1. VORONOI DIAGRAMS BY KATHARINE TISCHER Coordinating Seminar Spring 2013

2. WHAT ARE VORONOI DIAGRAMS? A way of dividing space into a number of regions A set of points (called sites) is specified beforehand and for each site there will be a corresponding region consisting of all points closer to that site than to any other. Each of these regions is called a Voronoi cell All regions are convex polygons Some are finite, others infinite The convex hull is formed by making all of the infinite convex polygons into finite convex polygons *Think of stretching a rubber band around the cluster of sites.

3. HISTORY OF VORONOI DIAGRAMS Descartes Dirichlet John Snow (a British physician) Named after Ukrainian mathematician Georgy Fedosievych Voronyi

4. HOW ARE VORONOI DIAGRAMS MADE? In-between two sites is a perpendicular bisector. Collinear sites form a series of parallel lines Non-collinear sites form Voronoi half lines that meet at a vertex. Color can be used to distinguish different cells Voronoi diagram has at most m = 2n - 4 Voronoi vertices (Toussaint, 1983)

5. EVERYDAY PROBLEMS BEING SOLVED WITH VORONOI DIAGRAMS Cell Tower Coverage Where to build fast food restaurants/malls/ etc. Map making School Districts Post Office problem

6. FINDING THE CLOSEST SITE

7. APPLICATIONS OF VORONOI DIAGRAMS Voronoi diagrams are used for a wide variety of applications in many different fields. Some include: Astronomy Biology Cartography Robotics Geometry- Largest Empty Circle** ETC.

8. ALGORITHMS FOR COMPUTING VORONOI DIAGRAMS There are hundreds of different algorithms for constructing various types of Voronoi diagrams. Here are two of the simplest ones: A) In this algorithm the sites are inserted one at a time into the diagram 1)New Point = 1)Figure out which of the existing Voronoi cells contains the new site 2)“Walk Around” the boundary of the new site’s Voronoi region, inserting new edges 3)Finally, delete all the old edges sticking into the new region.

9. ALGORITHMS CONT. B ) It has been proven that if you insert the points in random order, the expected time is only O(n log n), regardless of which set of points we’re given. A divide and conquer algorithm for constructing Voronoi diagrams was discovered by Shamos and Hoey. Split the points into two halves, the leftmost n/2 points, which we’ll color blue, and the rightmost n/2 points, which we’ll color red. Recursively compute the Voronoi diagram of the two halves. Finally, merge the two diagrams by finding the edges that separate the blue points from the red points. The last step can be done in linear time by the “walking ant” method. (Drysdale, 1993)

10. BIG ‘O’ NOTATION For every algorithm, any operation used in that algorithm, and any input given to the algorithm, there is a mathematical expression that describes how many times the operation is used on that input by that algorithm. (McGuire, 2011) The expression normally depends on the size of the input: if that simple algorithm is given a list of length n (the variable traditionally used for such things), then it will use that operation n times. The big-o notation is designed to highlight the most important part of the expression while hiding irrelevant details. It describes the asymptotic behavior of the expression as the inputs of the expression get larger. Asymptotic – A line that continually approaches a given curve but does not meet at any finite distance

11. BIG ‘O’ NOTATION HIERARCHY O(1) is constant-time; such an algorithm does not depend on the size of its inputs. O(n) is linear-time; such an algorithm looks at each input element once and is generally pretty good. O(n log n) is also pretty decent (that is n times the logarithm base 2 of n). O(n^2), O(n^3), etc. These are polynomial-time, and generally starting to look pretty slow, although they are still useful. O(2^n) is exponential-time, which is common for artificial intelligence tasks and is really quite bad. Exponential-time algorithms begin to run the risk of having a decent-sized input not finish before the person wanting the result retires. There are worse; like O(2^2^...(n times)...^2). (McGuire, 2011)

12. APPLICATION: GEOMETRY THE LARGEST OPEN CIRCLE PROBLEM The largest open circle problem is also known as the toxic waste dump problem. Given n sites in the plane, find the largest circle centered within the convex hull and enclosing none of the sites.

13. PROPERTIES OF VORONOI DIAGRAMS (Miu, 2001)

14. PROPERTIES OF VORONOI DIAGRAMS (Miu, 2001)

15. RESOURCES Drysdale, S. (1993, July 19). Geometry in action. Retrieved from http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html McGuire, T. M. (2011, July 25). O (n log n). Retrieved from www.crsr.net/Notes/BigO.html www.crsr.net/Notes/BigO.html Miu, A. (2001, September 27). Lecture 7: Voronoi diagrams. Retrieved from http://nms.csail.mit.edu/~aklmiu/6.838/L7.pdf http://nms.csail.mit.edu/~aklmiu/6.838/L7.pdf Toussaint, G. T. (1983). Computing largest empty circles with location constraints. International journal of computer and information sciences, 12(5), 349. Retrieved from http://asignatura.us.es/fgcitig/Articulos/11-Computing largest empty circles with location constraints.pdf Voronoi diagram. (2013, April 28). Retrieved from http://en.wikipedia.org/wiki/Voronoi_diagram