SEGMENT TREE NOTES
Suppose we are given a left-closed right-open interval I = [0,N). A line segment l w.r.t.
the interval I is represented as a pair [B,E) where 0 < B <= E < N. Suppose L is a finite
set of line segments w.r.t. I. Each line segment in L is assumed to have an associated id.
All end points are assumed to be integers.
A segment tree associated with I is a binary tree T having the following properties:
1. Each node n in the tree implicitly represents a line segment [n.B,n.E). Note that
the line segment implicitly associated with node n may not occur anywhere in L.
2. In addition, each node n has an associated list n.List of line id’s associated with
lines in L. The list may be empty.
3. The root of T satisfies the condition Root(T).B = 1, Root(T).E = N.
4. If n is a node in T and n.B + 1 < n.E then:
a. n may have upto two children
b. (n.left).B = n.B; (n.left).E = n.B + ceil((n.E – n.B)/2)
c. (n.right).B = (n.left).E; (n.right).E = n.E
5. Suppose l is a line-segment. n.LIST contains l.id iff:
a. There is no ancestor n’ of n in tree T such that n’.LIST contains l.id and
b. [n.B,n.E) is a subset (or equal to) the interval [l.B,l.E), i.e. the interval
associated with the node must be a subset of (or equal to) the line segment
l’s associated interval.
When inserting lines into a segment tree, we only create the necessary nodes (i.e. every
leaf in the segment tree must have at least one line-id labeling it at any given point in
time).
