Please answer questions A-D 21. (a) Suppose we are given two sorted arrays A[1..n] and B[1..n]. Describean algorithm to find the median element in the union of A and B inO(logn) time. You can assume that the arrays contain no duplicateelements.(b) Suppose we are given two sorted arrays A[1..m] and B[1..n] and aninteger k. Describe an algorithm to find the kth smallest element inAUB in O(log(m + n)) time. For example, if k = 1, your algorithmshould return the smallest element of AUB.) [Hint: Use your solutionto part (a).]"(c) Now suppose we are given three sorted arrays A[1..n], B[1..n], andC[1..n], and an integer k. Describe an algorithm to find the kth smallestelement in AUBUC in O(logn) time.55RECURSION(d) Finally, suppose we are given a two dimensional array A[1.. m, 1..n] inwhich every row A[i, ] is sorted, and an integer k. Describe an algorithmto find the kth smallest element in A as quickly as possible. How doesthe running time of your algorithm depend on m? [Hint: Solve problem16 first.]W

If n is 1, return the average of both arrays' elements. If n is 2, return the average of the first…

Suppose we are given two sorted arrays A[1..n] and B[1..n]. Describe an algorithm to find the median element in the union of A and B in O(log n) time. You can assume that the arrays contain no duplicate elements.

Suppose we are given two sorted arrays A[1..n] and B[1..n]. Describe an algorithm to find the median element in the union of A and B in O(log n) time. You can assume that the arrays contain no duplicate elements.

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

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1.Write a Θ(n) algorithm that sorts n distinct integers, ranging in size between 1 and kn inclusive, where k is a constant positive integer. (Hint: Use a knelement array.) 2.Algorithm A performs 10n 2 basic operations, and algorithm B performs 300 ln n basic operations. For what value of n does algorithm B start to show its better performance? 3. There are two algorithms called Alg1 and Alg2 for a problem of size n. Alg1 runs in n 2 microseconds and Alg2 runs in 100n log n microseconds. Alg1 can be implemented using 4 hours of programmer time and needs 2 minutes of CPU time. On the other hand, Alg2 requires 15 hours of programmer time and 6 minutes of CPU time. If programmers are paid 20 dollars per hour and CPU time costs 50 dollars per minute, how many times must a problem instance of size 500 be solved using Alg2 in order to justify its development
The ColumnChoice problem takes as input a two-dimensional array A[1..m, 1..n] of Os and 1s with m rows and n columns along with a non-negative integer k < n. It asks whether there exists a subset SC {1, ..., n} of k columns such that for each row i e {1,..., m} there exists at least one column j e S where A[i, j] = 1. Prove that ColumnChoice is NP- complete.
1. Asymptotics. Given an array A of n integers, you'd like to output a two-dimensional n x n array B in which B[i, j] = max {A[i], A[i + 1],..., A[j]} for each i < j. For i j the value of B[i, j] can be left as is. for i = 1,2, η for j = i + 1, n " Compute the maximum of the entries A[i], A[i + 1], , A[j]. Store the maximum value in B[i, j]. (a) Find a function of such that the running time of the algorithm is O(f(n)), and clearly explain why. (b) For the same function f argue that the running time of the algorithm is also (f(n)). (This establishes an asymptotically tight bound (f(n)).) (c) Design and analyze a faster algorithm for this problem. You should give an algorithm with running O(g(n)), where lim→∞ g(n)/f(n) = 0.
Question 2: Implement sequential search and binary search algorithms on your computer. Note down run time for each algorithm on arrays of size (1, 250, 500, 750, 1000, 1250, 1500, 5000). For both algorithms, store the values 0 throughn 1 in order in the array, and use a variety of random search values in the range 0 to n 1 on each size n. Graph the resulting times. When is sequential search faster than binary search for a sorted array? Solution: Execution Time for Input Size(N) Search Element Execution Time for Sequential Search(n) Binary Search(logn)
... Array A contains positive numbers that can be smaller than 1, larger than 1, or equal to 1. Find the sub-array A[..], such that the product of numbers in A[i..j] (i.e., A[i] × A[i+1] x x A[j - 1] x A[j]) is maximized. Of course, you should describe the fastest algorithm that you can find. A product of two numbers can be computed in time 0(1). Describe your algorithm in words, provide a pseudocode and briefly explain why your algo- rithm is correct.
An array is called a winner if it has two entries which together add up to a third entry, i.e. if A[i] + A[j] = A[k]for some i,j,k, where i≠j≠k. a)Write an algorithm (pseudocode preferred) to determine if a given array A is a winner. Your algorithm should not use any additional data structures. b)Analyze the running time of your algorithm using Big-O notation.
Problem 5. Let A be an array of n integers, each of which comes from the domain [1, U]. Note that some of the integers may be identical. Design an algorithm to compute, for each distinct integer x in A, how many integers in A are at most x. For example, if A stores the sequence of integers (35, 12, 28, 12, 35, 7, 63, 35), you should output an array ((7, 1), (12, 3), (28, 4), (35, 7), (63, 8)). Your algorithm should terminate in O(n log n) time.
) Consider an n x n array ARR stored in memory consisting of 0’s and 1’s such that, in a row of ARR, all 0’s comes before any of 1’s in the row. Write an algorithm having complexity O(n), if exists, that finds the row that contains the most 0’s. Step by step explain r algorithm with an illustrative example. 6
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Let A and B be two sorted arrays, each with n elements. For the purposes of this problem, assume that all 2n elements are distinct integers, and assumen= 2k + 1 for some integer k ≥ 1. In this problem, we wish to compute themedianof the 2nelements, which will be the average of the nth and (n+ 1)th smallest elements in the (combined) array. For example, suppose A= [1,2,4,8,16,32,64,128,256] and B= [30,40,50,60,70,80,90,100,110]. If we merge the arrays and sort it, we get C= [1,2,4,8,16,30,32,40,50,60,64,70,80,90,100,110,128,256]. From this we see that the middle two elements are 50 and 60 and so the median is 55. Question: Consider the following algorithm: merge the two arrays and sort it to produce a new array C with 2n elements. Then the median must be (C[n] +C[n+ 1])/2. Determine whether this algorithm is linear or linearithmic. Clearly justify your answer.
On 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.
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