Homework 7

General Comments

For full credit, provide context for each problem, show all calculations, and justify all answers by providing enough comments to explain your reasoning.
Homework assignments must be very neatly written or typeset (e.g. using Word or OpenOffice).
You can get up to 50% credit on a problem if you get significant outside assistance. Thus, if you are totally stuck on a problem it might be worth getting help. However, you must indicate any assistance/collaboration (See the Homework Assistance section on the Policies page). Failure to do so could result in a failing grade for the course! Note that getting help from the Help Center or me does not count as significant outside assistance, but talking with your classmates or searching on the Internet does!
If a problem asks for an algorithm, you should give the most efficient algorithm you can find to ensure full credit. You should also specify the complexity of the algorithm with justification, whether or not the problem asks for it.

Max Solvable Input Size: Assume your algorithm for Subeset Sum runs in \( 5n \cdot 2^n \) steps and your machine processes 100 million steps/second. Estimate the largest value of \( n \) solvable in 1 second, 1 hour, 1 day, 1 month, and 1 year. Show your work. (Hint: You might want to use a spreadsheet to help you with the calculations. If so, make sure what you submit has enough details printed so I can determine the formulas.)
Subarray Sum Match: Given an array of positive integers \( A[0 \dots n-1] \) and an integer target value \( k \), find all contiguous subarrays whose elements sum to exactly \( k \). (A contiguous subarray is a subset consisting of all of the elements from some index \(i\) to some index \(j\geq i\).)
1. Design and write a brute-force algorithm using a nested loop to solve this problem.
2. What are the best-case and worst-case time complexities?
3. Apply your algorithm to the example \( A = [1, 5, 2, 7, 4, 2, 5] \) with \( k = 10 \). Show all subarrays considered.