Problems In Computational Geometry Research Articles

Optimal Algorithm for the Planar Two-Center Problem

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set $S$ of $n$ points in the plane and the goal is to find two smallest congruent disks whose union contains all points of $S$. A longstanding open problem has been to obtain an $O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log n)$ lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in $O(n\log^2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Elastic-Degenerate String Matching with 1 Error or Mismatch

An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O~(nmω-1)+O(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\ ilde{O}}(nm^{\\omega -1})+\\mathcal {O}(N)$$\\end{document}-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} denotes the matrix multiplication exponent and the O~(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\ ilde{O}}(\\cdot )$$\\end{document} notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O(k2mG+kN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k^2mG+kN)$$\\end{document} time, under edit distance, or O(kmG+kN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(kmG+kN)$$\\end{document} time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N, and so even for k=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=1$$\\end{document}, the existing algorithms run in Ω(mN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varOmega (mN)$$\\end{document} time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O((nm2+N)logm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}((nm^2 + N)\\log m)$$\\end{document} or O(nm3+N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^3 + N)$$\\end{document} time under edit distance. For the decision version of the problem, we present a faster O(nm2logm+Nloglogm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^2\\sqrt{\\log m} + N\\log \\log m)$$\\end{document}-time algorithm. We also show that 1-EDSM can be solved in O(nm2+Nlogm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^2 + N\\log m)$$\\end{document} time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022.

Parallel intersection counting on shared-memory multiprocessors and GPUs

Computing intersections among sets of one-dimensional intervals is an ubiquitous problem in computational geometry with important applications in bioinformatics, where the size of typical inputs is large and it is therefore important to use efficient algorithms. In this paper we propose a parallel algorithm for the 1D intersection-counting problem, that is, the problem of counting the number of intersections between each interval in a given set A and every interval in a set B. Our algorithm is suitable for shared-memory architectures (e.g., multicore CPUs) and GPUs. The algorithm is work-efficient because it performs the same amount of work as the best serial algorithm for this kind of problem. Our algorithm has been implemented in C++ using the Thrust parallel algorithms library, enabling the generation of optimized programs for multicore CPUs and GPUs from the same source code. The performance of our algorithm is evaluated on synthetic and real datasets, showing good scalability on different generations of hardware.

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Parallel Vertex Diffusion for Unified Visual Grounding

Unified visual grounding (UVG) capitalizes on a wealth of task-related knowledge across various grounding tasks via one-shot training, which curtails retraining costs and task-specific architecture design efforts. Vertex generation-based UVG methods achieve this versatility by unified modeling object box and contour prediction and provide a text-powered interface to vast related multi-modal tasks, e.g., visual question answering and captioning. However, these methods typically generate vertexes sequentially through autoregression, which is prone to be trapped in error accumulation and heavy computation, especially for high-dimension sequence generation in complex scenarios. In this paper, we develop Parallel Vertex Diffusion (PVD) based on the parallelizability of diffusion models to accurately and efficiently generate vertexes in a parallel and scalable manner. Since the coordinates fluctuate greatly, it typically encounters slow convergence when training diffusion models without geometry constraints. Therefore, we consummate our PVD by two critical components, i.e., center anchor mechanism and angle summation loss, which serve to normalize coordinates and adopt a differentiable geometry descriptor from the point-in-polygon problem of computational geometry to constrain the overall difference of prediction and label vertexes. These innovative designs empower our PVD to demonstrate its superiority with state-of-the-art performance across various grounding tasks.

Intersection Properties of Finite Disk Collections

In this article, we study the intersection of a finite collection of disks in Euclidean space by examining spheres of various dimensions and their poles (extreme values with respect to canonical projections) contained within the intersection’s boundary. We derive explicit formulae for computing these extreme values and present two applications. The first application involves computing the smallest common rescaling factor for the radii of the disk system, which brings the system to a single point of intersection. This calculation allows us to compute the generalized Čech filtration, a crucial tool for the topological data analysis of weighted point clouds. The second application focuses on determining the minimal Axis-Aligned Bounding Box (AABB) for the intersection of a finite collection of disks in Euclidean space, addressing a significant problem in computational geometry. We consider that this work aims to contribute to the fields of topological data analysis and computational geometry by providing new tools for analyzing complex geometric structures.

A digital geometric approach for discrete Voronoi diagram construction using GPU

SummaryVoronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites.

THE MEDIAN TREE

Mathematical models in which objects have a finite discrete structure are used to solve a wide range of practical problems. Such are, for example, many problems in auto-matic design, operation research, control theory, network planning, and computational geometry. Often, oriented or non-oriented graphs are considered as such models, which can adequately describe the structure of the problem of interest to us. Among these tasks, a certain place is occupied by tasks related to network issues of optimal placement of ware-houses, certain service points, emergency, medical services, etc. When studying such issues, it becomes necessary to identify, in a certain sense, the central objects in the networks un-der consideration, and in order to find them, the concept of the shortest path between these two given elements plays an essential role. A definition of the medial vertex of a graph is provided in the article – the vertex whose sum of distances to the others is minimal. The median of a non-oriented tree con-sisting of medial vertices is considered, and it is proved that the median of each such tree has either one vertex or two connected by an edge. This result is a definite generalization of a similar property of the center of a tree consisting of central vertices. The question of the relationship between the center and median of the same tree is also studied. The examples of trees with ten vertices show that such a direct connection does not exist, and there can be all four logically possible cases: the same tree can contain one or two central and one or two medial vertices.

