Bruce E. Litow

Division in logspace-uniform NC1

Beame, Cook and Hoover were the first to exhibit a log-depth, polynomial size circuit family for integer division. However, the family was not logspace-uniform. In this paper we describe log-depth, polynomial size, logspace-uniform, i.e. , NC 1 circuit family for integer division. In particular, by a well-known result this shows that division is in logspace. We also refine the method of the paper to show that division is in dlogtime-uniform NC 1 .

Partition-distance via the assignment problem

MOTIVATION Accuracy testing of various pedigree reconstruction methods requires an efficient algorithm for the calculation of distance between a known partition and its reconstruction. The currently used algorithm of Almudevar and Field takes a prohibitively long time for certain partitions and population sizes. RESULTS We present an algorithm that very efficiently reduces the partition-distance calculation to the classic assignment problem of weighted bipartite graphs that has known polynomial-time solutions. The performance of the algorithm is tested against the Almudevar and Field partition-distance algorithm to verify the significant improvement in speed. AVAILABILITY Computer code written in java is available upon request from the first author.

Modified SIMPSON O(n3) algorithm for the full sibship reconstruction problem

MOTIVATION The problem of reconstructing full sibling groups from DNA marker data remains a significant challenge for computational biology. A recently published heuristic algorithm based on Mendelian exclusion rules and the Simpson index was successfully applied to the full sibship reconstruction (FSR) problem. However, the so-called SIMPSON algorithm has an unknown complexity measure, questioning its applicability range. RESULTS We present a modified version of the SIMPSON (MS) algorithm that behaves as O(n(3)) and achieves the same or better accuracy when compared with the original algorithm. Performance of the MS algorithm was tested on a variety of simulated diploid population samples to verify its complexity measure and the significant improvement in efficiency (e.g. 100 times faster than SIMPSON in some cases). It has been shown that, in theory, the SIMPSON algorithm runs in non-polynomial time, significantly limiting its usefulness. It has been also verified via simulation experiments that SIMPSON could run in O(n(a)), where a > 3. AVAILABILITY Computer code written in Java is available upon request from the first author. CONTACT Dmitry.Konovalov@jcu.edu.au.

Similarity Enrichment in Image Compression through Weighted Finite Automata

We propose and study in details a similarity enrichment scheme for the application to the image compression through the extension of the weighted finite automata (WFA). We then develop a mechanism with which rich families of legitimate similarity images can be systematically created so as to reduce the overall WFA size, leading to an eventual better WFA-based compression performance. A number of desirable properties, including WFA of minimum states, have been established for a class of packed WFA. Moreover, a codec based on a special extended WFA is implemented to exemplify explicitly the performance gain due to extended WFA under otherwise the same conditions.

Why step when you can run? Iterative line digitization algorithms based on hierarchies of runs

One challenge for computer graphics and scientific visualization is to develop techniques to more effectively display and analyze the vast amounts of information we are collating, a race we are currently losing. To meet this goal, a key initiative suggests moving away from small-scale, low-resolution displays to immersive, high-fidelity systems. These environments permit higher levels of interactivity and exploration, but require more power and bandwidth than current graphics systems can deliver. The move to larger and/or higher resolution displays will place a greater emphasis on the digitization algorithms we employ for even the simplest geometrical primitives. In this tutorial, we present a technique to describe and digitize the line as a set of runs, runs of runs, runs of runs of runs, and so on, in fact, as any level of runs within the full hierarchy of runs in the digital line. The digitization algorithms we present apply to a broad range of resolutions and applications, including geometric scan conversion, ray traversal, linear mapping and interpolation.

Running the line: Line drawing using runs and runs of runs

Abstract The efficiency of line drawing algorithms underwrites the performance of many rendering and visualisation systems. Therefore to significantly improve the process of line drawing, techniques have been developed to draw the line not pixel by pixel but by using higher order primitives such as runs and run length slices. In this paper we present a line drawing algorithm based on runs of runs, which is the next step in this progression. We will also discuss a number of special cases in the structure of runs and runs of runs within the line that can be used to short circuit the drawing process. These special cases can be used to help counter a common criticism that run-based techniques are less applicable for very short lines. In fact we will argue that the use of higher order primitives provides additional structural information that can be used to accelerate secondary processes within the graphics system, such as within the raster memory management.

Unification and extension of weighted finite automata applicable to image compression

Weighted finite automata (WFA), including the linear WFA due to Culik and Kari and the acyclic WFA due to Hafner, have been under investigation over the years for their applications to image compression. We shall in this work first examine in great details the underlying WFA structure and propose the most systematic extension, along with its full legitimacy analysis, to the WFA that are applicable to image compression. A new mechanism based on the concept of resolution-wise and resolution-driven image mappings is developed to create rich families of legitimate similarity images so as to reduce the overall WFA size, a property that is critically related the performance of WFA-based compression codecs. Moreover, we shall also unify the relevant WFA by showing an acyclic WFA can always be merged into a linear WFA but not vice versa.

On iterated integer product

Abstract IT denotes computation of z = ∏ni=1zi where the zi are n-bit integers. IT is not known to be in DSPA CE(log n). We show that computation of both the high order and low order log n bits of z is in DSPACE(log n).

Making the DDA run: two-dimensional ray traversal using runs and runs of runs

Iterative algorithms based on runs, and runs of runs are presented to calculate the cells of the two-dimensional lattice intersected by a line of real slope and intercept. The technique is applied to the problem of traversing a ray through a two-dimensional grid. Using runs or runs of runs provides a significant improvement in the efficiency of ray traversal for all but very short path lengths when compared to the DDA algorithm implemented using floating or fixed point arithmetic.

Fast arithmetics using Chinese remaindering

In this paper, some issues concerning the Chinese remaindering representation are discussed. A new converting method, which is an efficient probabilistic algorithm based on a recent result of von zur Gathen and Shparlinski [J. von zur Gathen, I. Shparlinski, GCD of random linear forms, Algorithmica 46 (2006) 137-148], is described. An efficient refinement of the NC^1 division algorithm of Chiu, Davida and Litow [A. Chiu, G. Davida, B. Litow, Division in logspace-uniform NC^1, Theoret. Informatics Appl. 35 (2001) 259-275] is given, where the number of moduli is reduced by a factor of logn.