Given an array A of n distinct numbers, we say that a pair of numbers i,...

Given an array A of n distinct numbers, we say that a pair of numbers i, j ∈ {0, . . . , n − 1} form an inversion of A if i < j and A[i] > A[j]. Let inv(A) = {(i, j) | i < j and A[i] > A[j]}. Define the Inversion problem as follows: • Input: an array A consisting of distinct numbers. • Output: the number of inversions of A, i.e. |inv(A)|. Answer the following: (a) How small can the number of inversions be? Give an example of an array of length n with the smallest possible number of inversions. (b) Repeat the last exercise with ‘small’ replaced by ‘large’. (c) Show that the running time of insertion sort on array A is Θ(|inv(A)| + n). (d) Assume that the array A contains the number 1, . . . , n in a uniformly random order (among all the n! many orderings). In this case, the running time of insertion sort on A becomes a random variable T. Calculate T up to constant factors.

Define the Exponentiation problem as follows. • Input: A number a and an integer n. • Output: a n . Design an algorithm to solve the problem with as few multiplications as possible

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orchestra answered 1 year ago

Given an array A of n distinct real numbers, we say that a pair of numbers...

Given an array A of n distinct real numbers, we say that a pair of numbers i, j ∈ {0, . . . , n−1} form an inversion of A if i < j and A[i] > A[j]. Let inv(A) = {(i, j) | i < j and A[i] > A[j]}. Answer the following: (a) How small can the number of inversions be? Give an example of an array of length n with the smallest possible number of inversions. (b)...

Given an array of n numbers, not necessarily in sorted order, we wish to find: (i)...

Given an array of n numbers, not necessarily in sorted order, we wish to find: (i) the range (max - min) and (ii) the interquartile range (IQR) of the numbers. IQR is defined as the difference between the number at the 25% percentile and the number at the 75% percentile. What is the exact number of comparisons needed for computing the range and why? Using any algorithm from the chapters of the textbook covered so far as a subroutine, give...

Given an array A[1..n], with distinct values and k with 1 ≤ k ≤ n. We...

Given an array A[1..n], with distinct values and k with 1 ≤ k ≤ n. We want to return the k smallest element of A[1.....n], in non-decreasing order. For example: A = [5, 4, 6, 2, 10] and k = 4, the algorithm returns [2, 4, 5, 6]. There are at least the following four approaches: a. heapify A and then extract k elements one by one b. sort the array (e.g. using MergeSort or HeapSort) and then read the...

An array A of n distinct numbers are said to be unimodal if there exists an...

An array A of n distinct numbers are said to be unimodal if there exists an index k, 1 ≤ k ≤ n, such that A[1] < A[2] < · · · < A[k − 1] < A[k] and A[k] > A[k + 1] > · · · > A[n]. In other words, A has a unique peak and are monotone on both sides of the peak. Design an efficient algorithm that finds the peak in a unimodal array of...

Project: Given an array numbers. We define a running sum of an array as runningSum[i] =...

Project: Given an array numbers. We define a running sum of an array as runningSum[i] = sum(nums[0]…nums[i]). Return the running sum of numbers. Example: Input: nums = [1,2,3,4] Output: [1,3,6,10] Explanation: Running sum is obtained as follows: [1, 1+2, 1+2+3, 1+2+3+4]. You need to do: Create a class called RunningSumArray to perform the main method. Create a class called ArrayOperationto hold the actions for array. In this project, you will need 4methods: 1. Public static Int[] getArray() --- inside ArrayOperationclass,...

You are given an array of length n that is filled with two symbols (say 0...

You are given an array of length n that is filled with two symbols (say 0 and 1); all m copies of one symbol appear first, at the beginning of the array, followed by all n − m copies of the other symbol. You are to find the index of the first copy of the second symbol in time O(logm). include: code, proof of correctness, and runtime analysis

5. Suppose you are given an n-element array containing distinct integers that are listed in increasing...

5. Suppose you are given an n-element array containing distinct integers that are listed in increasing order. Given a number k, describe a recursive algorithm to find two integers in A that sum to k, if such a pair exists. What is the running time of your algorithm? Java Language....

Suppose you are given an n-element array containing distinct integers that are listed in increasing order....

Suppose you are given an n-element array containing distinct integers that are listed in increasing order. Given a number k, describe a recursive algorithm to find two integers in A that sum to k, if such a pair exists. What is the running time of your algorithm?

Given an array A[1..n] of integers - all of whose numbers are in the range [0,...

Given an array A[1..n] of integers - all of whose numbers are in the range [0, n^3 − 1] give an algorithm which sorts them in O(n) time.

. As input you are given two arrays: an array of numbers ? and an array...

. As input you are given two arrays: an array of numbers ? and an array ? of queries where each query is a target number. The array ? is unsorted and may contain duplicates. Your goal is, for each query ? in the array ?, count the number of pairs in the array ? that sums up to ?; that is, the number of distinct pairs of indices [?, ?], with ? < ?, such that ?[?] + ?[?]...

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