Parallel window propagation on mesh surfaces

The task of calculating the shortest distance with minimal curvature between two points on a mesh surface is a well-known problem in computational geometry. This paper aims to develop a effective algorithm for computing geodesics on large, complex models with fast processing times to facilitate interactive applications. Our contribution is developing of the parallel window propagation (PWP) algorithm, which divides the sequential MMP algorithm into four phases: front node selection, window list propagation, window list merging, and vertex update. We use two separate buffers for incoming windows on each edge to avoid data dependency and conflicts in each phase, allowing for parallel processing on a CPU. As a result, our PWP algorithm can propagate a large number of windows simultaneously and independently, leading to significant improvements in performance for real-world models.

Hexagon-based adaptive hierarchies for efficient point-in-spherical-polygon tests on GPUs

Point-in-spherical-polygon tests are a fundamental problem in computational geometry. For such tests, efficiency is much required for many global information processing matters, especially their real-time processing. Though many efforts have been made, it is still challenging due to the constraints from the non-Euclidean space of the sphere. Recently, a method is proposed to construct an Adaptive Hexagonal Hierarchical Grid (AHHG) to manage spherical polygon edges, by which the non-Euclidean space constraint of the sphere can be well handled to improve point-in-spherical-polygon tests. However, this method is very expensive in the construction of AHHGs, which prevents its use in practice. In this paper, we present novel measures to divide the task of AHHG construction into several subtasks to solve the problems that arise in the parallel construction of incongruent adaptive hierarchies, so we can exploit the parallel computing potential of GPUs for acceleration. We also develop measures to adaptively optimize the hierarchical levels of an AHHG for high efficiency. Experimental results show that we can answer 1,000,000 query points against dynamically varying spherical polygons in over 100,000 edges in real time on a personnel computer, where AHHGs for spherical polygons are individually constructed. This is much superior to the existing methods.

Approximating the parallel transport of an induced connection

Efficient numerical methods to approximate the parallel transport operators of the induced connection on a sub-bundle of a vector bundle are presented. These methods are simpler than naive applications of a Runge–Kutta algorithm and have accuracy up to order 4. They have the desirable property of being insensitive to choices of trivialization of the sub-bundle. The methods were developed to solve a problem of computing skyrmions using the Atiyah–Manton–Sutcliffe and Atiyah–Drinfeld–Hitchin–Manin constructions, but are applicable to a broader range of problems in computational geometry.

A New Algorithm for Euclidean Shortest Paths in the Plane

Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (inSIAM Journal on Computing, 1999) gave an algorithm ofO(nlogn) time andO(nlogn) space, wherenis the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space toO(n)while the runtime of the algorithm is stillO(nlogn). In this article, we present a new algorithm ofO(n+hlogh) time andO(n)space, provided that a triangulation of the free space is given, wherehis the number of obstacles. The algorithm is better than the previous work whenhis relatively small. Our algorithm builds a shortest path map for a source pointsso that given any query pointt, the shortest path length fromstotcan be computed inO(logn) time and a shortests-tpath can be produced in additional time linear in the number of edges of the path.

An Efficient GNSS Coordinate Recognition Algorithm for Epidemic Management

Many highly contagious infectious diseases, such as COVID-19, monkeypox, chickenpox, influenza, etc., have seriously affected or currently are seriously affecting human health, economic activities, education, sports, and leisure. Many people will be infected or quarantined when an epidemic spreads in specific areas. These people whose activities must be restricted due to the epidemic are represented by targets in the article. Managing targets by using targeted areas is an effective option for slowing the spread. The Centers for Disease Control (CDC) usually determine management strategies by tracking targets in specific areas. A global navigation satellite system (GNSS) that can provide autonomous geospatial positioning of targets by using tiny electronic receivers can assist in recognition. The recognition of targets within a targeted area is a point-in-polygon (PtInPy) problem in computational geometry. Most previous methods used the method of identifying one target at a time, which made them unable to effectively deal with many targets. An earlier method was able to simultaneously recognize several targets but had the problem of the repeated recognition of the same targets. Therefore, we propose a GNSS coordinate recognition algorithm. This algorithm can efficiently recognize a large number of targets within a targeted area, which can provide assistance in epidemic management.

