{"version":"https:\/\/jsonfeed.org\/version\/1","title":"mathspp.com feed","home_page_url":"https:\/\/mathspp.com\/blog\/tags\/pigeonhole-principle","feed_url":"https:\/\/mathspp.com\/blog\/tags\/pigeonhole-principle.json","description":"Stay up-to-date with the articles on mathematics and programming that get published to mathspp.com.","author":{"name":"Rodrigo Gir\u00e3o Serr\u00e3o"},"items":[{"title":"Problem #054 \u2013 imperfect compression","date_published":"2022-01-23T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/problems\/imperfect-compression","url":"https:\/\/mathspp.com\/blog\/problems\/imperfect-compression","content_html":"

Can you show that perfect compression is impossible?<\/p>\n\n

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Photo by Rui Matayoshi on Unsplash.<\/figcaption><\/figure>

Problem statement<\/a><\/h2>\n

Compression is great: it is what lets you take that giant folder you have and reduce its size to save some memory on your laptop.\nOf course, you only do these compressions happily because you know you don't lose information when you compress things.\nThe data is just... compressed!<\/p>\n

For compression to be useful, it has to be bidirectional: you must be able to recover the original data from the compressed version.\nThis is only possible if two different pieces of data never get compressed into the same thing.\n(In mathematical terms, we say that the compression must be injective.)<\/p>\n

Now, on top of that, we are interested in compression that actually works, right?\nThat is, in compression that reduces the size of things.\nRight?<\/p>\n

Right!\nNow, the challenge is for you to show that no compression mechanism is perfect.\nIn other words, show that if a compression mechanism is bidirectional and it manages\nto take some pieces of data and transform them into something smaller,\nthen, there are pieces of data that will become larger<\/strong> by the action of the compression mechanism.<\/p>\n

If it makes it easier for you,\nwe can suppose that the data we are talking about are just sequences of letters.\nSo, we are talking about compression mechanisms that take sequences of letters and try to build smaller\nsequences of letters, the compression.\nFor example, maybe the sequence aaaaaa<\/code> gets compressed into Aaab<\/code>,\nbut maybe the mechanism fails on AAAAAA<\/code> because it “compresses” it into arghfewtoen<\/code>.<\/p>\n

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Give it some thought!<\/p>\n<\/div>\n

If you need any clarification whatsoever, feel free to ask in the comment section below.<\/p>\n

Solvers<\/a><\/h2>\n

Congratulations to the ones that solved this problem correctly and, in particular, to the ones\nwho sent me their correct solutions:<\/p>\n