Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 5.3, Problem 3E
Program Plan Intro
To decide whether the code PERMUTE-WITH_ALL
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Given an array Arr[] of N integers. Find the contiguous sub-
array(containing at least one number) which has the maximum sum
and return its sum.
Example 1:
Input:
N = 5
Arr[]
{1,2,3,-2,5}
%D
Output:
9.
Explanation:
Max subarray sum is 9
of elements (1, 2, 3, -2, 5) which
is a contiguous subarray.
Given an array, find the total number of inversions of it. If (i < j) and (A[i] > A[j]), then pair (i, j) is called an inversion of an array A. We need to count all such pairs in the array.
For example,
Input: A[] = [1, 9, 6, 4, 5] Output: The inversion count is 5 There are 5 inversions in the array: (9, 6), (9, 4), (9, 5), (6, 4), (6, 5)
Given an integer n and an array a of length n, your task is to apply the following mutation to a:
Array a mutates into a new array b of length n. For each i from 0 to n - 1, b[i] = a[i - 1] + a[i] + a[i + 1]. If some element in the sum a[i - 1] + a[i] + a[i + 1] does not exist, it should be set to 0. For example, b[0] should be equal to 0 + a[0] + a[1].
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Introduction to Algorithms
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- Array x is [a, b, c, . . . , y] has 25 integers. To find median of medians we form five sub-arrays: [d, c, b, e, a], [i, f, h, j, g], [1, m, k, n, o], [p, s, q, r, t], [w, x, v, u, y]. We find median from these sub-arrays and generate new sorted array [k, h, q, b, v]. Median of new sorted array is q, so median of medians is q. List all elements guaranteed to be greater or equal to q.arrow_forwardGiven an array A containing N numbers. The problem is to find the minimum gcd among all non-decreasing subarrays. Note: Gcd of k number is the greatest number which divides all k numbers. Example 1: Input : arr[] = {1, 2, 3, 2, 4, 1, 7} Output : 1 Explanation : Some non decreasing subarrays are - {1, 2, 3} gcd = 1, {2, 4} gcd = 2, {1, 7} gcd = 1, minimum gcd = 1arrow_forwardCorrect answer will be upvoted else downvoted. Computer science. You are a given an array a of length n. Find a subarray a[l..r] with length at least k with the largest median. A median in an array of length n is an element which occupies position number ⌊n+12⌋ after we sort the elements in non-decreasing order. For example: median([1,2,3,4])=2, median([3,2,1])=2, median([2,1,2,1])=1. Subarray a[l..r] is a contiguous part of the array a, i. e. the array al,al+1,…,ar for some 1≤l≤r≤n, its length is r−l+1. Input The first line contains two integers n and k (1≤k≤n≤2⋅105). The second line contains n integers a1,a2,…,an (1≤ai≤n). Output Output one integer m — the maximum median you can get.arrow_forward
- Find an 8-element array that require most number of Comparisons. Let`s assume S(n) = [?/2 + 1, 1, ?/2 + 2, 2, ?/2 + 3, ... ,?/2 + ?/2, ?/2]where n is a power of 2, for a decreasing powers of two gap sequence find a formula for the total number of comparisons required for Shell sorting S(n). Finally show that this S(n) has a running time of O(n^2).arrow_forwardConsider this alogrithm:// pre: array A of length n, each A[i] is picked randomly uniformly// from the set {0,1,2,3,4,5,6,7,8,9}.// post: return the index of the first occurrence of "check digit" of A// (already computed in the code)// return -1 if check digit is not foundint FindCheckDigit(A) {check_digit = 0for (i=0; i < n; i++) check_digit = check_digit + A[i]check_digit = check_digit % 10 # % means remainder of divisionfor (i = 0; i < n; i++) { if (A[i] == check_digit) { return i }}return -1;}Compute the average runtime for this algorithm. Show all details of your computation for both loops.arrow_forwardGiven an array arr[] of N non-negative integers representing the height of blocks. If width of each block is 1, compute how much water can be trapped between the blocks during the rainy season. Example 1: Input: N = 6 arr[] Output: 10 = {3,0,0,2,0,4}arrow_forward
- Given an array of integers arr, sort the array by performing a series of pancake flips. In one pancake flip we do the following steps: Choose an integer k where 1 <= k <= arr.length. Reverse the sub-array arr[0...k-1] (0-indexed). For example, if arr = [3,2,1,4] and we performed a pancake flip choosing k = 3, we reverse the sub-array [3,2,1], so arr = [1,2,3,4] after the pancake flip at k = 3. Print out the k-values corresponding to a sequence of pancake flips that sort arr. Example 1: Input: arr = [3,2,4,1] Output: 4, 2, 4, 3 Explanation: We perform 4 pancake flips, with k values 4, 2, 4, and 3. Starting state: arr = [3, 2, 4, 1]. After 1st flip (k = 4): arr = [1, 4, 2, 3] After 2nd flip (k = 2): arr = [4, 1, 2, 3] After 3rd flip (k = 4): arr = [3, 2, 1, 4] After 4th flip (k = 3): arr = [1, 2, 3, 4], which is sorted. Another potential solution is: Output = 3, 4, 2, 3, 1, 2, 1, 1 with a similar explanation. All potential solutions that solve the problem are…arrow_forwardGiven an integer array (data type: int) arr[10] with the initialized values for all its elements. For all the elements from arr[0] to arr[9], i.e., arr[0], arr[1], arr[2], ..., arr[8], arr[9], we call this array to be special, if it satisfies either one of two cases below: Case 1: arr[0] arr[2], arr[2] arr[4], .. Case 2: arr[0] > arr[1], arr[1] arr[3], arr[3] arr[1] for arr[0]. Part-(a). Write a program to determine whether this array is special or not. Part-(b). In this part, we can decrease any element by one each time. If the array is not special, determine the minimum number of decreases needed to change the array to be special. For example, if the array contains the following ten numbers "1, 2, 3, 4, 5, 6, 7, 8, 9, 10", it is not a special array. However, by decreasing "3 to 1", "5 to 3", "7 to 5" and "9 to 7", the array with these ten new numbers "1, 2, 1, 4, 3, 6, 5, 8, 7, 10" becomes special. The total number of decreases is 8 (= 2 + 2 +2 + 2), which is the minimum number…arrow_forwardConsider an Array arr= {2, 3, 4, 1, 5}, what are the pivots that are returned as a subsequent partitioning: 1 and 2 1 and 6 2 and 6 1 and 3arrow_forward
- In a non-empty array of integers, the priority of a number is determined by the frequency of its occurrence. Elements are added as they come but the most frequent element is deleted first. If two elements have the same priority, then the element which came first will be deleted first. For example: nums = [1,2,3,1,3,1,3,2,3]. After the first deletion now the array will be num=[2,3,3,3,2,3]. Implement a priority queue for the aforementioned scenario. give the answer in brief and the correct one.arrow_forwardGiven a sorted array: int arr[] = {-5,-2,-1,1,5,8,12,18}. Square every element of the array such that squares are also in sorted order. Print the square array. Do it in linear time and space complexity.arrow_forwardYou are given an array of positive integers S[1, . . . , n]. You are initially positioned at the array’s first index 1, and each element S[i] represents your maximum length of jump at position i. For example, you can jump from position i to any position among i + 1, . . . , i + S[i]. Return true if you can reach the last index, or false otherwise. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.arrow_forward
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