The algorithmics of solitaire-like games

One-person solitaire-like games are explored with a view to using them in teaching algorithmic problem solving. The key to understanding solutions to such games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a novel class of one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games, which we call ``replacement-set games'', inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We present an algorithm to solve arbitrary instances of replacement-set games and we show various ways of constructing infinite (solvable) classes of replacement-set games

Scalable Task Schedulers for Many-Core Architectures

This thesis develops schedulers for many-cores with different optimization objectives. The proposed schedulers are designed to be scale up as the number of cores in many-cores increase while continuing to provide guarantees on the quality of the schedule

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Simulation in Algorithmic Self-assembly

Winfree introduced a model of self-assembling systems called the abstract Tile Assembly Model (aTAM) where square tiles with glues on their edges attach spontaneously via matching glues to form complex structures. A generalization of the aTAM called the 2HAM (two-handed aTAM) not only allows for single tiles to bind, but also for supertile assemblies consisting of any number of tiles to attach. We consider a variety of models based on either the aTAM or the 2HAM. The underlying commonality of the work presented here is simulation. We introduce the polyTAM, where a tile system consists of a collection of polyomino tiles, and show that for any polyomino P of size greater than or equal to 3 and any Turing machine M , there exists a temperature-1 polyTAM system containing only shape-P tiles that simulates M . We introduce the RTAM (Reflexive Tile Assembly Model) that works like the aTAM except that tiles can nondeterministically flip prior to binding. We show that the temperature-1 RTAM cannot simulate a Turing machine by showing the much stronger result that the RTAM can only self-assemble periodic patterns. We then define notions of simulation which serve as relations between two tile assembly systems (possibly belonging to different models). Using simulation as a basis of comparison, we first show that cellular automata and the class of all tile assembly systems in the aTAM are equivalent. Next, we introduce the Dupled aTAM (DaTAM) and show that the temperature-2 aTAM and the temperature-1 DaTAM are mutually exclusive by showing that there is an aTAM system that cannot be simulated by any DaTAM system, and vice versa. Third, we consider the restricted glues Tile Assembly Model (rgTAM) and show that there is an aTAM system that cannot be simulated by any rgTAM system. We introduce the Dupled restricted glues Tile Assembly Model (DrgTAM), and show that the DrgTAM is intrinsically universal for the aTAM. Finally, we consider a variation of the Signal-passing Tile Assembly Model (STAM) called the STAM+ and show that the STAM+ is intrinsically universal and that the 3-D 2HAM is intrinsically universal for the STAM+

An optimizational approach for an algorithmic reassembly of fragmented objects

In Cambodia close to the Thai border, lies the Angkor-style temple of Banteay Chhmar. Like all nearly forgotten temples in remote places, it crumbles under the ages. By today most of it is only a heap of stones. Manually reconstructing these temples is both complex and challenging: The conservation team is confronted with a pile of stones, the original position of which is generally unkown. This reassembly task resembles a large-scale 3D puzzle. Usually, it is resolved by a team of specialists who analyze each stone, using their experience and knowledge of Khmer culture. Possible solutions are tried and retried and the stones are placed in different locations until the correct one is found. The major drawbacks of this technique are: First, since the stones are moved continuously they are further damaged, second, there is a threat to the safety of the workers due to handling very heavy weights, and third because of the high complexity and labour-intensity of the work it takes several months up to several years to solve even a small part of the puzzle. These risks and conditions motivated the development of a virtual approach to reassemble the stones, as computer algorithms are theoretically capable of enumerating all potential solutions in less time, thereby drastically reducing the amount of work required for handling the stones. Furthermore the virtual approach has the potential to reduce the on-site costs of in-situ analysis. The basis for this virtual puzzle algorithm are high-resolution 3D models of more than one hundred stones. The stones can be viewed as polytopes with approximately cuboidal form although some of them contain additional indentations. Exploiting these and related geometric features and using a priori knowledge of the orientation of each stone speeds up the process of matching the stones. The aim of the current thesis is to solve this complex large-scale virtual 3D puzzle. In order to achieve this, a general workflow is developed which involves 1) to simplify the high-resolution models to their most characteristic features, 2) apply an advanced similarity analysis and 3) to match best combinations as well as 4) validate the results. The simplification step is necessary to be able to quickly match potential side-surfaces. It introduces the new concept of a minimal volume box (MVB) designed to closely and storage efficiently resemble Khmer stones.Additionally, this reduced edge-based model is used to segment the high-resolution data according to each side-surface. The second step presents a novel technique allowing to conduct a similarity analysis of virtual temple stones. It is based on several geometric distance functions which determine the relatedness of a potential match and is capable of sorting out unlikely ones. The third step employs graph theoretical methods to combine the similarity values into a correct solution of this large-scale 3D puzzle. The validation demonstrates the high quality and robustness of this newly constructed puzzle workflow. The workflow this thesis presents virtually puzzles digitized stones of fallen straight Khmer temple walls. It is able to virtually and correctly reasemble up to 42 digitized stones requiring a minimum of user-interaction

