Consider array A of n numbers. We want to design a dynamic programming algorithm to find the maximum sum of consecutive numbers with at least one number. Clearly, if all numbers are positive, the maximum will be the sum of all the numbers. On the other hand, if all of them are negative, the maximum will be the largest negative number. The complexity of your dynamic programming algorithm must be O(n²). However, the running time of the most efficient algorithm is O(n). Design the most efficient algorithm you can and answer the following questions based on it. To get the full points you should design the O(n) algorithm. However, if you cannot do that, still answer the following questions based on your algorithm and you will get partial credit.

Consider array A of n numbers. We want to design a dynamic programming algorithm to find the maximum sum of consecutive numbers with at least one number. Clearly, if all numbers are positive, the maximum will be the sum of all the numbers. On the other hand, if all of them are negative, the maximum will be the largest negative number. The complexity of your dynamic programming algorithm must be O(n²). However, the running time of the most efficient algorithm is O(n). Design the most efficient algorithm you can and answer the following questions based on it. To get the full points you should design the O(n) algorithm. However, if you cannot do that, still answer the following questions based on your algorithm and you will get partial credit.

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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Consider array A of n numbers. We want to design a dynamic programming algorithm to find the maximum sum of consecutive numbers with at least one number. Clearly, if all numbers are positive, the maximum will be the sum of all the numbers. On the other hand, if all of them are negative, the maximum will be the largest negative number. The complexity of your dynamic programming algorithm must be O(n2). However, the running time of the most efficient algorithm is O(n). Design the most efficient algorithm you can and answer the following questions based on it. To get the full points you should design the O(n) algorithm. However, if you cannot do that, still answer the following questions based on your algorithm and you will get partial credit. Write the recursion that computes the optimal solution recursively based on the solu- (a) tion(s) to subproblem(s). Briefly explain how it computes the solution. Do not forget the base case(s) of your recursion.
You are given a collection of n bolts of different widths and n corresponding nuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However, there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to each nut. Design an algorithm for this problem with Θ(n log n) average-case complexity (in terms of nut-bolt comparisons). Explain why your algorithm has Θ(n log n) average case complexity.
You are given a collection of n bolts of different widths and n corresponding nuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However, there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to each nut. Design an algorithm for this problem with Θ(n log n) average-case complexity (in terms of nut-bolt comparisons). Explain why your algorithm has Θ(n log n) average-case complexity. You may use any result without explicitly solving recurrence equations.
You are given an arrayof colors (reds, whites, and blues), in no particular order. Sort them into the order of redsfollowed by whites followed by blues.a. Devise an efficient algorithm to solve this problem.b. What is your algorithm's time complexity?For example, if red is represented by 2, white represented by 4 and blue is represented by 6,your input and output array could be of following form:Input: [ 4 6 2 4 2 6 4 2 6 4 4 2 4 ]Output: [ 2 2 2 2 4 4 4 4 4 4 6 6 6]
We have N jobs and N workers to do these jobs. It is known at what cost each worker will do each job (as a positive numerical value). We want to assign jobs to workers in such a way that the total cost of completion of all jobs is minimal among other possible alternative assignments. For this problem, write the algorithm as pseudocode, whose input is a matrix representing worker/job costs, and the output is a list of tuples showing which work will be done by which worker, and that tries to reach the solution with GREEDY technique. Explain in what sense your algorithm exhibits greedy behavior. What is the time complexity of your algorithm? Interpret if your algorithm always produces the best (optimum) result for each instance of the problem.
You are given an array A[1..n] of positive numbers where Ai] is the stock price on day i. You are allowed to buy the stock once and sell it at some point later. For each day you own the stock you pay S1 fee. Design a divide-and-conquer algorithm that will return a pair (i,j) such that buying the stock on day i and selling it on day j will maximize your gain The complexity of the algorithm has to be O
You are given an array A[1..n] of positive numbers where Ai] is the stock price on day i. You are allowed to buy the stock once and sell it at some point later. For each day you own the stock you pay S1 fee. Design a divide-and-conquer algorithm that will return a pair (i,j) such that buying the stock on day i and selling it on day j will maximize your gain The complexity of the algorithm has to be O(n log n) You are given an array A[1..n] of positive numbers where Ai] is the stock price on day i. You are allowed to buy the stock once and sell it at some point later. For each day you own the stock you pay S1 fee. Design a divide-and-conquer algorithm that will return a pair (i,j) such that buying the stock on day i and selling it on day j will maximize your gain The complexity of the algorithm has to be O(n log n)
Let m be a matrix with n rows and n columns whose entries are either 1 or 0. recall that the element of m on row i and column j is denoted by mij . the diagonal entries of m are {mii} for 1 ≤ i ≤ n. We call M permutable if it is possible to swap some of the rows and some of columns so that all diagonal entries are 1. Design a polynomial time algorithm that decides whether a binary matrix M is permutable or not. Note that we can swap any two rows or two columns. Also, the order in which these swaps are done is not important.
You have an array of 10,000 subject scores for a high-school exam and you want to find the scores that are ranked from 4900 to 5100 that is the 201 most middle-ranked scores. Precisely describe two alternative algorithmic ap proaches to doing this - one of which is, in terms of time complexity, more efficient than the other.
4 points Given an n-element array X of integers, Algorithm A executes an O(n3.4)-time computation for each even positive number in X, an Oin2.3-time computation for each odd positive number in X, and an O(n2.5)-time computation for each negative number in X. What are the best-case and worst-case running times of Algorithm A? Justify your answer. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac) BIUS Paragraph V Arial 10pt EVE 2 IXO QFS3E
Given a positive integer number n (i.e., n>0), we can use the following algorithm to find the nth fibonacci number:fibonacci(n){If n=1, return 1;If n=2, return 1;return fibonacci(n-1) + fibonacci(n-2);}Given that there exists overlaps in the sub-problems of the problem, write a Dynamic programmingalgorithm to overcome the overlapping. Implement both algorithms in C++. You need to submit source code. Give the number of additions needed for each algorithm when n=10, 15, 20, 25, 30, 35, 40.
Consider the problem of finding the distance between the two closest numbers in an array of n and y is computed as x - y.) numbers. (The distance between two numbers x Design a presorting-based algorithm in pseudo code for solving this problem and determine its efficiency class. Answer: // your algorithm goes here Provide the the algorithm efficiency analysis here: