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IntervalTrees

A data structure for efficient manipulation of sets of intervals

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IntervalTrees

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API

IntervalTrees exports one type: IntervalTree{K, V}. It implements an associative container mapping (K, K) pairs to to values of type V. K may be any ordered type, but only pairs (a, b) where a ≤ b can be stored.

Intervals in this package are always treated as end-inclusive, similar to the Julia Range type.

Types

IntervalTrees exports an abstract type AbstractInterval{K}. Types deriving from it are expected to implement first and last methods that return the values of type K giving the inclusive range of the interval.

There are also basic interval type provided:

immutable Interval{T} <: AbstractInterval{T}
    first::T
    last::T
end

immutable IntervalValue{K, V} <: AbstractInterval{K}
    first::K
    last::K
    value::V
end

The basic data structure implemented is IntervalTree{K, V}, which stores intervals of type V, that have start and end positions of type K.

IntervalMap{K, V} is a typealias for IntervalTree{K, IntervalValue{K, V}} to simplify associating data of type V with intervals.

Insertion and Initialization

New intervals can be added to an IntervalTree with the push! function.

xs = IntervalTree{Int, Interval{Int}}()
push!(xs, Interval{Int}(500, 1000))

A more efficient means of building the data structure by bulk insertion. If the intervals are knows up front and provided in a sorted array, an IntervalTree can be built extremely efficiently.

intervals = Interval{Int}[]
# construct a large array of intervals...

sort!(intervals)
xs = IntervalTree{Int, Interval{Int}}(intervals)

Standard Dictionary Operations

IntervalTree implements all the standard dictionary operations. You can use it as an efficient way to map (K, K) tuples to values.

using IntervalTrees

# Create an interval tree mapping (Int, Int) intervals to Strings.
xs = IntervalMap{Int, String}()

# Insert values
xs[(1,100)] = "Low"
xs[(101,1000)] = "Medium"
xs[(1001,10000)] = "High"

# Search for values
println(xs[(1001,10000)]) # prints "High"

# Get a value, returning a default value if not found
println(get(xs, (10001, 100000), "Not found")) # prints "Not found"

# Set a value if it's not already present
println(set(xs, (10001, 100000), "Not found"))

# Delete values
delete!(xs, (1,100))

Iteration

As with dictionaries, key/value pairs can be iterated through efficiently.

    for x in xs
        println("Interval $(x.first), $(x.last) has value $(x.value)")
    end

Some other iteration functions are provided:

from(t::IntervalTree, query): Return an iterator thats iterates through every key/value pair with an end position >= to query.

keys(t::IntervalTree): Return an iterator that iterates through every interval key in the tree.

values(t::IntervalTree): Return an iterator that iterates through every value in the tree.

Intersection

The primary thing an IntervalTree offers over a Dict is the ability to efficiently find intersections. IntervalTrees supports searching and iterating over intersections between two trees or between a tree and a single interval.

intersect(t::IntervalTree, query::(Any, Any)): Return an iterator over every interval in t that intersects query.

intersect(t1::IntervalTree, t2::IntervalTree): Return an iterator over every pair of intersecting entries (interval1, interval2), where interval1 is in t1 and interval2 is in t2.

hasintersection(t::IntervalTree, position): Return true if position intersects some interval in t.

Algorithms

Multiple data structures are refered to as "interval trees". What's implemented here is the data structure described in the Cormen, et al. "Algorithms" book, or what's refered to as an augmented tree in the wikipedia article. This sort of data structure is just an balanced search tree augmented with a field to keep track of the maximum interval end point in that node's subtree.

Many operations over two or more sets of intervals can be most efficiently implemented by jointly iterating over the sets in order. For example, finding all the intersecting intervals in two sets S and T can be implemented similarly to the merge function in mergesort in O(n+m) time.

Thus a general purpose data structure should be optimized for fast in-order iteration while efficiently supporting other operations like insertion, deletion, and single intersection tests. A B+-tree is nicely suited to the task. Since all the intervals and values are stored in contiguous memory in the leaf nodes, and the leaf nodes augmented with sibling pointers, in-order traversal is exceedingly efficient compared to other balanced search trees, while other operations are comparable in performance.

First Commit

04/16/2014

Last Touched

2 months ago

Commits

85 commits

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