Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 9.3, Problem 3E
Program Plan Intro
To show that quick sort can be made to run in
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We have two input arrays, an array A with n elements, and an array
B with m elements, where m > n. There may be
duplicate elements. We want to decide if every element of B is an
element of A.
a. Describe a brute-force algorithm. What is the worst-case time
complexity?
b. Describe an algorithm to solve this problem in O(m log n) worst
case time. (Hint: You may apply instance simplification.)
Let T be a sorted array of n elements. An element x is said to be a majority element in T if the number of elements i, with T[i] = x, is greater than n/2.
Give an algorithm (code or pseudo-code) that can decide whether T includes a majority element (it cannot have more than one), and if so, find it. Your algorithm must run
in linear time.
11. Given is a sequence X of n keys k1, k2, ..., kin. For each key k; (1 < i<
n), its position in sorted order differs from i by at most d. Present an
O(n log d)-time sequential algorithm to sort X. Prove the correctness
of your algorithm using the zero-one principle. Implement the same
algorithm on a yn x Vn mesh. What is the resultant run time?
Chapter 9 Solutions
Introduction to Algorithms
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- For the problem, give pseudocode for your solution, and remember to include a proof of correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute force O(n 2 ) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative. Let H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’t contribute an edge to the intersection of all half planes in H. Prove that for any redundant half-plane h ∈ H, there are two other half-planes h' , h00 ∈ H such that h' ∩ h'' contains h. Use this fact to give an algorithm (as fast as possible) to compute all redundant half-planes.arrow_forwardYou are given k > 2 linked lists, each containing n > k natural numbers sorted in increasing order. All n · k numbers in these lists are distinct. Describe an algorithm that finds the k-th smallest element among all n k numbers. Your algorithm should run in O(k log k) time. Analyze the running time of your algorithm and prove its correctness.arrow_forwardsolution should have O(l1.length + l2.length) time complexity, since this is what you will be asked to accomplish in an interview. Given two singly linked lists sorted in non-decreasing order, your task is to merge them. In other words, return a singly linked list, also sorted in non-decreasing order, that contains the elements from both original lists.arrow_forward
- Given an array of numbers X₁ = {x₁, x2, ..., n } an exchanged pair in X is a pair xi, xj such that i x¡ . Note that an element x; can be part of up to n - 1 exchanged pairs, and that the maximal possible number of exchanged pairs in X is n(n − 1)/2, which is achieved if the array is sorted in descending order. Give a divide-and-conquer algorithm that counts the number of exchanged pairs in X in O(nlogn) time.arrow_forwardIn n=4, what set(s) of cube(s) is(are) equivalent to the set {0000, 0010, 01x1, 0110, 1000, 1010}? (equivalent= they cover the same set of minterms)arrow_forwardFor the problem, give pseudocode for your solution, and remember to include a proof of correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute force O(n 2 ) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative. In IR2 , we define a slab to be a pair of parallel lines. Given a set of points P in IR2 , find the narrowest slab containing P, where the width of the slab is the vertical distance between its bounding lines.arrow_forward
- 5. Let g(n) = log10 (n). Prove that g(n) = (lgn). Please show step by step solution. Show the work.arrow_forwardSuppose you have m sorted arrays, each with n elements, and you want to combine them into a single sorted array with mn elements. 1. If you do this by merging the first two arrays, next with the third, then with the fourth, until in the end with the last one. What is the time complexity of this algorithm, in terms of m and n? 2. Give a more efficient solution to this problem, using divide-and-conquer. What is the running time?arrow_forwardLet’s assume that you want to find both the minimum and maximum of m numbers. Design an algorithm that achieves the goal in [3n/2]−2 comparisons in the worst case (you can write a pseudo code or a paragraph,don’t write code). Explain it.arrow_forward
- Let's pretend we're interested in checking whether there are any duplicates in an unsorted array of n entries, A. These components are whole numbers in the range [1, 2n]. Share the worst-case running time T(n) as an asymptotic order. Seek for the most effective algorithm.arrow_forwardGiven a linked list L storing n integers, present an algorithm (either in words or in a pseudocode) that decides whether L contains any 0 or not. The output of your algorithm should be either Yes or No. What is the running time of your algorithm in the worst-case, using O notation?arrow_forwardQuestion: use magic-pivot to obtain a deterministic quick-sort algorithim with the worst-case running time of O(n log n).arrow_forward
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