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Unformatted text preview: CS 373: Combinatorial Algorithms, Spring 2001 http://www-courses.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 1-unit graduate students are required to solve problems that are worth extra credit for other students, 1-unit grad students may not be on the same team as 3/4-unit 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, 3/4-unit grad student, or 1-unit 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 two-dimensional box. A kd-tree 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 kd-tree 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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