Problem 1. Suppose you are given an array A[1 n] of integers that can be positive or negative. A contiguous subarray A[i: j] of A is called a positive subarray if sum of its entries is strictly positive, i.e., j ΣA[k] > 0. k=i Design and analyze a dynamic programming algorithm that in O(n²) time finds the smallest number of positive contiguous subarrays of A such that every positive entry of A belongs to exactly one of these subarrays. You can assume that A has at least one positive entry and thus the answer is at least one.

Problem 1. Suppose you are given an array A[1 n] of integers that can be positive or negative. A contiguous subarray A[i: j] of A is called a positive subarray if sum of its entries is strictly positive, i.e., j ΣA[k] > 0. k=i Design and analyze a dynamic programming algorithm that in O(n²) time finds the smallest number of positive contiguous subarrays of A such that every positive entry of A belongs to exactly one of these subarrays. You can assume that A has at least one positive entry and thus the answer is at least one.

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

Write an efficient algorithm for the following problem, and describe your reasoning. Determine the Time complexity and if you cannot find any polynomial time algorithm, then give a backtracking algorithm. Problem: Binary Array Core Input: Two integers p and q and a binary array A[1...n], i.e., each entry contains either a 0 or 1. Output: Print Yes - if there exists a subarray A[i...k], where 1<= i < k <= n, with exactly p zeros in A[1...i-1] (there may be 1s) and exactly q ones in A[k+1...n] (there may be 0s). Print No - otherwise. Outputs with examples Input: A = [0,1,1,0,1], p = 2, q =1, Output: No. Input: A = [0, 1,1,0, 0,1,0,1], p = 1, q =1, Output: Yes because we can have A[1...i-1] = [0] and A[k+1 ... n] = [0,1,0,1]
In this exercise we will develop a dynamic programming algorithm for finding the maximum sum of consecutive terms of a sequence of real numbers. Thatis, given a sequence of real numbers a1, a2,… , an,the algorithm computes the maximum sum ∑k(top) i=j(bottom) a_iwhere 1 ≤ j ≤ k ≤ n. b) Let M(k) be the maximum of the sums of consecutiveterms of the sequence ending at ak. That is, M(k) =max1≤j≤k∑ki=j ai. Explain why the recurrence relationM(k) = max(M(k − 1) + ak, ak) holds for k = 2, ..., n.c) Use part (b) to develop a dynamic programming algorithm for solving this problem.d) Show each step your algorithm from part (c) uses tofind the maximum sum of consecutive terms of thesequence 2,−3, 4, 1,−2, 3.e) Show that the worst-case complexity in terms of thenumber of additions and comparisons of your algorithm from part (c) is linear.
Give a dynamic programming algorithm using the following optimal subproblem structureLCIS+(i; j) : is the longest closed to increasing subsequence that ends and XiXj , i.e., its last elementis Xj and its second to last element is Xi. Give a recursive formula, prove its correctness and write abottom-up implementation.
Problem 2: Search an ArrayConsidering the search problem, we have a list of n integers A = ⟨v1, v2, · · · vn⟩. We wantto design an algorithm to check whether an item v exists or not such that it should returneither the index, i, if it was found or −1 otherwise, when not found.a) Implement the algorithm in Pythonb) Empirically, show the performance curve of the algorithm using time measurements.c) Using the basics of the theoretical analysis, write the complexity of its worst-casetime.d) Prove your algorithm is correct—it terminates with the valid output in every case
You will analyze three algorithms to solve the maximum contiguous subsequence sum problem, and then evaluate the performance of instructor-supplied implementations of those three algorithms. You will compare your theoretical results to your actual results in a written report. What is the maximum contiguous subsequence sum problem? Given a sequence of integers A1, A2, ..., An (where the integers may be positive or negative), find a subsequence Aj, ... , Ak that has the maximum value of all possible subsequences. The maximum contiguous subsequence sum is defined to be zero if all of the integers in the sequence are negative. Consider the sequence shown below. A1: -2 A2: 11 A3: -4 A4: 13 A5: -5 A6: 2 The maximum contiguous subsequence sum is 20, representing the contiguous subsequence in positions 2, 3, and 4 (i.e. 11 + (-4) + 13 = 20). The sum of the values in all other contiguous subsequences is less than or equal to 20. Consider a second sequence, shown below. A1: 1…
(c) 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 your algorithm with an illustrative example.
For example, Input: {10, 12, 17, 14, 8, 3, 22} Output: {5, 4, 2, 3, 6, 7, 1} Given an array of different integers, replace each element by its corresponding descending order in the array. The maximum array element has the order 1; the second maximum element has order 2, and so on... a) Design a brute-force algorithm to solve this problem (4 marks), and analyse the complexity of your solution b) Design a more efficient algorithm to do the same task with less complexity , and analyse the complexity of your solution. c) Develop a python code to implement your efficient algorithm. depend on the correctness of the code, indentation, comments, test-case] d) Prepare a brief report (250 words) comparing the two algorithms
Construct a Divide and Conquer-based algorithm for the following problem andanalyze the same in the worst case.Input: An ordered array L with n elements and item X that is to be searched inthe array L.Output: Index j where L[j]=X, if X is found or else j = 0, if X not found.
Consider a simplified version of bucket-sort. You are given an array A of n random integers in the range [1...r] as input. The output data consist of r buckets, such that at the end of the algorithm, Bucket i contains indices of all the elements in A that are equal to i. Describe a decomposition based on partitioning the input data (i.e., the array A) and an appropriate mapping onto p processes. Describe briefly how the resulting parallel algorithm would work. Describe a decomposition based on partitioning the output data (i.e., the set of r buckets) and an appropriate mapping onto p processes. Describe briefly how the resulting parallel algorithm would work. Which of the two decompositions leads to a better parallel algorithm? Should the relative values of n and r have a bearing on the selection of one of the two decomposition schemes?
Consider the problem of counting, in a given text, the number of substrings that start with an A and end with a B. For example, there are four such substrings in CABAAXBYA.a. Design a brute-force algorithm for this problem and determine its efficiency class.b. Design a more efficient algorithm for this problem with complexity O (n)
Search a sorted matrix: The input consistsof a real number x and a matrix A[1..n, 1..m] of nm real numbers such that each rowA[i, 1..m] is sorted and each column A[1..n, j] is sorted. The goal is to find the maximum array entry A[i, j] that is less than or equal to x, or report that all elements of Aare larger than x. Design and analyze an iterative algorithm for this problem that examines as few matrix entries as possible. Becareful if you believe that a simple binarysearch solves the problem. make a lower bound, and recursive algorithm.
Search a sorted matrix: The input consistsof a real number x and a matrix A[1..n, 1..m] of nm real numbers such that each rowA[i, 1..m] is sorted and each column A[1..n, j] is sorted. The goal is to find the maximum array entry A[i, j] that is less than or equal to x, or report that all elements of Aare larger than x. Design and analyze an iterative algorithm for this problem that examines as few matrix entries as possible. Becareful if you believe that a simple binarysearch solves the problem.