Motion planning for self-reconfiguring robotic systems

Robots that can actively change morphology offer many advantages over fixed shape, or monolithic, robots: flexibility, increased maneuverability and modularity. So called self-reconfiguring systems (SRS) are endowed with a shape changing ability enabled by an active connection mechanism. This mechanism allows a mechanical link to be engaged or disengaged between two neighboring robotic subunits. Through utilization of embedded joints to change the geometry plus the connection mechanism to change the topology of the kinematics, a collection of robotic subunits can drastically alter the overall kinematics. Thus, an SRS is a large robot comprised of many small cooperating robots that is able to change its morphology on demand. By design, such a system has many and variable degrees of freedom (DOF). To gain the benefits of self-reconfiguration, the process of morphological change needs to be controlled in response to the environment. This is a motion planning problem in a high dimensional configuration space. This problem is complex because each subunit only has a few internal DOFs, and each subunit's range of motion depends on the state of its connected neighbors. Together with the high dimensionality, the problem may initially appear to be intractable, because as the number of subunits grow, the state space expands combinatorially. However, there is hope. If individual robotic subunits are identical, then there will exist some form of regularity in the resulting state space of the conglomerate. If this regularity can be exploited, then there may exist tractable motion planning algorithms for self-reconfiguring system. Existing approaches in the literature have been successful in developing algorithms for specific SRSs. However, it is not possible to transfer one motion planning algorithm onto another system. SRSs share a similar form of regularity, so one might hope that a tool from mathematical literature would identify the common properties that are exploitable for motion planning. So, while there exists a number of algorithms for certain subsets of possible SRS instantiations, there is no general motion planning methodology applicable to all SRSs. In this thesis, firstly, the best existing general motion planning techniques were evaluated to the SRS motion planning problem. Greedy search, simulated annealing, rapidly exploring random trees and probabilistic roadmap planning were found not to scale well, requiring exponential computation time, as the number of subunits in the SRS increased. The planners performance was limited by the availability of a good general purpose heuristic. There does not currently exist a heuristic which can accurately guide a path through the search space toward a far away goal configuration. Secondly, it is shown that a computationally efficient reconfiguration algorithms do exist by development of an efficient motion planning algorithm for an exemplary SRS, the Claytronics formulation of the Hexagonal Metamorphic Robot (HMR). The developed algorithm was able to solve a randomly generated shape-to-shape planning task for the SRS in near linear time as the number of units in the configuration grew. Configurations containing 20,000 units were solvable in under ten seconds on modest computational hardware. The key to the success of the approach was discovering a subspace of the motion planning space that corresponded with configurations with high mobility. Plans could be discovered in this sub-space much more readily because the risk of the search entering a blind alley was greatly reduced. Thirdly, in order to extract general conclusions, the efficient subspace, and other efficient subspaces utilized in other works, are analyzed using graph theoretic methods. The high mobility is observable as an increase in the state space's Cheeger constant, which can be estimated with a local sampling procedure. Furthermore, state spaces associated with an efficient motion planning algorithm are well ordered by the graph minor relation. These qualitative observations are discoverable by machine without human intervention, and could be useful components in development of a general purpose SRS motion planner compiler

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Hierarchical Variance Reduction Techniques for Monte Carlo Rendering

Ever since the first three-dimensional computer graphics appeared half a century ago, the goal has been to model and simulate how light interacts with materials and objects to form an image. The ultimate goal is photorealistic rendering, where the created images reach a level of accuracy that makes them indistinguishable from photographs of the real world. There are many applications ñ visualization of products and architectural designs yet to be built, special effects, computer-generated films, virtual reality, and video games, to name a few. However, the problem has proven tremendously complex; the illumination at any point is described by a recursive integral to which a closed-form solution seldom exists. Instead, computer simulation and Monte Carlo methods are commonly used to statistically estimate the result. This introduces undesirable noise, or variance, and a large body of research has been devoted to finding ways to reduce the variance. I continue along this line of research, and present several novel techniques for variance reduction in Monte Carlo rendering, as well as a few related tools. The research in this dissertation focuses on using importance sampling to pick a small set of well-distributed point samples. As the primary contribution, I have developed the first methods to explicitly draw samples from the product of distant high-frequency lighting and complex reflectance functions. By sampling the product, low noise results can be achieved using a very small number of samples, which is important to minimize the rendering times. Several different hierarchical representations are explored to allow efficient product sampling. In the first publication, the key idea is to work in a compressed wavelet basis, which allows fast evaluation of the product. Many of the initial restrictions of this technique were removed in follow-up work, allowing higher-resolution uncompressed lighting and avoiding precomputation of reflectance functions. My second main contribution is to present one of the first techniques to take the triple product of lighting, visibility and reflectance into account to further reduce the variance in Monte Carlo rendering. For this purpose, control variates are combined with importance sampling to solve the problem in a novel way. A large part of the technique also focuses on analysis and approximation of the visibility function. To further refine the above techniques, several useful tools are introduced. These include a fast, low-distortion map to represent (hemi)spherical functions, a method to create high-quality quasi-random points, and an optimizing compiler for analyzing shaders using interval arithmetic. The latter automatically extracts bounds for importance sampling of arbitrary shaders, as opposed to using a priori known reflectance functions. In summary, the work presented here takes the field of computer graphics one step further towards making photorealistic rendering practical for a wide range of uses. By introducing several novel Monte Carlo methods, more sophisticated lighting and materials can be used without increasing the computation times. The research is aimed at domain-specific solutions to the rendering problem, but I believe that much of the new theory is applicable in other parts of computer graphics, as well as in other fields