Project Suggestions

Algorithm Design and Implementation

• Aspect Ratio Computation.
  • Problem: Given a simple polygon P in the plane, determine the largest area axis-aligned square contained in P. Similarly, determine the smallest area square containing P. The ratio between the areas of the containing and the contained squares is called the aspect ratio of P.
  • Motivation: The aspect ratio plays an important role in the performance of certain graphics heuristics. For example, this paper by Suri, Hubbard, Hughes proves that the popular bounding box heuristic for collision detection is a constant factor optimal if the objects have constant aspect ration and scale factors.

• Multi-Colored Ball.
  • Let S be a set of n points in d-space (the dimension d can be anything, but you can focus on d=2 and d=3). Each point has a color, indexed 1 through k. Your task is to compute the smallest ball that contains at least one point of each color.

    A more general version asks for the smallest ball that contains points of at least p colors, where p < k.

    Consider both optimal and approximately optimal algorithms.

• Optimal Covering of Art Gallery.
  • Given a simple polygon P, determine the minimum number of guards that can watch P.

    This differs from the art gallery theorem in that we want the optimal answer for a specific polygon. What can you prove about the quality of your algorithm's solution?

• Coverage in ad hoc networks.
  • Determine the minimum number of antennae to cover a polygon. Each antennae's range is a circular disk, and the signal cannot penetrate the polygon boundary. What can you prove about the quality of your algorithm's solution?

• Auctions with Demand Curves.
  • The paper Optimal Clearing of Supply/Demand Curves by Sandholm and Suri describes geometry-based schemes for auctions and exchangs with supply demand curves. Implement these schemes.

• Voronoi Game or Beacon Placement.
  • Suppose we have n points scattered in the plane inside a rectangle. We want to place a new point X so as to maximize its Voronoi cell.

    For some recent work and motivation, you can read these papers: "On Finding a Guard that Sees Most and a Shop that Sells Most" and "Adaptive Beacon Placement".