IntervalTree-class {IRanges} | R Documentation

## Interval Search Trees

### Description

An `IntervalTree` object is an external representation of ranges (i.e. it is derived from `XRanges`) that is optimized for overlap queries.

### Details

A common type of query that arises when working with intervals is finding which intervals in one set overlap those in another. An efficient family of algorithms for answering such queries is known as the Interval Tree. This class makes use of the augmented tree algorithm from the reference below, but heavily adapts it for the use case of large, sorted query sets.

The usual workflow is to create an `IntervalTree` using the constructor described below and then perform overlap queries using the `overlap` method. The results of the query are returned as a `RangesMatching` object.

### Constructor

  IntervalTree(ranges): Creates an `IntervalTree` from the ranges in `ranges`, an object coercible to `IntervalTree`, such as an `IRanges` object.

### Finding Overlaps

This main purpose of the interval tree is to optimize the search for ranges overlapping those in a query set. The interface for this operation is the `overlap` function.

  `overlap(object, query = object, maxgap = 0, multiple = TRUE)`:

  Find the intervals in `query`, a `Ranges` or `integer` vector to be converted to length-one ranges, that overlap with the intervals `object`, an `IntervalTree` or, for convenience, a `Ranges` coercible to a `IntervalTree`. If `query` is omitted, `object` is queried against itself. if `query` is unsorted, it is sorted first, so it is usually better to sort up-front, to avoid a sort with each `overlap` call. Intervals with a separation of `maxgap` or less are considered to be overlapping. `maxgap` should be a scalar, non-negative, non-NA number. When `multiple` (a scalar non-NA logical) is `TRUE`, the results are returned as a `RangesMatching` object.

  If `multiple` is `FALSE`, at most one overlapping interval in `object` is returned for each interval in `query`. The matchings are returned as an integer vector of length `length(query)`, with `NA` indicating intervals that did not overlap any intervals in `object`. This is analogous to the return value of the `match` function.

  `query` may also be a `RangesList`, in which case `object` must also be a `RangesList`. If both lists have names, each element from the subject is paired with the element from the query with the matching name, if any. Otherwise, elements are paired by position. The overlap is then computed between the pairs as described above. If `multiple` is `TRUE`, a `RangesMatchingList` is returned, otherwise a list of integer vectors. Each element of the result corresponds to a space in `query`. For spaces that did not exist in `object`, the overlap is nil.

  `x %in% table`: Shortcut for finding the ranges in `x` that overlap any of the ranges in `table`. Both `x` and `table` should be `Ranges` objects. The result is a `logical` vector of the same length as `x`.

### Coercion

  `as(from, "IRanges")`: Imports the ranges in `from`, an `IntervalTree`, to an `IRanges`.
  `as(from, "IntervalTree")`: Constructs an `IntervalTree` representing `from`, a `Ranges` object that is coercible to `IRanges`.

### Accessors

  `length(x)`: Gets the number of ranges stored in the tree. This is a fast operation that does not bring the ranges into R.

### Notes on Time Complexity

The cost of constructing an instance of the interval tree is a `O(n*lg(n))`, which makes it about as fast as other types of overlap query algorithms based on sorting. The good news is that the tree need only be built once per subject; this is useful in situations of frequent querying. Also, in this implementation the data is stored outside of R, avoiding needless copying. Of course, external storage is not always convenient, so it is possible to coerce the tree to an instance of `IRanges` (see the Coercion section).

For the query operation, the running time is based on the query size `m` and the average number of hits per query `k`. The output size is then `max(mk,m)`, but we abbreviate this as `mk`. Note that when the `multiple` parameter is set to `FALSE`, `k` is fixed to 1 and drops out of this analysis. We also assume here that the query is sorted by start position (the `overlap` function sorts the query if it is unsorted).

An upper bound for finding overlaps is `O(min(mk*lg(n),n+mk))`. The fastest interval tree algorithm known is bounded by `O(min(m*lg(n),n)+mk)` but is a lot more complicated and involves two auxillary trees. The lower bound is `Omega(lg(n)+mk)`, which is almost the same as for returning the answer, `Omega(mk)`. The average is of course somewhere in between.

This analysis informs the choice of which set of ranges to process into a tree, i.e. assigning one to be the subject and the other to be the query. Note that if `m > n`, then the running time is `O(m)`, and the total operation of complexity `O(n*lg(n) + m)` is better than if `m` and `n` were exchanged. Thus, for once-off operations, it is often most efficient to choose the smaller set to become the tree (but `k` also affects this). This is reinforced by the realization that if `mk` is about the same in either direction, the running time depends only on `n`, which should be minimized. Even in cases where a tree has already been constructed for one of the sets, it can be more efficient to build a new tree when the existing tree of size `n` is much larger than the query set of size `m`, roughly when `n > m*lg(n)`.

### Author(s)

Michael Lawrence

### References

Interval tree algorithm from: Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford. Introduction to Algorithms, second edition, MIT Press and McGraw-Hill. ISBN 0-262-53196-8

### See Also

`XRanges`, the parent of this class, `RangesMatching`, the result of an overlap query.

### Examples

      query <- IRanges(c(1, 4, 9), c(5, 7, 10))
      subject <- IRanges(c(2, 2, 10), c(2, 3, 12))
      tree <- IntervalTree(subject)

      ## at most one hit per query
      overlap(tree, query, multiple = FALSE) # c(2, NA, 3)

      ## allow multiple hits
      overlap(tree, query)

      ## overlap as long as distance <= 1
      overlap(tree, query, maxgap = 1)

      ## shortcut
      overlap(subject, query)

      ## query and subject are easily interchangeable
      query <- IRanges(c(1, 4, 9), c(5, 7, 10))
      subject <- IRanges(c(2, 2), c(5, 4))
      tree <- IntervalTree(subject)
      t(overlap(tree, query))
      # the same as:
      overlap(query, subject)

      ## one Ranges with itself
      overlap(query)

      ## single points as query
      subject <- IRanges(c(1, 6, 13), c(4, 9, 14))
      overlap(subject, c(3L, 7L, 10L), multiple=FALSE)

[Package IRanges version 1.2.3 Index]