Lecture 19

Learning objectives

After this class, you should be able to:

1. Describe how a d-dimensional range tree can be constructed, derive the time and space complexity for constructing it, and derive the time complexity for querying it.
2. Explain where, in our previous algorithms for range search, we used the assumption that no two points have identical values for the same coordinate.
3. Given a set of points, give the equivalent composite points, and draw the range tree corresponding to it.

Reading assignment

1. Handout on Orthogonal range searching, sections 5.4 and 5.5.

Exercises and review questions

• Questions on current lecture's material
  1. Prove, using induction, the time complexity given in CG for constructing a d-dimensional range tree.
  2. (Post your answer on the discussion board) Give an application of orthogonal range searching.
  3. Given the following points: P = {(1.1,1.1), (2.1,1.2), (3.1,1.3), (1.2,2.1), (2.2,2.2), (3.2,2.3), (1.3,3.1), (1.2,3.3), (1.3,3.5)}, construct a 2-d range tree using composite points, and show composite range query equivalent to the following query: x = 1.2, x' = 3.15, y = 1.3, y' = 3.3.
  4. How will you define composite points in three dimensions, how will you define a total ordering on them, and how will you transform a range query to a range query on the composite points?

• Questions on next lecture's material
  1. None.