R: Interval Search Trees

IntervalTree-class {IRanges} | R Documentation

## Interval Search Trees

### Description

Efficiently perform overlap queries with an interval tree.

### 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 implementation 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 simplest approach is to call the `findOverlaps` function on a `Ranges` or other object with range information, as described in the following section.

An `IntervalTree` object is a derivative of `Ranges` and stores its ranges as a tree that is optimized for overlap queries. Thus, for repeated queries against the same subject, it is more efficient to create an `IntervalTree` once for the subject using the constructor described below and then perform the queries against the `IntervalTree` instance.

### 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 `findOverlaps` function.

  `findOverlaps(query, subject = query, maxgap = 0L, minoverlap = 1L, type = c("any", "start", "end", "within", "equal"), select = c("all", "first", "last", "arbitrary"), drop = FALSE, ignoreSelf = FALSE, ignoreRedundant = FALSE)`:

  Find the intervals in `query`, a `Ranges`, `RangesList`, `RangedData`, or `integer` vector (to be converted to length-one ranges), that overlap with the intervals `subject`, a `Ranges`, `RangesList`, or `RangedData`. If `query` is unsorted, it is sorted first, so it is usually better to sort up-front, to avoid a sort with each `findOverlaps` call.

  If `subject` is omitted, `query` is queried against itself. In this case, and only this case, the `ignoreSelf` and `ignoreRedundant` arguments are allowed. By default, the result will contain hits for each range against itself, and if there is a hit from A to B, there is also a hit for B to A. If `ignoreSelf` is `TRUE`, all self matches are dropped. If `ignoreRedundant` is `TRUE`, only one of A->B and B->A is returned.

  Intervals with a separation of `maxgap` or less and a minimum of `minoverlap` overlapping positions, allowing for `maxgap`, are considered to be overlapping. `maxgap` should be a scalar, non-negative, integer. `minoverlap` should be a scalar, positive integer.

  When `select` is {"all"}, the results are returned as a `RangesMatching` object. When `select` is `"first"`, `"last"`, or `"arbitrary"` the results are returned as an integer vector of length `query` containing the first, last, or arbitrary overlapping interval in `subject`, with `NA` indicating intervals that did not overlap any intervals in `subject`.

  If `query` is a `RangesList` or `RangedData`, `subject` must be a `RangesList` or `RangedData`. 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 `select` is `"all"`, a `RangesMatchingList` is returned. For all other `select` the return value depends on the `drop` argument. When `select != "all" && !drop`, an `IntegerList` is returned, where each element of the result corresponds to a space in `query`. When`select != "all" && drop`, an integer vector is returned containing indices that are offset to align with the unlisted `query`.

  By default, any overlap is accepted. By specifying the `type` parameter, one can select for specific types of overlap. The types correspond to operations in Allen's Interval Algebra (see references). If `type` is `start` or `end`, the intervals are required to have matching starts or ends, respectively. While this operation seems trivial, the naive implementation using `outer` would be much less efficient. Specifying `equal` as the type returns the intersection of the `start` and `end` matches. If `type` is `within`, the query interval must be wholly contained within the subject interval. Note that all matches must additionally satisfy the `minoverlap` constraint described above.

  The `maxgap` parameter has special meaning with the special overlap types. For `start`, `end`, and `equal`, it specifies the maximum difference in the starts, ends or both, respectively. For `within`, it is the maximum amount by which the query may be wider than the subject.

  `countOverlaps(query, subject, maxgap = 0L, minoverlap = 1L, type = c("any", "start", "end", "within", "equal"))`: Returns the overlap hit count for each range in `query` using the specified `findOverlaps` parameters. Both `query` and `subject` should be `Ranges`, `RangesList` or `RangedData` objects.
  `subsetByOverlaps(query, subject, maxgap = 0L, minoverlap = 1L, type = c("any", "start", "end", "within", "equal"))`: Returns the subset of `query` that has an overlap hit with a range in `subject` using the specified `findOverlaps` parameters. Both `query` and `subject` should be `Ranges`, `RangesList` or `RangedData` objects.
  `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`, `RangesList` or `RangedData` objects. For `Ranges` objects, the result is a `logical` vector of length equal to the number of ranges in `x`. For `RangesList` and `RangedData` objects, the result is a `LogicalList` object, where each element of the result corresponds to a space in `x`.
  `match(x, table, nomatch = NA_integer_, incomparables = NULL)`: Returns an integer vector of length `length(x)`, containing the index of the first overlapping range in `table` for each range in `x`. If a range in `x` does not overlap any ranges in `table`, its value is `nomatch`. The `x` and `table` arguments should either be both `Ranges` objects or both `RangesList` objects, in which case the indices are into the unlisted `table`. The `incomparables` argument is currently ignored.

### Constructor

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

### 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.
  `start(x)`: Get the starts of the ranges.
  `end(x)`: Get the ends of the ranges.

### 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 `findOverlaps` 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

Allen's Interval Algebra: James F. Allen: Maintaining knowledge about temporal intervals. In: Communications of the ACM. 26/11/1983. ACM Press. S. 832-843, ISSN 0001-0782

### See Also

`Ranges`, 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
      findOverlaps(query, tree, select = "first")
      findOverlaps(query, tree, select = "last")
      findOverlaps(query, tree, select = "arbitrary")

      ## allow multiple hits
      findOverlaps(query, tree)

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

      ## shortcut
      findOverlaps(query, subject)

      ## 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(findOverlaps(query, tree))
      # the same as:
      findOverlaps(subject, query)

      ## one Ranges with itself
      findOverlaps(query)

      ## single points as query
      subject <- IRanges(c(1, 6, 13), c(4, 9, 14))
      findOverlaps(c(3L, 7L, 10L), subject, select = "first")

      ## alternative overlap types
      query <- IRanges(c(1, 5, 3, 4), width=c(2, 2, 4, 6))
      subject <- IRanges(c(1, 3, 5, 6), width=c(4, 4, 5, 4))

      findOverlaps(query, tree, type = "start")
      findOverlaps(query, tree, type = "start", maxgap = 1L)
      findOverlaps(query, tree, type = "end", select = "first")
      findOverlaps(query, tree, type = "within", maxgap = 1L)

[Package IRanges version 1.6.17 Index]