# Minimum space needed to sort a stream of integers

This question has gotten a lot of attention on SO:
Sorting 1 million 8-digit numbers in 1MB of RAM

The problem is to sort a stream of 1 million 8-digit numbers (integers in the range $[0,\: 99\mathord{,}999\mathord{,}999]$) using only 1 MB of memory ($2^{20}$ bytes = $2^{23}$ bits) and no external storage. The program must read values from an input stream and write the sorted result to an output stream.

Obviously the entire input can't fit into memory, but clearly the result can be represented in under 1 MB since $2^{23} \geq \log_2 \binom{10^8}{10^6} \approx 8079302$ (it's a tight fit).

So, what is the minimum amount of space needed to sort n integers with duplicates in this streaming manner, and is there an algorithm to accomplish the specified task?

• @Juho: Hmm, I am not sure if the lower bound of Munro and Paterson applies in this case. Their model is quite different from the scenario we have here. In particular we have a fixed universe. Maybe one could use ideas from integer data structures, however the space bound is very tight. – A.Schulz Oct 22 '12 at 19:42
• The input is read-only and the output is write-only? Or are you allowed an intermediate "workspace" stream, that you can write to, and then read back from in the next round? – Joe Oct 22 '12 at 19:45
• Or is this question actually about buffered IO as in this blog post? (it might be, coming from SO). neopythonic.blogspot.com/2008/10/… – Joe Oct 22 '12 at 19:48
• hmm don't think the question allows writing back to the input stream... – Dave Tremont Oct 22 '12 at 22:40
• @Juho $\Theta$-bounds won't help to estimate whether we stay within constant memory. – Raphael Oct 25 '12 at 8:23

Ok, it's way past my bedtime, but this won't let me go, so here's a second try (deleted my first)...

Before going into the specifics of how to sort the numbers, I will start by representing them not as a sequence of integers $i_1, i_2, \dots , i_n$, but as a sequence of their differences, i.e. $\delta_k = i_k - i_{k-1}$ where, by convention, $i_0 = 0$.

Then, in order to represent these values, I will first group them into buckets depending on the number of bits needed for their representation, with the assumption that I need one bit to represent 0. Since the numbers are between 0 and 99'999'999, I will need 27 buckets.

For each bucket, I will choose a prefix from the set 1, 01, 001, 0001, etc... Up to 000000000000000000000000001. Obviously, I will choose the shortest prefix for the largest bucket, the second-shortest for the second largest, etc...

The sequence of $\delta_k$ can then be written as pairs of prefixes and values, whereby for the values $>1$, the leading bit can be omitted.

For any random set of numbers, then, how many bits do I need to represent the sequence of $\delta_k$?

For a constant sequence, e.g. $0, 0, 0, 0, \dots , 0$, I only have to store one million representations of the number $0$, or with the above encoding, 10, hence 2 million bits. For any other constant, it would be 2 million bits plus the number of bits of that value for the initial $\delta_1$.

The same computation goes for an increasing sequence, e.g. $0, 1, 2, 3, \dots , n$, where $\delta_k=1$ and the sequence is represented as a million copies of 11.

Now what is the worst-case, i.e. what sequence of $\delta_k$ will give the longest sequence of bits? The worst we can do is make sure that each bucket has one entry less than the previous one, choosing the smallest $\delta_k$ possible, thus assigning the largest prefix to the longest sequences. For simplicity, I'll assume that $K$ buckets have the same amount of entries, with the shortest prefix assigned to the shortest sequence. This means that we have $n$ ones, $n$ twos, $n$ fours, etc... To represent the $j$th bucket, we need $2j-1$ bits, except for the first bucket, that needs $2$ bits.

So if we put $m$ $\delta_k$s into each of the $K$ buckets, we need $m\left(2 + \sum_{j=2}^{K}2j-1\right) = m(K^2 + 1)$ bits. However, the sum of all the $\delta_k$ must not exceed 99'999'999. This sum, assuming we've kept the $\delta_k$ as small as possible, is $m\sum_{j=2}^{K}2^{j-1} = m(2^K-2)$.

So here's the question: is there a combination of $K$ and $m$ such that

• $m(2^K-2) < 100'000'000$, e.g. the largest value is $<100\,000\,000$,
• $Km < 1'000'000$, e.g. there are at most a million values, and
• $m(K^2+1) > 8'000'000$, e.g. we use more than $8\,000\,000$ bits?

Unfortunately, this does happen. Here's a plot of the number of bits as computed in Maple with plot( min( 100000000/(2^K-2) , 1000000/K ) * (K^2 + 1) , K=2..27 ):

So we can represent any sequence of up to one million numbers in $[0,99\,999\,999]$ using roughly 10 million bits. This is, unfortunately, still above the specification.

But how can we sort numbers like this? Well, remember that the numbers come in one by one, so as each number comes in, we update its bucket count ($\mathcal O(1)$) and compute the new optimal prefixes for each bucket ($\mathcal O(\log\log(i_\mathsf{max}))$ if the buckets are indexed in a heap). We then run through our encoded $\delta_k$, reading each entry, computing the respective $i_k$ from the cumulative sum, and comparing it to the new value. If the computed $i_k$ exceeds the new value, we encode the new value and write it back, along with the adjusted $\delta_k$, using the new encoding. Otherwise, we just write back the $\delta_k$ with the new encoding.

But write-back where? Well, if we treat our eight million bits as one large ring buffer, we can append the new, re-coded sequence of $\delta_k$ to the old sequence, re-writing the old values as we go along. We've got almost 1 million extra bits from the worse-case scenario, and I don't think the re-coding can cause that amount of stretching to make this not work.

All in all, this leaves us with a $\mathcal O(n)$ cost for inserting the $n$th number, which gives us an average-case cost of $\mathcal O(N^2)$ for a total of $N$ elements. This isn't really good, but speed did not seem to be a criteria.

• You say we can't find a combination of m and k for which m(k^2+1) > 8e6, then show a graph where m(k^2+1) peaks out around 10e6 and claim that your point is proved when it's exactly the opposite! By your own formula, if we choose k=10, the encoding takes 9882583 bits... or 1.2 megabytes. – Daniel Wagner Oct 23 '12 at 9:28
• @DanielWagner: Oops... Misread that completely, have amended my answer accordingly. Thanks for pointing that out! – Pedro Oct 23 '12 at 10:14