Interval Trees

1
Interval Trees

• Store intervals of the form li,ri, li lt ri.
• An interval is stored in exactly 1 node.
• O(n) nodes.
• 3 versions.
• Differing capability.

2
Version 1

• Store intervals of the form li,ri, li lt ri.
• At least 1 interval per node.
• Static interval set.
• Report all intervals that intersect/overlap a
  given interval l,r.

3
DefinitionVersion 1

• A binary tree.
• Each node v has a point v.pt and two lists v.left
  and v.right.
• u.pt lt v.pt for nodes u in left subtree of v.
• u.pt gt v.pt for nodes u in right subtree of v.
• So, it is a binary search tree on pt.

4
DefinitionVersion 1

• Intervals with ri lt v.pt are stored in the left
  subtree of v.
• Intervals with li gt v.pt are stored in the right
  subtree of v.
• Intervals with li lt v.pt lt ri are stored in v.
• v.left has these intervals sorted by li.
• v.right has these intervals sorted by ri.

5
Example

• v.pt 4
• L a, e
• R d

• v.left c, f, b
• v.right c,b,f

6
Properties

• Each interval is stored in exactly one node.
• Each node stores between 1 and n intervals.
• Number of nodes is O(n).
• Sum of sizes of left and right lists is O(n).
• Tree height depends on how you choose the points
  v.pt.

7
Selection of v.pt

• v is the median of the end points of the
  intervals stored in the subtree rooted at v.

• End points 1, 2, 3, 4, 5, 6, 7
• Use 4 as v.pt.

8
Selection of v.pt

• With median selection, tree height is O(log n).
• Median selection is possible only for static
  interval set. So, no inserts/deletes.
• Could relax to support insert/delete.

9
Find All Overlapping Intervals

• Query interval is l,r.
• v.pt e l,r
• All intervals in v overlap.
• Search L and R for additional overlapping
  intervals.

10
Find All Overlapping Intervals

• v.pt lt l
• Intervals in v with ri gt l overlap.
• No interval in L overlaps.
• Search R for additional overlapping intervals.

11
Find All Overlapping Intervals

• v.pt gt r
• Intervals in v with li lt r overlap.
• No interval in R overlaps.
• Search L for additional overlapping intervals.

12
Find All Overlapping Intervals

• Complexity
• O(log n output) nodes encountered.
• All intervals in v overlap.
• Intervals in v with ri gt l overlap.
• Intervals in v with li lt r overlap.
• O(log n output) when v.left and v.right are
  sorted arrays.

13
Version 2

• Store intervals of the form li,ri, li lt ri.
• Empty nodes permitted.
• Inserts and deletes.
• Answer queries of the form which intervals
  include the point d.

14
Inserts Deletes

• Difficult in version 1 because v.pt is median.
• Select v.pt (almost) arbitrarily and use a
  red-black tree.
• Each node stores between 0 and n intervals.
• At most 2n nodes permissible.
• Tree height is O(log n).
• Interval placement rule is same as for Version 1.

15
Need For Empty Nodes

• Deletion from a degree 2 node.

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Why Upto 2n Nodes Permissible

• When number of nodes gt 2n, at least 1 degree 0 or
  degree 1 node must be empty.
• Empty degree 0 and 1 nodes are easily removed.
• So, no need to keep them around.
• 2n suffices to avoid having to handle empty
  degree 2 nodes.

17
LL Rotation

• Intervals change only for A and B.
• Those intervals of A that include B.pt need to be
  moved into B.

18
Remaining Rotations

• All insert/delete rotations require relocating
  intervals from O(1) nodes.
• O(1) rotations per insert/delete.
• Complexity of insert/delete is O(f(n) log n),
  where f(n) is time needed to relocate O(n)
  intervals from one node to another.

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All intervals that contain d

• d v.pt
• All intervals in v.
• Done!

20
All intervals that contain d

• d lt v.pt
• Intervals in v with left end point lt d.
• No interval in R contains d.
• Search L for additional overlapping intervals.
• d gt v.pt
• Similar
• O(log n output)

21
Version 3

• Store intervals of the form li,ri, li lt ri.
• Exactly 1 interval per node.
• Inserts and deletes.
• Report just 1 overlapping interval.

22
Version 3Structure

• Red-black tree.
• Each node has exactly one interval v.int and one
  point v.max.
• v.max max (right) end point of intervals in
  subtree rooted at v.
• Binary search tree on intervals. So, need an
  ordering relation for intervals.

23
Interval Ordering

• Ordered by left end points. Tie breaker for equal
  left end points.
• i and j are two intervals.
• i lt j iff li lt lj or (li lj and ri gt rj)

24
Example
25
Version 3Search

• Search for an interval that has an overlap with Q
  l,r
• If v.interval and Q overlap, done.
• Otherwise, if v.leftChild.max gt l search
  v.leftChild.
• Otherwise search v.rightChild.

26
Version 3Search
v.leftChild.max gt l
l

• r

max
27
Version 3LL Rotation

• Max values changes only for A and B.
• A.max maxA.interval.right, BR.max, AR.max.
• B.max maxB.interval.right, BL.max, A.max.

28
Remaining Rotations

• All insert/delete rotations require computing max
  for O(1) nodes.
• O(1) rotations per insert/delete.
• Complexity of insert/delete is O(log n).