Effectiveness of A* Algorithm for Constrained Tiling

Abstract

The ‘Mondrian Tiling’ problem is a particular class of constraint optimization problem where a square grid is covered with some non-overlapping integer dimension rectangles, which must not be pairwise congruent. The objective is to use rectangles with similar areas so that difference between the largest and smallest rectangle area, known as score, is minimum. A brute-force approach towards solving this problem enumerates all possible solutions to find the optimal one and incorporates two prevalent NP-Complete problems, making it an exponential algorithm and computationally challenging to compute for large grids. In contrast, our proposed approach employs grids as states and applies a number of (specifically, 4) state transformation operations to improve the states in terms of score. The state-space representation is utilised to explore the states with some strategy to obtain the optimal one. A number of restrictions are applied with the purpose of obtaining a balance between exploration and exploitation of the state-space. The results of our experiments exhibit that the recommended approach is profoundly efficient compared to the former approach, and the obtained scores are close. In contrast to the brute force approach, the state-space search approach can lead to feasible solutions within a relatively small amount of run-time for large grid sizes. It can be deemed as a quick way to provide information about the position of the optimal score.