Search Results (8)

The general problem of tiling finite regions of the plane with polyominoes is $\mathcal{NP}$-complete, and so the associated computational geometry problem rapidly becomes intractable for large instances. Thus, the need to reduce algorithm complexity for tiling is important and continues as a fruitful area of research. Traditional approaches to tiling with polyominoes use backtracking, which is a refinement of the ‘brute-force’ solution procedure for exhaustively finding all solutions to a combinatorial search problem. In this work, we combine checkerboard colouring techniques with a recently introduced integer linear programming (ILP) technique for tiling with polyominoes. The colouring arguments often split large tiling problems into smaller subproblems, each represented as a separate ILP problem. Problems that are amenable to this approach are embarrassingly parallel, and our work provides proof of concept of a parallelizable algorithm. The main goal is to analyze when this approach yields a potential parallel speedup. The novel colouring technique shows excellent promise in yielding a parallel speedup for finding large tiling solutions with ILP, particularly when we seek a single (optimal) solution. We also classify the tiling problems that result from applying our colouring technique according to different criteria and compute representative examples using a combination of MATLAB and CPLEX, a commercial optimization package that can solve ILP problems. The collections of MATLAB programs PARIOMINOES (v3.0.0) and POLYOMINOES (v2.1.4) used to construct the ILP problems are freely available for download. Full article

Checkerboard colouring arguments for proving that a given collection of polyominoes cannot tile a finite target region of the plane are well-known and typically applied on a case-by-case basis. In this article, we give a systematic mathematical treatment of such colouring arguments, based on the concept of a parity violation, which arises from the mismatch between the colouring of the tiles and the colouring of the target region. Identifying parity violations is a combinatorial problem related to the subset sum problem. We convert the combinatorial problem into linear Diophantine equations and give necessary and sufficient conditions for a parity violation. The linear Diophantine equation approach leads to an algorithm implemented in MATLAB for finding all possible parity violations of large tiling problems, and is the main contribution of this article. Numerical examples illustrate the effectiveness of our algorithm. The collection of MATLAB programs, POLYOMINO_PARITY (v2.0.0) is freely available for download. Full article

In this study, we aim to optimize and improve the efficiency of a Tetris-inspired reconfigurable cleaning robot. Multi-criteria decision making (MCDM) is utilized as a powerful tool to target this aim by introducing the best solution among others in terms of lower energy consumption and greater area coverage. Regarding the Tetris-inspired structure, polyomino tiling theory is utilized to generate tiling path-planning maps which are evaluated via MCDM to seek a solution that can deliver the best balance between the two mentioned key issues; energy and area coverage. In order to obtain a tiling area that better meets the requirements of polyomino tiling theorems, first, the whole area is decomposed into five smaller sub-areas based on furniture layout. Afterward, four tetromino tiling theorems are applied to each sub-area to give the tiling sets that govern the robot navigation strategy in terms of shape-shifting tiles. Then, the area coverage and energy consumption are calculated and eventually, these key values are considered as the decision criteria in a MCDM process to select the best tiling set in each sub-area, and following the aggregation of best tiling path-plannings, the robot navigation is oriented towards efficiency and improved optimality. Also, for each sub-area, a preference order for the tiling sets is put forward. Based on simulation results, the tiling theorem that can best serve all sub-areas turns out to be the same. Moreover, a comparison between a fixed-morphology mechanism with the current approach further advocates the proposed technique. Full article

The efficiency of autonomous systems that tackle tasks such as home cleaning, agriculture harvesting, and mineral mining depends heavily on the adopted area coverage strategy. Extensive navigation strategies have been studied and developed, but few focus on scenarios with reconfigurable robot agents. This paper proposes a navigation strategy that accomplishes complete path planning for a Tetris-inspired hinge-based self-reconfigurable robot (hTetro), which consists of two main phases. In the first phase, polyomino form-based tilesets are generated to cover the predefined area based on the tiling theory, which generates a series of unsequenced waypoints that guarantee complete coverage of the entire workspace. Each waypoint specifies the position of the robot and the robot morphology on the map. In the second phase, an energy consumption evaluation model is constructed in order to determine a valid strategy to generate the sequence of the waypoints. The cost value between waypoints is formulated under the consideration of the hTetro robot platform’s kinematic design, where we calculate the minimum sum of displacement of the four blocks in the hTetro robot. With the cost function determined, the waypoint sequencing problem is then formulated as a travelling salesman problem (TSP). In this paper, a genetic algorithm (GA) is proposed as a strong candidate to solve the TSP. The GA produces a viable navigation sequence for the hTetro robot to follow and to accomplish complete coverage tasks. We performed an analysis across several complete coverage algorithms including zigzag, spiral, and greedy search to demonstrate that TSP with GA is a valid and considerably consistent waypoint sequencing strategy that can be implemented in real-world hTetro robot navigations. The scalability of the proposed framework allows the algorithm to produce reliable results while navigating within larger workspaces in the real world, and the flexibility of the framework ensures easy implementation of the algorithm on other polynomial-based shape shifting robots. Full article

Whilst Polyomino tiling theory has been extensively studied as a branch of research in mathematics, its application has been largely confined to multimedia, graphics and gaming domains. In this paper, we present a novel application of Tromino tiling theory, a class of Polyomino with three cells in the context of a reconfigurable floor cleaning robot, hTromo. The developed robot platform is able to automatically generate a global tiling set required to cover a defined space while leveraging on the Tromino tiling theory. Specifically, we validated the application of five Tromino tiling theorems with our hTromo robot. Experiments performed clearly demonstrate the efficacy of the proposed approach resulting in very high levels of area coverage performance in all considered experimental cases. This paper also presents the system architecture of our hTromo robot and a detailed description of the five tiling theorems applied in this study. Full article

In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure. Full article

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3]. Full article

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D₂ among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5]. Full article