Faster Algorithms for Largest Empty Rectangles and Boxes

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time \(O(n\log ^2n)\) for \(d=2\) (Aggarwal and Suri 1987) and near \(n^d\) for \(d\ge 3\). We describe faster algorithms with the following running times (where \(\varepsilon >0\) is an arbitrarily small constant and \(\widetilde{O}\) hides polylogarithmic factors): \(n2^{O({\log }^*n)}\log n\) for \(d=2\), \(O(n^{2.5+\varepsilon })\) for \(d=3\), and \(\widetilde{O}(n^{(5d+2)/6})\) for any constant \(d\ge 4\). To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee’s measure problem to optimize certain objective functions over the complement of a union of orthants.

Algorithm to support decision-making regarding the placement of reserve units and logistical support of military formations in the area according to the minimax criterion

The scientific issues of the article are aimed at the development of an algorithm for solving one of the important in practice of the class of problems of computational geometry - optimization. This task is related to the optimal placement of reserve units and logistical support of the military formation in the area according to the minimax criterion, which will be the basis for making a decision on the formation of a battle order and will ensure the minimization of the maximum response time to a change in the situation and the time of evacuation (carriage) in the conditions of increasing dynamic and spatial indicators of hostilities.
 The essence of the study is to choose a method of optimizing the function of the set of concentric circles based on its majorization by a paraboloid to solve the problem of placing a reserve unit and logistical support according to the minimax criterion and developing, based on it, an algorithm to support decision-making regarding the formation of the order of battle of a military formation in the area.
 Methods of optimization, computational geometry and algorithmization were used to solve the problem.
 The result of the work is a developed algorithm for finding the minimax point on the terrain in the area of hostilities relative to the units of the military formation and the defined boundaries.
 The implementation of the algorithm in the decision-making support system will ensure an increase in the efficiency and reasonableness of decision-making regarding the positioning of reserve units and logistical support in the overall construction of the order of battle of the military formation.

Time and space efficient collinearity indexing

The collinearity testing problem is a basic problem in computational geometry, in which, given three sets A, B, C in the plane, of n points each, the task is to detect a collinear triple of points in A×B×C or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the collinearity indexing problem, in which we are given two sets A and B, each of n points in the plane, and our goal is to preprocess A and B into a data structure, so that, for any query point q∈R2, we can determine whether q is collinear with a pair of points (a,b)∈A×B. We provide a solution to the problem for the case where the points of A, B lie on an integer grid, and the query points lie on a vertical line, with a data structure of subquadratic storage and sublinear query time. We then extend our result to the case where the query points lie on the graph of a polynomial of constant degree. Our solution is based on the function-inversion technique of Fiat and Naor [11].

Conditioning problems for invariant sets of expanding piecewise affine mappings: application to loss of ergodicity in globally coupled maps

We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes’ localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of piecewise linear inequalities for some matrices which encapsulate geometric constraints. In that way, the original topological puzzle is converted into a standard problem in computational geometry. This transformation involves an optimisation procedure that ensures that both problems are equivalent. As a proof of concept, the approach is applied to the study of the loss of ergodicity in basic examples of globally coupled maps. The study explains, completes and substantially extends previous achievements about asymmetric IUP in these systems. Comparison with the numerics reveals sharp existence conditions depending on the map parameters, and accurate fits of the empirical ergodic components. In addition, this application also reveals unanticipated features about conditioning problem solutions, especially as the dependence on the set of admissible face directions is concerned.

A Minimax Model for Generalized Penetration Distance Between Convex Sets by Directed Hausdorff Distance

The penetration distance, which characterizes the depth of intersection between two sets, is a fundamental computational geometry problem with a wide variety of applications in image processing, robotics, and circuit design. However, the penetration distance (even in Euclidean metric) is typically challenging to handle because it is essentially a nonconvex nonlinear optimization involving the projection onto the complement of a convex set. In this letter, we explore the penetration distance in non-Euclidean metrics (e.g., Manhattan, Chebyshev, and hexagon distances) by deploying the gauge function in convex analysis, and reformulate it as a nonlinear equation by virtue of the directed Hausdorff distance. More precisely, the generalized penetration distance amounts to seeking the largest zero point of a nondecreasing convex “black-box” function, whose function values can be obtained by the recent algorithmic advances in minimax optimization. We develop a solver for calculating generalized penetration distance between compact convex sets (possibly non-polyhedron) in arbitrary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> dimension. Numerical experiments on variant types of compact convex sets demonstrate that the proposed solver is compelling to render solutions with high accuracy.

Efficient optimization-based quadrature for variational discretization of nonlocal problems

Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius δ, where δ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element–ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius δ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of h∼δ→0, featuring at least first-order convergence in L2 for all dimensions, using both uniform and nonuniform grids. Moreover, in the case of uniform grids, the proposed method passes a patch test and, according to numerical evidence, exhibits an optimal, second-order convergence rate. Our numerical tests also indicate that, even for nonuniform grids, second-order convergence can be observed over a substantial pre-asymptotic regime.

A Parallelizable Integer Linear Programming Approach for Tiling Finite Regions of the Plane with Polyominoes

The general problem of tiling finite regions of the plane with polyominoes is 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.