Brian Hurt wrote:

While we're blue skying things, I've had an idea for a sorting
algorithm kicking around for a couple of years that might be
interesting. It's a variation on heapsort to make it significantly
more block-friendly. I have no idea if the idea would work, or how
well it'd work, but it might be worthwhile kicking around.

Now, the core idea of heapsort is that the array is put into heap
order- basically, that a[i] >= a[2i+1] and a[i] >= a[2i+2] (doing the
0-based array version here). The problem is that, assuming that the
length of a is larger than memory, then a[2i+1] is likely going to be
on a different page or block than a[i]. That means every time you
have to bubble down a new element, you end up reading O(log N) blocks-
this is *per element*.

The variation is to instead work with blocks, so you have a block of
entries b[i], and you change the definition of heap order, so that
min(b[i]) >= max(b[2i+1]) and min(b[i]) >= max(b[2i+2]). Also, during
bubble down, you need to be carefull to only change the minimum value
of one of the two child blocks b[2i+1] and b[2i+2]. Other than that,
the algorithm works as normal. The advantage of doing it this way is
that while each bubble down still takes O(log N) blocks being touched,
you get a entire block worth of results for your effort. Make your
blocks large enough (say, 1/4 the size of workmem) and you greatly
reduce N, the number of blocks you have to deal with, and get much
better I/O (when you're reading, you're reading megabytes at a shot).

Now, there are boatloads of complexities I'm glossing over here. This
is more of a sketch of the idea. But it's something to consider.
Following up to myself (my apologies), but it's occurred to me that
there are three advantages to this proposal that I've since thought of:

1) The two child blocks b[2i+1] and b[2i+2]- the one with the larger
minimum element is the one we might replace. In other words, if
min(b[2i+1]) > min(b[2i+2]) and min(b[i]) < min(b[2i+1]), then we know
we're going to want the blocks b[4i+3] and b[4i+4]- before we're done
with blocks b[2i+1] and b[2i+2]. The point here is that this would work
wonders with the posix_fadvise/asyncio ideas kicking around. It'd be
easy for the code to keep 2 large writes and 2 large reads going pretty
constantly.

2) There is some easy parallelization available. I'm not sure how much
worth this is, but the bubble down code is fairly easy to parallelize.
If we have two bubble-downs going on in parallel, once they go down
different branches (one thread goes to block b[2i+1] while the other
goes to b[2i+2]) they no longer interact. Blocks near the root of the
heap would be contended over, and multiple threads means smaller blocks
to keep the total memory foot print the same. Personally, I think the
asyncio idea above is more likely to be worthwhile.

3) It's possible to perform the sort lazily. You have the initial O(N)
pass over the list, but then each block is only O(log N) cost. If it's
likely that only the first part of the result is needed, then much of
the work can be avoided.

Brian

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