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 35.5, Problem 1E
Program Plan Intro
To determine that after the execution of the line
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et X = {1, 2, 3, 4, 5} and Y = {7, 11, 13} are two sets. find R = {(x, y): x ∈ X and y ∈ Y and (y – x) is divisible by 6}
b) Your lecturer is a funny guy; he wants you to find out from a sorted list that contains
distinct integers, whether there is an index i for which the value of the element i is also
i. For example, given a sorted list of distinct integers A[1], A[2], A[3],..., A[n], find
whether there is an index i for which A[i] = i. Your lecturer wants you to device an
algorithm, that performs the described function, that runs in complexity time 0(log n).
An example, A = [-1, 0, 1, 3, 7, 8, 9, 10] does have such an index, that is, A[3] = 3.
Suppose that we implement a union-find structure by representing each set using a balanced search tree. Describe and analyze algorithms for each of the methods for a union-find structure so that every operation runs in at most O(logn) time in worst case.
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Let X = {1, 2, 3, 4, 5} and Y = {7, 11, 13} are two sets. find R = {(x, y): x e X and y e Y and (y – {x) is divisible by 6arrow_forwardUse the Transform-and-Conquer algorithm design technique with Instance Simplification variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode. Given a set S of n integers and another integer x, determine whether or not there exist two elements in S whose sum is exactly x.arrow_forwardGiven 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_forward
- Let A[1...N] be a maxheap. Design an algorithm to change the array A into a minheap as theoutput. Briefly explain the main idea of your algorithm and then present the pseudocode of it.Additional space consumptions for carrying out the algorithm, e.g., additional arrays or linkedlists, should be minimized. You may assume that the standard function swap() is given. How do I solve this question?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_forwardConsider a list of n positive integers a1, a2, 9 an. Given a query positive integer m, we are required to determine if there is a prefix (that is, a₁, a2, ..., a;) of this list that sums to m. To solve such query problems, we precompute all prefix sums (there are n of them) and keep them in an array whose ith element is a₁ + a₂ + + ai. Now, the most efficient way of searching this list for m is to do a (a) Linear search (b) Binary search (c) Random search (d) None of the above Ans:arrow_forward
- Make K-map as followsarrow_forwardOn his way back from detention, Harry runs into his friend Hermione. He is upset because Professor Snape discovered that his sorting spell failed. Instead of sorting the papers correctly, each paper was within k slots of the proper position. Hermione immediately suggests that insertion sort would have easily fixed the problem. In this problem, we show that Hermione is correct (as usual). As before, A[1 ..n] in an array of n distinct elements. Show that sorting a list in which each paper is within k slots of its proper position takes N (n log k) comparisons. Hint: Use the decision-tree technique.arrow_forwardConsider the set X = {2, 3, 4, 5, 6, 7, 8, 9}, which contains 8 elements. How many subsets of X are such that exactly two of the elements from {2,3,4} are present in X?arrow_forward
- The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4arrow_forwardQUESTION 5 Given A=(nez: In| <2} and B-P(A), how many permutations of the elements in B start with (1 )? Answer:arrow_forwardConsider the following variant of Mergesort. If the array has more thantwo elements, it recursively sorts the first two-third and the last one-third. Then, it mergesthe first two-third with the last one-third. If the array has at most two elements, then ittrivially sorts the array. For completeness, here is the pseudocode for sorting the arrayA[l..r]. Write a recurrence relation for the worst-case running time of Newsort. Solve the recurrencerelation to obtain aΘ-bound on the running time of the algorithm, in terms of the lengthnof the array. You must use the tree method to solve the recurrence. You may also ignorefloors and ceilings in your calculations.arrow_forward
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