Hard and Easy Instances of L-Tromino Tilings

We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle and an Aztec diamond has an L-tromino tiling. Then, we study tilings of arbitrary regions where only $180^\circ$ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contains certain so-called "forbidden polyominoes" as sub-regions, then there exists a polynomial time algorithm for deciding a tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019

Hard and easy instances of L-Tromino tilings

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle has an L-tromino tiling, and hence also an Aztec diamond; if an Aztec rectangle has an unknown number of defects or holes, however, the problem of deciding a tiling is NP-complete. Then, we study tilings of arbitrary regions where only 180∘ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contain so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm for deciding a tiling

Reversible Image Watermarking Using Modified Quadratic Difference Expansion and Hybrid Optimization Technique

With increasing copyright violation cases, watermarking of digital images is a very popular solution for securing online media content. Since some sensitive applications require image recovery after watermark extraction, reversible watermarking is widely preferred. This article introduces a Modified Quadratic Difference Expansion (MQDE) and fractal encryption-based reversible watermarking for securing the copyrights of images. First, fractal encryption is applied to watermarks using Tromino's L-shaped theorem to improve security. In addition, Cuckoo Search-Grey Wolf Optimization (CSGWO) is enforced on the cover image to optimize block allocation for inserting an encrypted watermark such that it greatly increases its invisibility. While the developed MQDE technique helps to improve coverage and visual quality, the novel data-driven distortion control unit ensures optimal performance. The suggested approach provides the highest level of protection when retrieving the secret image and original cover image without losing the essential information, apart from improving transparency and capacity without much tradeoff. The simulation results of this approach are superior to existing methods in terms of embedding capacity. With an average PSNR of 67 dB, the method shows good imperceptibility in comparison to other schemes

Towards Developing Efficient Area Coverage Approach in Domestic Robots

東京電機大学201