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Unformatted text preview: CS 373: Combinatorial Algorithms, Spring 2001 http://wwwcourses.cs.uiuc.edu/~cs373 Homework 4 (due Thu. March 29, 2001 at 11:59:59 pm) Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Homeworks may be done in teams of up to three people. Each team turns in just one solution, and every member of a team gets the same grade. Since 1unit graduate students are required to solve problems that are worth extra credit for other students, 1unit grad students may not be on the same team as 3/4unit grad students or undergraduates. Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3/4unit grad student, or 1unit grad student by circling U, 3 / 4 , or 1, respectively. Staple this sheet to the top of your homework. Required Problems 1. Suppose we have n points scattered inside a twodimensional box. A kdtree recursively subdivides the rectangle as follows. First we split the box into two smaller boxes with a vertical line, then we split each of those boxes with horizontal lines, and so on, always alternating between horizontal and vertical splits. Each time we split a box, the splitting line partitions the rest of the interior points as evenly as possible by passing through a median point inside the box ( not on the boundary). If a box doesn’t contain any points, we don’t split it any more; these final empty boxes are called cells . = ⇒ = ⇒ = ⇒ Successive divisions of a kdtree for 15 points. The dashed line crosses four cells. CS 373 Homework 4 (due 3/29/01) Spring 2001 An example staircase as in problem 3. (a) How many cells are there, as a function of n ? Prove your answer is correct....
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