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Design and analyze an algorithm (as fast as possible) that returns the k smallest numbers in A in sorted order.

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Exercise 1.4. Let A[1..n] € R" be an array of n comparable elements. Let kЄ N be an input parameter where k ≤ n. (We are particularly interested in the case where k is much smaller than n). Consider the problem where we want to obtained a sorted list of the k smallest elements.

1. Design and analyze an algorithm (as fast as possible) that returns the k smallest numbers in A in sorted order.

2. Prove a lower bound (large as possible, up to constants) on the number of comparisons required by any correct algorithm.

Food for thought: How does your lower bound compare to your upper bound obtained in exercise 1.4?

Exercise 1.5. Consider coordinates in the plane; i.e., pairs (a,b) where a, b € R. Recall that two coordinate pairs (a,b) and (c,d) are comparable if either (a) a≤c and b ≤d, or (b) c ≤ a and d < b.

Given a set of n coordinate pairs (a1, b1),..., (an, bn), consider the problem of computing the number of pairs of coordinate-pairs that are comparable. You may assume all pairs are distinct." Design and analyze an algorithm (as fast as possible) for counting the number of comparable pairs.

"We added this assumption on Wednesday, January 19, to make things a little simpler. Of course if you already solved the problem and can already handle duplicates then no adjustment is required.

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