Finding the most similar documents (nearest neighbours) from a set of documents
You should learn about hashing mechanisms that can be used to calculate similarity between documents. 
Typical hash functions are designed to minimize collision mapping near duplicates to very different hash keys. In cryptographic hash functions, if the data is changed with one bit, the hash key will be changed to a completely different one. 
The goal of similarity hashing is to create a similarity hash function. Hash based techniques for near duplicate detection are designed for the opposite intent of cryptographic hash algorithms. Very similar documents map to very similar hash keys, or even to the same key. The difference between bitwise hamming distance of keys is a measure of similarity. 
After calculating the hash keys, keys can be sorted to increase the speed of near duplicate detection from O(n2) to O(nlog(n)). A threshold can be defined and tuned by analysing accuracy of training data. 
Simhash, Minhash and Local sensitive hashing are three implementations of hash based methods. You can google and get more information about these. There are a lot of research papers related to this topic...
I have 80,000 documents that are about a very vast number of topics. What I want to do is for every article, provide links to recommend other articles (something like top 5 related articles) that are similar to the one that a user is currently reading. If I don't have to, I'm not really interested in classifying the documents, just similarity or relatedness, and ideally I would like to output a 80,000 x 80,000 matrix of all the documents with the corresponding distance (or perhaps correlation? similarity?) to other documents in the set.
I'm currently using NLTK to process the contents of the document and get ngrams, but from there I'm not sure what approach I should take to calculate the similarity between documents.
I read about using tf-idf and cosine similarity, however because of the vast number of topics I'm expecting a very high number of unique tokens, so multiplying two very long vectors might be a bad way to go about it. Also 80,000 documents might call for a lot of multiplication between vectors. (Admittedly, it would only have to be done once though, so it's still an option).
Is there a better way to get the distance between documents without creating a huge vector of ngrams? Spearman Correlation? or would a more low-tech approach like taking the top ngrams and finding other documents with the same ngrams in the top k-ngrams be more appropriate? I just feel like surely I must be going about the problem in the most brute force way possible if I need to multiply possibly 10,000 element vectors together 320 million times (sum of the arithmetic series 79,999 + 79,998... to 1).
Any advice for approaches or what to read up on would be greatly appreciated.