CSCE 221: Data Structures and Algorithms

Sections: 507, 508, 509
Fall 2015

Homework 7

Due Date: Oct. 30th (before class)

Assignment

  1. R11.8 (Pg. 545)

    Suppose we are given two n-element sorted sequences A and B that should not be viewed as sets (that is, A and B may contain duplicate entries). Describe an O(n)-time method for computing a sequence representing the set A ∪ B (with no duplicates). Provide pseudocode for the algorithm, an informal argument as to its correctness, complexity and its justification.

  2. R11.20 (Pg. 546)

    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. Provide pseudocode for the algorithm and its correctness. 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]? State what and where in your pseudocode you will be making the necessary changes for the generalized algorithm with d tuples. Justify its correctness and state its complexity with justification.

  3. R11.26 (Pg. 547)

    Show that the worst-case running time of quick-select on an n-element sequence is Ω(n2).

  4. C11.2 (Pg. 547)

    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. Justify the correctness and complexity of your algorithm.
    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.

Bonus

    C11.29 (Pg. 549)

    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(nlog n/log k) time.

Grading Rubric

Each homework composes 2% of your final grade. Each homework will be graded out of 100 possible points. In any problem asking you to design an algorithm please also provide proper pseudocode (mathematical notation), complexity analysis (both space and time), and brief justification of complexity and correctness.

All assignments must be turned in with a signed coverpage. If no coverpage is provided, the assignment will not be graded.

If there are any discrepancies in grades please see Jory Denny during his office hours (not the TA or Grader).