Final answer:
No compression scheme can guarantee a shorter output for every input file, due to the pigeonhole principle which dictates that there are more possible input files than compressed outputs, making it impossible to uniquely represent each input by a shorter output.
Step-by-step explanation:
The question asks us to demonstrate why no compression scheme can guarantee that every input file will be smaller once compressed. To understand this, we must first grasp the concept of compression and the principle known as the pigeonhole principle. Compression algorithms work by finding patterns and redundancies in data and encoding them in a more space-efficient manner, but they must always account for the possibility of new data that doesn't follow any recognized pattern.
Consider an algorithm designed to compress files of a certain length: there are a finite number of shorter lengths for the output files. According to the pigeonhole principle, if we have more input files than possible output file lengths, then at least two different input files must compress into the same output file length. This is a problem because you must be able to uniquely decompress the output file back into the original input file. If two files compress into the same output, information is lost and decompression cannot be unique, which is unacceptable in lossless compression.
After all, for lossless compression, each different input file must have a unique compressed version, or else the original file cannot be reconstructed accurately. Since the number of possible shorter output files is always less than the number of possible input files (which is virtually infinite), not every file can be compressed to be shorter; for some files, the compressed version must be as long as, or even longer than, the original.
As an example, imagine trying to compress every possible three-letter sequence using a two-letter alphabet. There are 32 = 9 possible two-letter sequences but 33 = 27 three-letter sequences. As you can see, not all 27 three-letter sequences can be uniquely represented by only 9 two-letter sequences. Thus, for at least some three-letter input sequences, the compressed output must be at least three letters long, demonstrating that it is impossible for a compression scheme to guarantee a shorter output for every possible input.