6. Consider the following basic problem. You're given an array A consisting of n integers A[1],...
7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n, we have |Ali]- Ali+1]| < 1. Let A[1] = and Alny such that r < y. Using the divide-and-conquer technique, describe in English algorithm to find j such that Alj] z for a given value z, xz < y. Show that your algorithm's running time is o(n) and that it is correct o(n) search an 2 8. (10) Solve the recurrence in asymptotically tight...
need help in this algorithm question Let A be an array containing n numbers (positive and negative). Develop a divide and conquer algorithm that finds the two indices 1 sisjsn such that A[k] (the sum of the elements from i to j) is maximized. For example, in the array A [10,-5,-6,5, 7,-2,4, -11], the sub-array A[4:6] has the sum 5+ 7-2+4-14 and no other sub-array contains elements that sum to a value greater than 14, so for this input the...
ALGORITHM PROBLEM: A) Significant Inversions: We are given a sequence of n arbitrary but distinct real numbers <a1 , a2 ,..., an>. We define a significant inversion to be a pair i < j such that ai > 2 aj . Design and analyze an O(n log n) time algorithm to count the number of significant inversions in the given sequence. [Hint: Use divide-&-conquer. Do the “combine” step carefully] B) The Maximum-Sum Monotone Sub-Array Problem: Input: An array A[1..n] of...
Consider the following algorithm. ALGORITHM Enigma(A[0.n - 1]) //Input: An array A[0.n - 1] of integer numbers for i leftarrow 0 to n - 2 do for j leftarrow i +1 to n - 1 do if A[i] = = A[j] return false return true a) What does this algorithm do? b) Compute the running time of this algorithm.
Let A = [A[1], A[2],…..,A[n]] be an array of n distinct integers. For 1 <= j <= n, the index j is a happy index if A[i] < A[j] for all 1 <= i < j. Describe an O(n)- time algorithm that finds all the happy indices in the array A. Partial credit will be given for an O(n log(n))-time algorithm and a minimal credit will be given for an O(n^2) –time algorithm. What is the running time of your...
Given an array A[1..n] of positive integers and given a number x, find if any two numbers in this array sum upto x. That is, are there i,j such that A[i]+A[j] = x? Give an O(nlogn) algorithm for this. Suppose now that A was already sorted, can you obtain O(n) algorithm?
4 Problem 4 -Extra Credit Given an array A, we say that elements A and Al are swapped if J >「 but Alj]< A[i]. For example, if A - [8,5,9,7], then there are a total of3 swapped pairs, namely 8 and 5; 8 and 7; and 9 and 7 Describe a recursive algorithm that given an array A, determines the number of swapped pairs in the array in O(n log(n)) time. To analyze the algorithm, you must state the recurrence...
Consider the following problem: Input: a list of n-1 integers and these integers are in the range of 1 to n. There are no duplicates in list. One of the integers from 1 to n is missing in the list. Output: find the missing integer Let the input array be [2, 4, 1, 6, 3, 7, 8]. Elements in this list are in the range of 1 to 8. There are no duplicates, and 5 is missing. Your algorithm needs...
(13 pts) Given an array AlI,2,. .. ,n] integers, design and analyze an efficient Divide-and-Conquer algorithm to find some i and j, where j > 1, such that A[j]-Ali] is maximized. For example, given A 6, 1,3,8,4,5, 12,6], the maximum value of AL] - Ali] for j > i is 12-1 11 where j -7 and i 2. Give the underlying recurrence relation for your algorithm and analyze its running time. You should carefully state all details of your algorithm:...
Lower bound arguments. In class, we proved that any comparison-based algorithm that has to sort n numbers runs in Ω (nlogn) time. Let’s change the problem a bit. Suppose S[1. . . n] is a sorted array. We want to know if S contains some element x. a. How would you solve this problem and what is the running time of your algorithm? (Note: you can just say what algorithm you will use.) b. Show that any comparison-based algorithm(i.e., one...