Data Compression

The best method would be Lossy compression. This technique reduces the file by eliminating less important data. This can significantly reduce file size without degrading the visual perception. This is ideal for e-commerce websites where the focus is on reducing storage space while maintaining reasonable image quality. On the other hand, Lossless compression shrinks the file without losing any information, which means the image quality will be high but the size reduction would not be as significant. Fixed-size compression and Huffman coding without compression are not effective in this scenario as they don't offer the balance between size reduction and image quality that is required for profile pictures on an e-commerce website.

The best method for compressing 1080p HD video in this scenario is H.264. This method, also known as Advanced Video Coding (AVC), is commonly used for high-definition video because it provides a good trade-off between video quality and bandwidth efficiency. It is recommended for streaming services because it reduces the amount of data while maintaining high-quality video. On the other hand, LZ77 and Huffman coding are general data compression methods and not specifically designed for video. They are not as efficient for video compression as H.264. Run-length encoding is a simple form of data compression, however, it's not typically used for video compression due to its simplicity and lower efficiency.

The best data compression for transmitting sensitive data across a network regularly is Lossless compression. This type of compression reduces the file size without losing any original data when it is compressed and then decompressed, therefore ensuring the integrity of data. On the other hand, Lossy compression, which reduces file size by removing unnecessary or less important information, may lead to data loss and is therefore not suitable for sensitive data transmission. H.264 is a video compression standard, not a data compression type. Fractal compression is also primarily used for images and is not the most effective method for preserving the integrity of sensitive data being transmitted across a network.