School of Computing Clemson University Fall, 2013

1. Sweep Line Algorithms, Problem Visualization, Etc. CpSc 212: Algorithms and Data Structures Brian C. Dean School of Computing Clemson University Fall, 2013

2. Warm-Up • You are told the locations of N houses on a long one-dimensional street, along with the quality of candy each one gives out on Halloween. • Where should you start “trick-or-treating” so that you maximize the total candy quality within a one-mile radius of your starting point? 9 4 6 7

3. Warm-Up • You are told N intervals on the number line, each with an associated value. • Find a point of maximum overlap (i.e., maximizing the sum of interval values overlapping at that point). 9 4 6 7

4. A “Sweep Line” Approach • You are told N intervals on the number line, each with an associated value. • Find a point of maximum overlap (i.e., maximizing the sum of interval values overlapping at that point). 9 4 6 7

5. Give N rectangular buildings sharing a common base, find the area of the “skyline” they form. Sweep line + binary heap (or balanced BST). N inserts, N deletes, 2N find-max: O(N log N) Example: The Skyline Problem

6. Counting Intersection Points Among Axially-Aligned Segments

7. Counting Intersection Points Among Axially-Aligned Segments 1, 3, 5, 7 7 5 3 1 • Sweep line on x + balanced BST of “active” horizontal segments, keyed on y. • ≤ 2N inserts, ≤ 2N deletes, ≤ N range counting queries: O(N log N)

8. Non-Dominated Points

9. Non-Dominated Points • Reverse sweep line on x + running max on y. • O(N log N)

10. Non-Dominated Points • Forward sweep line on x + balanced BST on y. • N inserts, ≤ N deletes: O(N log N)

11. Given a set of points in 2D, compute the “domination count” for each point. Given N intervals on the number line, compute for each one the number of intervals contained within it. Count the number of crossings to the right… Three Interesting Problems 13 3 1 2 2 3 3 4 4 1

12. Visualizing Problems Graphically b a b c a a2 + b2 = c2 sum of first n odd numbers = n2

13. Robot Arms, Databases, and “Parameter Space” • Suppose a robot arm has 4 movable joints, each with one degree of freedom. You want to move the arm from its current state to a new “goal” state, where certain joint configurations are not valid.  path finding in a 4D space. • Consider a database of N records where each contains fields (height, age, income).  set of points in 3D space.

14. Ballroom Matching • You are given the height and weight of N boys and N girls. • In a ballroom matching class, you want to match each girl with a boy who is taller and heavier than he is (to allow for easy lifts). • How quickly can you determine a valid matching between the boys and girls (or that none exists)?

15. Pumpkin Catching • You are told when and where N pumpkins will fall from a pumpkin tree onto a number line. (e.g., “Pumpkin #1 lands at time t = 4 at the point x = 5). • You are also told where your N friends are standing on the number line at time t = 0. • Each friend can move at one unit of distance per second and can catch at most one pumpkin (they are heavy, after all!) • Can your team catch all the pumpkins?