13.5. Combining Results

Try these first

1. 1.

   What is the time complexity of the brute-force approach to the closest pair problem?

   Hint

   Consider the number of comparisons needed.

2. 2.

   What distance formula is used to determine the distance between two points?

   Hint

   Think about the right triangles involved in measuring distance.

3. 3.

   What is the time complexity of the closest pair algorithm using divide and conquer?

   • O(n)
   • O(n log n)
   • O(n²)

   Hint

   Recall how the algorithm splits and sorts points.

4. 4.

   True or False: The divide and conquer algorithm guarantees to find the closest pair of points.

   • True
   • False

   Hint

   Think of how it divides the problem into parts.

5. 5.

   Given the points (0,2), (1,3), (4,2), and (1,1), apply the closest pair algorithm to determine the closest pair and show your calculations.

   Hint

   You may need to compare all points.

6. 6.

   Propose an optimization for cases where there are multiple points with the same x or y coordinates and describe its impact on performance.

   Hint

   Think about how clustering can impact your choices.