1. Let S be a set of n arbitrary but distinct numbers. (a) Give a deterministic algorithm to output the largest n³/4 of the numbers in S in sorted order. Your algorithm must use O(n) comparisons. (b) Show that any deterministic comparison-based algorithm that correctly outputs the median on inputs of n distinct elements must use at least n - 1 comparisons. To be completely precise, the median is defined to be the [n/2]th smallest element.

1. Let S be a set of n arbitrary but distinct numbers. (a) Give a deterministic algorithm to output the largest n³/4 of the numbers in S in sorted order. Your algorithm must use O(n) comparisons. (b) Show that any deterministic comparison-based algorithm that correctly outputs the median on inputs of n distinct elements must use at least n - 1 comparisons. To be completely precise, the median is defined to be the [n/2]th smallest element.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

Suppose arrays A and B are both sorted in increasing order and both contain n elements. What is the time complexity to find the median of A U B, i.e., the set of elements in A or B?
Problem 3. Consider the following recurrence. T(n) = {(n) = 37(n T(n) = 3T(n/2) + n² if n=1 otherwise. (a) Solve this recurrence exactly by the method of substitution. You may assume n is a power of 2. (b) Solve it using the recursion tree method.
Devise a divide-and-conquer algorithm that multiplies a matrix A and a matrix B (both of them are n-by-n) in time O(nlo927).This improves over the naive O(n³) algorithm we learned in high school. Your answer should be a clear exposition of the steps needed to perform the multiplication, and also a proof of its runtime. If it helps feel free to assume that n is a power of 2.
Question: Suppose we sample m indices from {1, . . . , n} uniformly at random and with repetition. What is theprobability that none of these indices is a correct answer?
Question A: Mapping. The two-sum problem is a popular algorithm problem. Given an array of integers and an integer target, return indices of the two numbers such that they add up to the target. You may assume that each input would have exactly one solution, and you may not use the same element twice. For example, suppose we have an array arr = [1,10,100]. If the target is 11, you should return [0,1] because arr[0] + arr[1] = 1+ 10 = 11. If the target is 101, you should return [0,2] because arr[0] + arr[2] = 1+ 100 = 101. Complete this problem in O(n). Hint: using a hash table.
Input: An odd integer B, and a set A= {a_1, . .. , a_2n} of 2n distinct positive integers. Question: Decide whether A can be partitioned inton disjoint pairs (a_i, a_j), where 1 s i
Problem 3: Median of two sorted arrays. Finding the median of a sorted array is easy:return the middle element. Suppose we are given two sorted arrays A and B; the array Aconsists of m not necessary different integers, while B consists of n not necessary differentintegers. Suppose that A and B are disjoint, that is, there does not exist an element whichis in A and B.(i) Describe a naive algorithm which in time O(m + n) finds the median of all integersin A and B.(ii) If m = n, describe an algorithm which in time O(log n) finds the median of all integersin A and B.
How did my textbook get 2(n − 1) + 1 = 2n − 1 comparisons are used whenever this algorithm is applied: (textbook solves the time complexity of finding the maximum element in a finite set using the following algorithm) Algorithm below: procedure max(a1, a2,…, an: integers) max := a1 for i := 2 to n if max < ai then max := ai return max{max is the largest element}
An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.
please answer with proper explanation and step by step solution.
Problem 2. Suppose we have an array A[1 : n] which consists of numbers {1,...,n} written in some arbitrary order (this means that A is a permutation of the set {1,...,n}). Our goal in this problem is to design a very fast randomized algorithm that can find an index i in this array such that A[i] mod 8 € {1,2}, i.e., the reminder of dividing A[i] by 8 is either 1 or 2. For simplicity, in the following, we assume that n itself is a multiple of 8 and is at least 8 (so a correct answer always exist). For instance, if n = 8 and the array is A = [8,7, 2,5, 4, 6,3, 1], we want to output either of indices 3 or 8. (a) Suppose we sample an index i from {1,...,n} uniformly at random. What is the probability that i is a correct answer, i.e., A[i] mod 8 E {1,2}? (b) Suppose we sample m indices from {1,...,n} uniformly at random and with repetition. What is the probability that none of these indices is a correct answer? Now, consider the following simple algorithm for this problem: Find-Index-1(A[1:…