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COMP90038 Algorithms And Complexity

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Questions:

Objectives

To improve your understanding of data structures and algorithms for sorting and search. To consolidate your knowledge of trees and tree-based algorithms. To develop problem-solving and design skills. To develop skills in analysis and formal reasoning about complex concepts. To improve written communication skills; in particular the ability to use pseudo-code and present algorithms clearly, precisely and unambiguously.

Problems

  1. Consider the data sequence S = [82; 91; 13; 92; 64; 10; 28; 55; 96; 97]. Draw a valid AVL tree for it, assuming that the data has arrived one at the time. Show detailed steps by giving the AVL tree after inserting each element.
  1. Consider two sets of integers, X = [x1; x2; : : : ; xn] and Y = [y1; y2; : : : ; yn]. Write two versions of a FindSetIntersection(X; Y ) algorithm to nd the common elements in both sets. Each of your algorithms should return an array with the common elements, or an empty array if there are no common elements.

You may make use of any algorithm introduced in the lectures to help you develop your solution. That is, you do not have to write the ‘standard’ algorithms { just use them. Therefore, you should be able to write each algorithm in about 10 lines of code. You must include appropriate comments in your pseudocode.

  • Write a pre-sorting based algorithm of FindSetIntersection(X; Y ). Your algorithm should strictly run in O (n log n).
  • Write a Hashing based algorithm of FindSetIntersection(X; Y ). Your algorithm should run in O (n).
  1. Sloppy Inc. has a very unusual way to communicate the decisions made by its CEO to all employees. Each day, any employee that knows the decision can disclose it to at most one of its direct subordinates. Design an e cient algorithm to compute the minimum number of days required for the decision to be disclosed to all employees. What is the time complexity of your algorithm?

To help you design your algorithm, assume that Sloppy Inc. has a hierarchical structure resembling an n-ary tree. Each employee is labeled f0; 1; : : : ; n 1g, where 0 corresponds to root of the tree (the CEO). To store the tree you can use a two-dimensional array C [n] [n], where k = C [i] [0] is the number of direct subordinates of employee i, and C [i] [1 : : : k] contains the labels of each direct subordinate employee. Any other entry in the array has value of 1. Note that the order in which the messages are distributed matters, e.g., employees with deeper subordinate trees should probably receive the message rst.

  1. Given an array of n numbers A [0 : : : n 1]. Write an e  cient algorithm for below cases:

(a) For each of the element A [i], fifind the minimum j so that A [j] > A [i] and j > i. Your algorithm should return an array of length n. If such j does not exist for some i, that entry should be -1. What is the complexity of your algorithm?

(b) For each of the element A [i], fifind the minimum A [j] so that A [j] > A [i] and j > i. Your algorithm should return an array of length n. If such j does not exist for some i, that entry should be -1. Your algorithm should have O (n log n) complexity.

To help you verify your algorithm, for the sequence [80; 19; 49; 45; 65; 71; 76; 28; 68; 66] the results are:

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My Assignment Help. Algorithms And Complexity [Internet]. My Assignment Help. 2021 [cited 11 April 2021]. Available from: https://myassignmenthelp.com/free-samples/comp90038-algorithms-and-complexity/understanding-of-data-structures.html.


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