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1. Suppose A[1..n] is an array of n distinct integers. Each integer A[i] could be positive, negative, or zero. Find

a contiguous subarray which has the largest sum. For example, if the A = [−2, 1,−3, 4,−1, 2, 1,−5, 4], then

the contiguous subarray with the largest sum is 4,−1, 2, 1, with sum 6. Design a recursive and an iterative

algorithm runs in O(n) time. (Hint: use induction)

2. Given n numbers, find the maximum and the second maximum in about n + log n steps. (Fact: if the range

of the numbers is unbounded, the only thing you can do is using comparison. Therefore, you can’t use radix

sort here)

3. The Element Uniqueness problem is to determine whether all elements in an array are distinct. If you are

only allowed to use comparison, the problem will need at least O(n log n) time. Based on this fact, prove that

there is no algorithm for closest pair problem less than O(n log n) time if the points are not from a bounded

domain (see comment in problem 2), because then you’ll be able to answer the Element Uniqueness problem

in less than O(n log n) time, also called o(n log n)- little O of n log n, i.e. function whose asymptotic grows

slower than a rate of a constant from n log n, e.g., n

p

log n.

4. Design a convex hull algorithm in O(n log n) time using divide and conquer.

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Æ Unless specified, you should justify your algorithm with proof of correctness and time complexity.

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