Given a sorted array A of n distinct integers, some of which may be negative, give an O(log(n)) algorithm to find an index i such that 1 ≤ i ≤ n and A[i] = i provided such an index exists.  If there are many such indices, the algorithm can return any one of them.

Given n arrays, each array contain n positive integers. Write an O (n² log n) algorithm to find the smallest n sums out of nn possible sums that can be obtained by picking one positive integer from each of n arrays. For example, given three arrays as follows: [5, 1, 8], [5, 2,9], and [6,7, 10]. The smallest n sums of the given array is [9, 10, 12].

The algorithm below searches for the maximum in an input array A. Assume A is a random sequence containing n distinguishable real numbers, what is the probability for line #4 to be executed exactly once? (The phrase "random sequence" here means the n numbers can appear in any order with equal chance.) mymax(A, n) { 1: max = A[1]; 2: for i = 2:n 3: if A[i] > max 4: max = A[i); 5: end; 6: end; 7: return max; }

Consider the following algorithm that uses a sorted list of n elements (alist). What is the worst case runtime of this algorithm? for each element in alist    1. ask the user for an input, call it value    2. search value in alist using binary search    3. if value exists in alist, print "successful"       otherwise print "unsuccessful"      Question options: a. O(log n) b. O(n) c. O(n log n) d. O(2^n) e. O(n^2) f. O(1)

Given an unsorted array of integers, write a function in Python to find the length of the longest increasing subsequence (LIS) in the array. For example, given the array [10, 9, 2, 5, 3, 7, 101, 18], the LIS is [2, 3, 7, 101], which has a length of 4. Your solution should have a time complexity of O(n log n), where n is the length of the input array. Here's some code to get you started: def longest increasing_subsequence(arr): # TODO: implement function pass # example usage arr = [10, 9, 2, 5, 3, 7, 101, 18] print(longest_increasing_subsequence(arr)) # should print 4

Let A[1..n] be an array of n integers that each is larger than 1. give an O(n lg n)-time algorithm that decides if there are two integers x,y in A such that x=y² (multiplication takes O(1) time)

Let M(n) be the minimum number of comparisons needed to sort an array A with exactly n ele- ments. For example, M(1) = 0, M(2) = 1, and M(4) = 4. If n is an even number, clearly explain why M(n) = 2M(n/2) + n/2.

Problem 2. Suppose we have an array A[1 : n] which consists of numbers {1,...,n} written in some arbitrary order (this means that A is a permutation of the set {1,...,n}). Our goal in this problem is to design a very fast randomized algorithm that can find an index i in this array such that A[i] mod 8 € {1,2}, i.e., the reminder of dividing A[i] by 8 is either 1 or 2. For simplicity, in the following, we assume that n itself is a multiple of 8 and is at least 8 (so a correct answer always exist). For instance, if n = 8 and the array is A = [8,7, 2,5, 4, 6,3, 1], we want to output either of indices 3 or 8. (a) Suppose we sample an index i from {1,...,n} uniformly at random. What is the probability that i is a correct answer, i.e., A[i] mod 8 E {1,2}? (b) Suppose we sample m indices from {1,...,n} uniformly at random and with repetition. What is the probability that none of these indices is a correct answer? Now, consider the following simple algorithm for this problem: Find-Index-1(A[1:…