As HOMEWORKLIB Guidelines, I'll answer one question (first question) although two questions are given.
Divide the array of n elements into two parts: one of length 1 and another one of length (n-1). If the element in the part of size 1 is equal to z, then it's done and return the index of that element, otherwise recurs in the (n-1) element part. The algorithm can be written as follows:
FindZ (A, 1);
FindZ (Array A, j)
if (A[j] = z)
return j;
if (j = N)
Can't find Z and return;
FindZ (A,j+1);
The running time of the algorithm can be represented bu using a recurrence equation as follows:
T(n) = T(n-1) + 1 when n > 1
= 1 when n = 1
The recurrence equation can be solved as follows:
T(n) = T(n-1) + 1
= [T(n-2) + 1] + 1
= n = O(n)
The above explanation shows that the algorithm has running time of O(n).
7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n,...
(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:...
1. Design and write a Divide& Conquer algorithm that, given an array A of n distinct integers which is already sorted into ascending order, will find if there is some i such that Ali] in worst-case 0(log n) time.
6. Consider the following basic problem. You're given an array A consisting of n integers A[1], A[2], , Aln]. You'd like to output a two-dimensional n-by-n array B in which B[i, j] (for i <j) contains the sum of array entries Ali] through Aj]-that is, the sum A[i] Ai 1]+ .. +Alj]. (The value of array entry B[i. Λ is left unspecified whenever i >j, so it doesn't matter what is output for these values.) Here's a simple algorithm to...
1. (16 pts.) Sorted Array Given a sorted array A of n (possibly negative) distinct integers, you want to find out whether there is an index i for which Al = i. Give a divide-and-conquer algorithm that runs in time O(log n). Provide only the main idea and the runtime analysis.
An m×n array A of real numbers is a Monge array if for all i,j,k, and l such that 1≤i<k≤m and 1≤j<l≤n , we have >A[i,j]+a[k,l]≤A[i,l]+A[k,j]> In other words, whenever we pick two rows and two columns of a Monge array and consider the four elements at the intersections of the rows and columns, the sum of the upper-left and lower-right elements is less than or equal to the sum of the lower-left and upper-right elements. For example, the following...
QUESTION 3 (10 Marks) Suppose you are given an array A[0..n 1 of integers, each of which may be positive, negative, or zero. A contiguous subarray A|i..j] which includes all the items starting at array index i and ending at array index j is called a positive interval if the sum of its entries is at least zero. For example let A-{-1, 3,-5, 2, 0,-4, 3,-6,-2). Ten A[3, 6) is a positive interval since its sum is 2+0+(-4)+3-1, but Al4,7isnot...
Suppose we are given two sorted arrays (nondecreasing from index 1 to index n) X[1] · · · X[n] and Y [1] · · · Y [n] of integers. For simplicity, assume that n is a power of 2. Problem is to design an algorithm that determines if there is a number p in X and a number q in Y such that p + q is zero. If such numbers exist, the algorithm returns true; otherwise, it returns false....
(20 points) You are given an array A of distinct integers of size n. The sequence A[1], A[2], ..., A[n] is unimodal if for some index k between 1 and n the values increase up to position k and then decrease the reminder of the way until position n. (example 1, 4, 5, 7, 9, 10, 13, 14, 8, 6, 4, 3, 2 where the values increase until 14 and then decrease until 1). (a) Propose a recursive algorithm to...
I already solved part A and I just need help with part B 1. Matrix Multiplication The product of two n xn matrices X and Y is a third n x n matrix 2 = XY, with entries 2 - 21; = xixYk x k=1 There are n’ entries to compute, each one at a cost of O(n). The formula implies an algorithm with O(nº) running time. For a long time this was widely believed to be the best running...
We are given an array A holding n integers, for some large n. The array is sorted, and the values in A range from -2147483648 to 2147483647, evenly distributed. Give Θ expressions for the following tasks: A. Running the insertion sort algorithm on the array A: B. Running the selection sort algorithm on the array A: C. Performing binary search for integer k which is not in A: D. Performing interpolation search for integer k not in A: