A “divide and conquer” algorithm is one that divides a problem into smaller subproblems, solves those subproblems, and then combines the “sub-solutions” into a solution to the original problem. In this assignment, you’ll apply the divide and conquer approach to solve two variations on a problem.
Deliverables:
Part 1: Finding the Singleton Element
A singleton is a single item, as opposed to two (or more) items grouped together, as in the “singleton pattern”, describing an object that may only be instantiated once, or a “singleton set”, a set containing only one element.
Consider the following problem: you are given a sorted sequence of integers. Within this sequence, one and only one element is unique; the other elements are all duplicated once. For example:
(−2,−2,5,5,12,12,67,67,72,80,80)
In your programming language of choice per Assignment 1, implement an algorithm to find that lone, unique, singleton element. In the example above, your algorithm should return 72.
Your program must accept as a command line argument the name of a file containing a sequence as described above, then print to stdout the singleton element. For example:
$ ./compile.sh
$ ./run.sh in1.txt
72
Your program will be tested using diff, so its printed output must match exactly.
As you solve this problem, recall that there are two broad types of divide and conquer algorithms: those that look like binary search and those that look like merge sort. Which of those approaches will lead to a more efficient algorithm for this problem?
Part 2: Dealing with Multiple Copies
Consider the following generalization of the problem: you are given a sorted sequence of integers. Within this sequence, one and only one element is unique; the other elements are duplicated at least once. For example:
(−2,−2,5,5,5,67,67,72,80,80,80,80)
Does this alteration affect your algorithm? If so, write new pseudocode accordingly, however, do not adjust your implementation to match — your program need only solve the original variation of the problem.
