is a powerful tool for symbolic computations. Applications include term rewriting systems, as well as the manipulation of symbolic expressions, abstract syntax trees, and XML and JSON data. It also allows for an intuitive description of algorithms in the form of rewrite rules. We present the open source Python module MatchPy, which offers functionality and expressiveness similar to the pattern matching in Mathematica. In particular, it includes syntactic pattern matching, as well as matching for commutative and/or associative functions, sequence variables, and matching with constraints. MatchPy uses new and improved algorithms to efficiently find matches for large pattern sets by exploiting similarities between patterns. The performance of MatchPy is investigated on several real-world problems