# Not-Just-Sorted-Arrays

It is fascinating, how simple data structures can be used to build web-scale systems (Related: A funny video on MongoDB). If this doesn’t make sense to you yet, allow me to slowly build up to the story. One of the most simple, and yet powerful algorithm a programmer has in his toolbox is the Binary Search. There are far too many applications to it. Consider reading this Quora answer for simple examples. I personally use it in git bisect to hunt down bad commits in a repository with tens of thousands of commits.

The humble sorted array is a beautiful thing. You can search over it in $O(\log n)$ time. There is one trouble though. You cannot modify it. I mean, you can, but then you will spoil the nice property of it being sorted, unless you pay an $O(n)$ cost to copy the array to a new location, and insert the new element. If you have reserved a large enough array before hand, you don’t need to copy to a new array, but still have to shift elements and that will still be an $O(n)$ cost.

Also, if we were allowed to plot complexities on a graph, we can plot the insert complexity on the X-axis and search complexity on the Y-axis. Then all the suitable data-structures would hopefully be bound by the square with edges on <$O(1)$, $O(1)$> and <$O(n)$, $O(n)$>. The sorted array with <$O(n)$, $O(\log n)$> would lie somewhere on the bottom right corner, whereas, a simple unsorted array would be on the top-left with <$O(1)$, $O(n)$>. You can’t do insertions better than $O(1)$ and you can’t do searches better than $O(\log n)$ (although the bases and constants matter a lot, in practice).

Now, how do we use a static structure, so that we retain the goodness of a sorted array, but allow ourselves the ability to add elements in an online fashion? What we have here, is a ‘static’ data-structure, and we are trying to use it for a ‘dynamic’ usecase. Jeff Erickson’s notes on Static to Dynamic Transformation are of good use here. The notes present results related to how to use static data-structures to build dynamic ones. In this case, you compromise a bit on the search complexity, to get much better insert complexity.

The notes present inserts-only and inserts-with-deletions static to dynamic transformations. I haven’t read the deletions part of it, but the inserts-only transformation is easy to follow. The first important result is:

If the static structure has a space complexity of $S(n)$, query complexity of $Q(n)$, and insert complexity of $P(n)$, then the space complexity of the dynamic structure would be $O(S(n))$, with query complexity of $O(\log n).Q(n)$, and insert complexity of $O(\log n).\frac{P(n)}{n}$ amortized.

Then the notes present the lazy-rebuilding method by Overmars and van Leeuwen. Which improves the first result’s insertion complexity by getting the same complexity in the worst case instead of amortized. (Fun fact: Overmars is the same great fellow who wrote Game Maker, a simple game creating tool, which I used when I was 12! Man, the nostalgia :) I digress..)

The inserts-only dynamic structure, is pretty much how LSM trees work. The difference is the $L_0$ array starts big (hundreds of MBs, or a GB in some cases), and resides in memory, so that inserts are fast. This $L_0$ structure is later flushed to disk, but does not need to be immediately merged with a bigger file. That is done by background compaction threads, which run in a staggered fashion, so as to minimize disruption to the read workload. Read the BigTable paper, to understand how simple sorted arrays sit at the core of the biggest databases in the world.

Next Up: Fractal Trees and others.