CMSC 221: Data Structures Section: 02 Spring 2017

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Homework Assignment 3

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Due: Feb. 2 before class starts (now Feb. 4 by 5pm)

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Assignment

1. (30 points) Provide statements and proofs for the following:
   1. What is the minimum number of internal nodes for an improper binary tree with n nodes?
   2. What is the maximum number of internal nodes for an improper binary tree with n nodes?
   3. What is the minimum number of external nodes for an improper binary tree with n nodes?
   4. What is the maximum number of external nodes for an improper binary tree with n nodes?
   5. Let T be a proper binary tree with height h and n nodes. Show that log(n+1)-1 ≤ h ≤ (n-1)/2.
   6. For the prior question what structure of trees yield the upper and lower bounds?
2. (40 points) Show how to use an Euler tour traversal to compute the level number f(p), as defined in Section 8.3.2, of each position in a binary tree T.
   1. Provide and explain pseudocode
   2. Statement and proof of time complexity
   3. Statement and proof of memory complexity
3. (30 points) Let T be a tree with n positions. Define the lowest common ancestor (LCA) between two positions p and q as the lowest position in T that has both p and q as descendants (where we allow a position to be a descendant of itself). Given two positions p and q, describe and analyze an efficient algorithm for finding the LCA of p and q.
   1. Provide and explain pseudocode
   2. Statement and proof of time complexity
   3. Statement and proof of memory complexity
4. Bonus. (10 points) Two ordered trees T' and T" are said to be isomorphic if one of the following holds:
   • Both T' and T" are empty.
   • Both T' and T" consist of a single node.
   • The roots of T' and T" have the same number k≥1 of subtrees, and the ith such subtree of T' is isomorphic to the ith such subtree of T" for i=1,...,k.
   Design and analyze an algorithm that tests whether two given ordered trees are isomorphic.
   1. Provide and explain pseudocode
   2. Statement and proof of time complexity
   3. Statement and proof of memory 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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