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 = 0, multiple = TRUE, drop = 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 subject is omitted, query 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 findOverlaps 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 subject 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 subject. This is analogous to the default behavior of the match function.

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 multiple is TRUE, a RangesMatchingList is returned, otherwise a list of integer vectors or, if drop is TRUE, an integer vector with indices offset to align with the unlisted query. When drop is FALSE, an IntegerList is returned, where each element of the result corresponds to a space in query. For spaces that did not exist in subject, 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, 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.
countOverlaps(query, subject): Return the count of the number of ranges in query that overlap a range in subject.

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

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, multiple = FALSE) # c(2, NA, 3)

  ## allow multiple hits
  findOverlaps(query, tree)

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

  ## 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, multiple=FALSE)

[Package IRanges version 1.4.16 Index]