Work Note 12, DM819, fall 2015

Lecture November 17

• Quadtrees (chapter 14).

Exercises November 19

• Exercises 12.3, 12.4, 12.5.
• Given a set of n line segments, I₁, ..., I_n, a one-dimensional matryoshka is a subset I_j1, ..., I_jk such that I_j1 ⊂ I_j2, ..., I_jk-1 ⊂ I_jk and k is the size of the matryoshka.
  • Design and analyze an algorithm for finding the size of the largest matryoshka.
  • Define two-dimensional matryoshkas as a generalization of the above. Can you think of restrictions on input, similar to restrictions from the course, that would enable you to design an effective algorithm for this case?
• A polygon path is the same as a polygon, except that the curve can start and end at different vertices, i.e., it does not have to be closed. As for polygons, we only consider paths that do not self-intersect. Angles of 180° at vertices is allowed.
  • Design and analyze an algorithm for computing a polygon path connecting n input points.
  • Can any polygon path over n point be obtained from a polygon by removing an edge?
  • Design and analyze an algorithm for computing a polygon connecting n input points.
• A point q in the plane is dominated by a point p if p.x ≥ q.x and p.y ≥ q.y. A point from an input set of n points is called dominating if it is not dominated by any other point. The dominating set of a set of points S is the set of all dominating points in S.
  • Design a divide-and-conquer algorithm for computing the dominating set.
  • Assume that somebody will provide a lexicographically sorted input set. Design a linear time algorithm for computing the dominating set.
• Assume that we have n mice located under the x-axis and n cheeses located above. You may assume that the mice and cheeses are in general position. We would like to assign one cheese to each mouse so that when the mouse runs straight to its cheese, it will not bump into another mouse, independent of their relative speeds.
  • Does it work to sort both mice and cheeses on their x-coordinates and assign them pair-wise in that order?
  • Design and analyze and algorithm that solves the problem.
  • Make reasonable assumptions and show that this problem cannot be solved in time faster than Θ(n log n).