Question 13 Given n numbers in hexadecimal format, each number has m digits. We call this data set, "data set A". We encode (convert) the numbers into Binary format. Let's ignore the encoding cost. The number in Binary format is called "data set B". We sort the numbers in both data sets using Radix sort. Proof that the time complexity (0) of Radix sort of data set B always is greater than the Radix sort for the data set A.

Question 13 Given n numbers in hexadecimal format, each number has m digits. We call this data set, "data set A". We encode (convert) the numbers into Binary format. Let's ignore the encoding cost. The number in Binary format is called "data set B". We sort the numbers in both data sets using Radix sort. Proof that the time complexity (0) of Radix sort of data set B always is greater than the Radix sort for the data set A.

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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Except for one number, an array A includes all the integers from O to n. We cannot access a full integer in A with a single operation in this scenario. The elements of A are encoded in binary, and the sole way to access them is to "get the jth bit of A[i];" this operation takes constant time. Create code to locate the missing integer. Can you do it in O(n) time?
Prove that the maximum number of compares used for a binary search in a table of size N is precisely the number of bits in the binary representation of N, because the operation of shifting 1 bit to the right converts the binary representation of N into the binary representation of ⎣N/2⎦.
Modify the quicksort function so that it calls insertion sort to sort any sublistwhose size is less than 50 items. Compare the performance of this version withthat of the original one, using data sets of 50, 500, and 5,000 items. Then adjust thethreshold for using the insertion sort to determine an optimal setting. Use python’s time module to calculate the duration of the original quicksort version and the modified version. Do this for 3 different data sets of 50, 500, and 5000 items. These datasets are not going to be provided, so you have to come up with them. You can use python’s random module to help come up with the random item. Experiment with a different threshold value for the size of the sublist that indicates a switch to insertion sort, and report which value was optimal. Use this template: import timedef original_quicksort(input_list):sorted_list = []#TODO: Your work here# Return sorted_listreturn sorted_list def modified_quicksort(input_list):sorted_list = []#TODO: Your…
Alex had an array A of length N such that 1≤Ai≤N for all 1≤i≤N. He constructed another binary array B of length N in the following manner: Bi=1 if the frequency of element i in A is odd. Bi=0 if the frequency of element i in A is even. Such an array B is called the parity encoding array of A. For example, if A=[1,1,2,3], then B=[0,1,1,0]. Unfortunately, he completely forgot the array A and vaguely remembers the parity encoding array B. He is now wondering whether there exists any valid array A for which the parity encoding array is B. Can you help himf? Input Format The first line contains a single integer T — the number of test cases. Then the test cases follow. The first line of each test case contains an integer N — the size of the arrays A and B. The second line of each test case contains N space-separated integers B1,B2,…,BN denoting the parity encoding array B. Output Format For each test case, output YES if there exists any valid array A for which the parity encoding…
Prove that the number of compares used for a binary search in a table of size N is exactly the number of bits in the binary representation of N, because shifting 1 bit to the right transforms the binary representation of N into the binary representation of N/2.
In an array A, every integer from O to n is present with the exception of one. We are unable to access a full integer in A in this case with a single operation. The sole operation we can use to access the elements of A, which are represented in binary, is "get the jth bit of A[i]," which has a constant time cost. To discover the missing integer, write some code. Do you have O(n) time?
The parity function is a very simple hash function producing a one-bit digest, which is 0 if the number of bits in the message is even and 1 if it is odd. Consider a communication system that uses 7 bits out of every byte to transmit data, and reserves the 8th bit (the one on the right) to transmit the parity of the other seven bits, for purposes of error checking. Below is the binary representation of some bytes that were transmitted using this system. For each one, tell whether there is a reason to suspect an error in transmission, giving the reason for your answer. 10110011 11101111 10101100 00110011
The Burrows-Wheeler transform (BWT) is a transformation that is used in data compression algorithms, including bzip2, and in high-throughput sequencing in genomics. Write a SuffixArray client that computes the BWT in linear time, as follows: Given a string of length N (terminated by a special end-of-file character $ that is smaller than any other character), consider the N-by-N matrix in which each row contains a different cyclic rotation of the original text string. Sort the rows lexicographically. The Burrows-Wheeler transform is the rightmost column in the sorted matrix. For example, the BWT of Mississippi is ipssm$pissii. The Burrows-Wheeler inverse transform (BWI) inverts the BWT. For example, the BWI of ipssm$pissii is mississippi$. Also write a client that, given the BWT of a text string, computes the BWI in linear time.
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Consider the following problem on a dictionary of n words, W1...Wn, each with exactly k characters. You can transform a word Wi into word Wj if they differ in at most d≤k characters. (both dd and kk are specified as part of the input, along with n and the words) For example, if the dictionary is: W1 = 'hit', W2 = 'cog', W3 = 'hot', W4 = 'dot', W5 = 'dog', W6 = 'lot', W7 = 'log', and d=1d=1, one way to change 'hit' to 'cog' is: 'hit' →→ 'hot' →→ 'dot' →→ 'dog' →→ 'cog'. We want to find the fewest number of steps to transform W1 to W2. Q1.1 I claim that this problem be formulated as a shortest path problem. Please provide a strategy to formulate this as the shortest path problem and give a graph visualization of the problem Q1.2 Show that your graph (in Q 1.1) can be constructed in O(n^2) time, and its size is up to O(n^2).
Given two arrays X and Y of positive integers, find the number of pairs such that xy > yx (raised to power of) where x is an element from X and y is an element from Y. Example 1: Input: M = 3, X[] = [2 1 6] N = 2, Y[] = [1 5]Output: 3Explanation: The pairs which follow xy > yx are as such: 21 > 12, 25 > 52 and 61 > 16 .Example 2: Input: M = 4, X[] = [2 3 4 5]N = 3, Y[] = [1 2 3]Output: 5Explanation: The pairs for the given input are 21 > 12 , 31 > 13 , 32 > 23 , 41 > 14 , 51 > 15 . Your Task:This is a function problem. You only need to complete the function countPairs() that takes X, Y, M, N as parameters and returns the total number of pairs. Expected Time Complexity: O((N + M)log(N)).Expected Auxiliary Space: O(1). Constraints:1 ≤ M, N ≤ 1051 ≤ X[i], Y[i] ≤ 103
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