Data structure for non overlapping ranges of integers?

Question

I remember learning a data structure that stored a set of integers as ranges in a tree, but it's been 10 years and I can't remember the name of the data structure, and I'm a bit fuzzy on the details. If it helps, it's a functional data structure that was taught at CMU, I believe in 15-212 (Principles of Programming) in 2002.

Basically, I want to store a set of integers, most of which are consecutive. I want to be able to query for set membership efficiently, add a range of integers efficiently, and remove a range of integers efficiently. In particular, I don't care to preserve what the original ranges are. It's better if adjacent ranges are coalesced into a single larger range.

A naive implementation would be to simply use a generic set data structure such as a HashSet or TreeSet, and add all integers in a range when adding a range, or remove all integers in a range when removing a range. But of course, that would waste a lot of memory in addition to making add and remove slow.

I'm thinking of a purely functional data structure, but for my current use I don't need it to be. IIRC, lookup, insertion, and deletion were all O(log N), where N was the number of ranges in the set.

So, can you tell me the name of the data structure I'm trying to remember, or a suitable alternative?

Answer 1

I found the old homework and the data structure I had in mind were Discrete Interval Encoding Trees or diets for short. They are described in detail in Diets for Fat Sets, Martin Erwig. Journal of Functional Programming, Vol. 8, No. 6, 627-632, 1998. It is basically a tree of intervals with the invariant that all of the intervals are non-overlapping and non-touching. There is a Haskell implementation in Hackage. I was hoping there would be an existing implementation for Scala, but I'm not seeing any.

The homework also included another data structure they called a Recursive Interval-Occluding Tree (RIOT), which rather than keeping only an interval at each node keeps an interval and another (possibly empty) RIOT of things removed from the interval. The assignment included benchmarks showing it did better than diets for random insertions and deletions. AFAICT it is simply something the TAs made up and never published as it no longer seems to exist anywhere on the Internets, at least not under that name.

Answer 2

You probably are looking for segment trees. This might be helpful: http://www.topcoder.com/tc?d1=tutorials&d2=lowestCommonAncestor&module=Static

You can also use binary search trees for the same, for which each node will have two data fields: min_val and max_val.

During insertion algorithm, you just need to call another merging operation to check if the left-child,parent,right-child create a sequence, so as to club them into a single node. This will take O(log n) time.

Other operations like deletion and look-up will take O(log n) time as usual, but special measures need to be taken while deletion.