Sorting 1 million 8-decimal-digit numbers with 1 MB of RAM

Question

I have a computer with 1 MB of RAM and no other local storage. I must use it to accept 1 million 8-digit decimal numbers over a TCP connection, sort them, and then send the

Answer 1

The trick is to represent the algorithms state, which is an integer multi-set, as a compressed stream of "increment counter"="+" and "output counter"="!" characters. For example, the set {0,3,3,4} would be represented as "!+++!!+!", followed by any number of "+" characters. To modify the multi-set you stream out the characters, keeping only a constant amount decompressed at a time, and make changes inplace before streaming them back in compressed form.

Details

We know there are exactly 10^6 numbers in the final set, so there are at most 10^6 "!" characters. We also know that our range has size 10^8, meaning there are at most 10^8 "+" characters. The number of ways we can arrange 10^6 "!"s amongst 10^8 "+"s is (10^8 + 10^6) choose 10^6, and so specifying some particular arrangement takes ~0.965 MiB` of data. That'll be a tight fit.

We can treat each character as independent without exceeding our quota. There are exactly 100 times more "+" characters than "!" characters, which simplifies to 100:1 odds of each character being a "+" if we forget that they are dependent. Odds of 100:101 corresponds to ~0.08 bits per character, for an almost identical total of ~0.965 MiB (ignoring the dependency has a cost of only ~12 bits in this case!).

The simplest technique for storing independent characters with known prior probability is Huffman coding. Note that we need an impractically large tree (A huffman tree for blocks of 10 characters has an average cost per block of about 2.4 bits, for a total of ~2.9 Mib. A huffman tree for blocks of 20 characters has an average cost per block of about 3 bits, which is a total of ~1.8 MiB. We're probably going to need a block of size on the order of a hundred, implying more nodes in our tree than all the computer equipment that has ever existed can store.). However, ROM is technically "free" according to the problem and practical solutions that take advantage of the regularity in the tree will look essentially the same.

Pseudo-code

• Have a sufficiently large huffman tree (or similar block-by-block compression data) stored in ROM
• Start with a compressed string of 10^8 "+" characters.
• To insert the number N, stream out the compressed string until N "+" characters have gone past then insert a "!". Stream the recompressed string back over the previous one as you go, keeping a constant amount of buffered blocks to avoid over/under-runs.
• Repeat one million times: [input, stream decompress>insert>compress], then decompress to output