9.1 Windowing Queries 9.2 Interval Trees

Windowing queries answer important questions. Primarily, windowing queries answer information involving a given line segment and a windowing region. Given a set of n line segments S and a query window W := [x : x ]x[y : y ], windowing queries answer which segments in S intersect W. The primary difference between windowing queries and range queries is the data. While range queries are associated with points, windowing queries are associated with line segments, polygons , curves, etc. Typically, windowing queries exist in 2-or 3-dimensions, whereas range queries deal with queries in higher-dimensional spaces. In the examples below, we will visit three types of windowing queries, two of which consider the specific case regarding axis-aligned segments, and one which analyzes arbitrarily oriented line segments. In every structure, all segments are assumed to be non-intersecting. Interval trees [9.1] present suitable windowing queries for axis-parallel line segments and an axis-parallel query window. Given the set S of n axis-parallel line segments and an axis-parallel query window W := [x : x ] x [y : y ] the segments S intersecting W can be reported efficiently. Intersecting segments in S are defined by having at least one endpoint inside W. There different cases which satisfy this condition include: the segment can lie entirely inside W , it can the boundary of W once, twice, or partially overlap the boundary of W. In the cases where one endpoint lies inside W , we can use a 2-dimensional range query, which uses O(n log n) storage and can be answered in O(log 2 n + k) time. The use of fractional cascading can reduce query time to O(log n + k). In the other cases, where a segment in S crosses the boundary of W twice or lies on the edge of a boundary, another approach must be used. This is where the interval tree is needed.