CMSC 221: Data Structures Section: 02 Spring 2017

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Homework Assignment 7

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Due: Mar. 28 before class starts

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Assignment

1. (30 points) Suppose we are given two n-element sorted sequences A and B each with distinct elements, but potentially some elements that are in both sequences. Describe an O(n)-time method for computing a sequence representing the union A ∪ B (with no duplicates) as a sorted sequence.
   1. Provide and explain pseudocode
   2. Statement and proof of time complexity
   3. Statement and proof of memory complexity
2. (30 points) Describe a radix-sort method for lexicographically sorting a sequence S of triplets (k, l, m), where k, l, and m are integers in the range [0, N - 1], for some N ≥ 2. How could this scheme be extended to sequences of d-tuples (k1, k2, ..., kd), where each ki is an integer in the range [0, N - 1]?
   1. Provide and explain pseudocodes
   2. Statement and proof of time complexities
   3. Statement and proof of memory complexities
3. (40 points) Linda claims to have an algorithm that takes an input sequence S and produces an output sequence T that is a sorting of the n elements in S.
   1. Give an algorithm, isSorted, for testing in O(n) time if T is sorted.
      1. Provide and explain pseudocode
      2. Statement and proof of time complexity
      3. Statement and proof of memory complexity
   2. Explain why the algorithm isSorted is not sufficient to prove a particular output T of Linda's algorithm is a sorting of S.
   3. Describe what additional information Linda's algorithm could output so that her algorithm's correctness could be established on any given S and T in O(n) time.
4. Bonus. (10 points) Space aliens have given us a program, alienSplit, that can take a sequence S of n integers and partition S in O(n) time into sequences S1, S2, ..., Sk of size at most ⌈n/k⌉ each, such that the elements in Si are less than or equal to every element in Si+1, for i = 1, 2, ..., k - 1, for a fixed number, k < n. Show how to use alienSplit to sort S in O(n log n / log k) time.
   1. Provide and explain pseudocode
   2. Statement and proof of time complexity

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General Instructions, Turning in assignments, and Grading

General Instructions

• All homework assignments must be typeset using LaTex. Using another program (e.g., Microsoft Word) or handwriting assignments will result in grade of 0. Here is an example homework problem worked out to see the expected format and the tex file that generated it to help you start.
• If you are unfamiliar with LaTex, here is a short tutorial and the tex file that generated it. This tutorial shows how to write common mathematics, how to write pseudocode, how to cite sources, how to include images, and how to include tables.
• Use pseudocode and good mathematical style to describe algorithms and data structures. Do not specify Java code unless otherwise stated. Described more in example homework document above.
• Complete every part of every problem!
• Follow turn-in instructions precisely.
• Failure to complete any of these steps will result in a significant loss of points.

Turn in Instructions

Each assignment will be turned in in class (hard copy). Assignments are due BEFORE, let me repeat, before class starts. This does not mean five minutes after class starts. Details:

• The first page of your hard copy must be a signed coverpage.
• Next put the problems in order as described in the description. This may seem silly but you would be surprised sometimes.
• If you do not know how to print you may have to consult University of Richmond webpages to learn how to use campus printers. If possible, please print 2-sided.
• Staple all pages together.
• Turn in packet before class begins.
• I reserve the right to assign a 0 to any assignment failing to comply with these instructions. Even for something as small as a missing staple.

Points

• Each assignment is graded out of 100 points (not including bonus).
• Criteria and point distribution (This distribution is a general guideline that may change depending on the specific problem - exceptions for example include but are not limited to proof-only problems.)
  • Approximately 50% on a problem is devoted to the algorithm. This includes items like describing a correct and efficient algorithm, using proper and clear pseudocode, and describing your algorithm clearly.
  • Approximately 25% on a problem is devoted to time complexity. This includes stating a lemma or theorem of your result and providing a proof and justification of the result.
  • Approximately 25% on a problem is devoted to memory complexity. This includes stating a lemma or theorem of your result and providing a proof and justification of the result.
  • If the problem is simply a proof, 100% is split between a proper statement (lemma or theorem) of the result and proving a clear and concise proof of the result.
  • You can receive points off for not following formatting guidelines.
• If there are any discrepencies in grades please see the instructor during his office hours or by appointment (do not discuss with the lab assistants or graders).

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