Segment Tree - Structure Description

Structure Description

This section describes the structure of a segment tree in a one-dimensional space.

Let S be a set of intervals, or segments. Let p₁, p₂, ..., p_m be the list of distinct interval endpoints, sorted from left to right. Consider the partitioning of the real line induced by those points. The regions of this partitioning are called elementary intervals. Thus, the elementary intervals are, from left to right:

That is, the list of elementary intervals consists of open intervals between two consecutive endpoints p_i and p_i+1, alternated with closed intervals consisting of a single endpoint. Single points are treated themselves as intervals because the answer to a query is not necessarily the same at the interior of an elementary interval and its endpoints.

Given a set I of intervals, or segments, a segment tree T for I is structured as follows:

• T is a binary tree.
• Its leaves correspond to the elementary intervals induced by the endpoints in I, in an ordered way: the leftmost leaf corresponds to the leftmost interval, and so on. The elementary interval corresponding to a leaf v is denoted Int(v).
• The internal nodes of T correspond to intervals that are the union of elementary intervals: the interval Int(N) corresponding to node N is the union of the intervals corresponding to the leaves of the tree rooted at N. That implies that Int(N) is the union of the intervals of its two children.
• Each node or leaf v in T stores the interval Int(v) and a set of intervals, in some data structure. This canonical subset of node v contains the intervals from I such that contains Int(v) and does not contain Int(parent(v)). That is, each segment in I stores the segments that span through its interval, but do not span through the interval of its parent.