Homework Assignment 1

Introduction to Algorithms and Analysis

CS 3530
Design and Analysis of Algorithms
Spring Semester 2014

Due: Thursday, January 30, at 12:30 PM

Tasks

1. Given two sorted lists of values, we would like to find the list of common elements, also sorted. For example, given [2, 2, 3, 7] and [2, 3, 3, 3, 5, 7], the list of common elements is [2, 3, 7].

   In Session 4, we saw one algorithm for solving this problem.

   
   
   • Design either a new top-down algorithm or a new bottom-up algorithm to solve this problem.
   
   
   • Explain why your algorithm is top-down or bottom-up.
   
   
   • Let m and n be the lengths of the input lists. State the maximum number of comparisons that your algorithm makes in terms of m and n.


2. Consider the Comparison Counting Sort algorithm below, which appears in Levitin's text. Given an unsorted array of values, it produces a new, sorted array. It does so by counting for each item v the number of items in the array that are less than v, and using that information to put v in its appropriate spot.
   
   
   • Demonstrate how this algorithm operates on this input array: [93, 13, 37, 14].
   
   
   • Explain why this algorithm is (or isn't) stable.
   
   
   • Explain why this algorithm is (or isn't) in-place.


3. Read the first two paragraphs of the section labeled What is a patent? on US Patent and Trademark Office's patents web page.

   Should the inventor of an algorithm be eligible to receive a utility patent? Why, or why not? (Limit yourself to three to five sentences.)

Deliverables

By the due date and time, submit a hardcopy of your solutions to my mailbox in the department office, or to the front desk in our classroom before class begins.

Comparison Counting Sort

    // Input : Array A[0..n-1] of orderable values
    // Output: Array S[0..n-1] of A's elements,
    //         sorted in nondecreasing order

    for i ← 0 to n-1 do
        Count[i] ← 0

    for i ← 0 to n-2 do
        for j ← i+1 to n-1 do
            if A[i] < A[j]
               then Count[j] ← Count[j] + 1
               else Count[i] ← Count[i] + 1

    for i ← 0 to n-1 do
        S[Count[i]] ← A[i]

    return S

(This algorithm appears on Page 34 of The Design and Analysis of Algorithms by Anany Levitin. I have made a few trivial typographical changes to the presentation.)