Question

Greedy Algorithms) Describe an efficient algorithm that, given a set {X1, X2,...,xn} of points on the real line, determines the smallest set of unit-length closed intervals that contains all of the given points. Argue that your algorithm is correct.

Answer 1

The efficient algorithm for determining the smallest set of unit-length closed intervals that contains all the given points on the real line is the greedy algorithm.

Step-by-step explanation:

An efficient algorithm for finding the smallest set of unit-length closed intervals that contains all the given points on the real line is to use the greedy algorithm approach.

Here is the step-by-step process using the greedy algorithm:

1. Sort the set of points in ascending order.
2. Initialize an empty set of intervals.
3. Iterate through the sorted set of points:

• If the current point is not covered by any interval, create a new interval with that point as its left endpoint and unit length. Add the interval to the set of intervals.
• If the current point is covered by an interval, do nothing and move to the next point.

Return the set of intervals.

The greedy algorithm is correct because at each step, it chooses the point that covers the maximum number of uncovered points. By iteratively selecting the point that covers the most number of uncovered points, the algorithm guarantees that the smallest set of unit-length closed intervals containing all the given points is found.