Considering the sorting problem, we have a list of n integers A = (v1, v2, · ·…· Vn). We vant to sort the array using insertion sort whose pseudo-code is as follows: note 2 := A.length denotes the size of the array, number of items. !! Algorithm 2 INSERTION-SORT(A) for i = 1 to n – 1 do j+ i - 1 while A[j] > A[j+ 1] ^ j 2 0 do swap (A[j], A[j +1]) j+j- 1 - end while end for

Considering the sorting problem, we have a list of n integers A = (v1, v2, · ·…· Vn). We vant to sort the array using insertion sort whose pseudo-code is as follows: note 2 := A.length denotes the size of the array, number of items. !! Algorithm 2 INSERTION-SORT(A) for i = 1 to n – 1 do j+ i - 1 while A[j] > A[j+ 1] ^ j 2 0 do swap (A[j], A[j +1]) j+j- 1 - end while end for

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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You are given an unsorted array A[1, ..., n].We know that the maximum (or minimum) element in the array can be found in (n-1) comparisons.Write an algorithm that finds both the maximum and the minimum element in the array in ~(3n/2) comparisons, i.e., (3n/2) + THETA(1) comparisons.
Problem 3: Array ReversalWe have come through an old algorithm whose name indicates that it reverses the arrayelements; its pseudo-code was defined as follows: note n := A.length denotes the size ofthe array, number of items.Algorithm 2 Reverse-Array(A)s ← ⌊n−12 ⌋, e ← ⌊n2 ⌋while s ≥ 0 ∧ e ≤ (n − 1) doswap (A[s],A[e])s ← s − 1e ← e + 1end whilea) Implement the algorithm in Python, and test itb) Empirically, show the performance curve of the algorithm using time measurementsc) Using the basics of the theoretical analysis, write the complexity of its worst-casetime.d) Prove that the algorithm is actually correct
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An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.
You are given an arrayof colors (reds, whites, and blues), in no particular order. Sort them into the order of redsfollowed by whites followed by blues.a. Devise an efficient algorithm to solve this problem.b. What is your algorithm's time complexity?For example, if red is represented by 2, white represented by 4 and blue is represented by 6,your input and output array could be of following form:Input: [ 4 6 2 4 2 6 4 2 6 4 4 2 4 ]Output: [ 2 2 2 2 4 4 4 4 4 4 6 6 6]
Suppose you are given a very long array A, whose first n elements are integers (in arbitrary order) and the remaining elements are the special symbol 1. You can access any position I in A by referring to A[i]. However, you do not known. Give a n algorithm for determining n. Analyze the number of accesses to A made by your algorithm ,in terms of n. For full credit, your algorithm must make O(logn) accesses to A
1. Consider a problem of computing the prefix average of a sequence of numbers stored in an array P consisting of P integers. We want to compute an array P such that P is the average of P for P . Consider the two algorithms related to the above problem. Answer the questions given below
The recursive algorithm below takes as input an array A of distinct integers, indexed between s andf, and an integer k. The algorithm returns the index of the integer k in the array A, or ?1 if the integerk is not contained within A. Complete the missing portion of the algorithm in such a way that you makethree recursive calls to subarrays of approximately one third the size of A.• Write and justify a recurrence for the runtime T(n) of the above algorithm.• Use the recursion tree to show that the algorithm runs in time O(n).FindK(A,s,f,k)if s < fif f = s + 1if k = A[s] return sif k = A[f] return felseq1 = b(2s + f)=3cq2 = b(q1 + 1 + f)=2c... to be continued.else... to be continued.
Suppose we are given an array of numbers A[0], A[2], . . . , A[n - 1]. You may assume that n is a multiple of6. We wish to find a number x that simultaneously satisfies two properties: First, x should be in the middle two-thirds ofthe array (in other words, x = A[i] for some n/6 <= i < 5n/6). Second, after sorting the array, x should still be in the middletwo-thirds.2a) Prove that such an x exists. (Hint: how many values are not in the middle two-thirds of the original array? How manyare not in the middle two-thirds of the sorted array?)2b) Describe a linear time algorithm for finding such an x. (Note, if your solution calls one of the algorithms described inthe lectures, you should just refer to it by name, e.g. “quicksort”, rather than explaining how it works.)
Write an efficient algorithm for the following problem, and describe your reasoning. Determine the Time complexity and if you cannot find any polynomial time algorithm, then give a backtracking algorithm. Problem: Binary Array Core Input: Two integers p and q and a binary array A[1...n], i.e., each entry contains either a 0 or 1. Output: Print Yes - if there exists a subarray A[i...k], where 1<= i < k <= n, with exactly p zeros in A[1...i-1] (there may be 1s) and exactly q ones in A[k+1...n] (there may be 0s). Print No - otherwise. Outputs with examples Input: A = [0,1,1,0,1], p = 2, q =1, Output: No. Input: A = [0, 1,1,0, 0,1,0,1], p = 1, q =1, Output: Yes because we can have A[1...i-1] = [0] and A[k+1 ... n] = [0,1,0,1]
Describe an efecient in-place algorithm called Partition-Even-Odd(A) that partitions an array A in even and odd numbers. The algorithm must terminate with A containing all its even elements preceding all its odd elements. For example, for input A = [7, 17, 74, 21, 7, 9, 26, 10], the result might be A = [74, 10, 26, 17, 7, 21, 9, 7]. Partition-Even-Odd must be an in-place algorithm. What does this mean about your algorithm? (a) Write the pseudo-code for Partition-Even-Odd. (b) Characterize the complexity of Partition-Even-Odd. Brie y justify your answer. (c) Formalize the correctness of the partition problem as stated above, and prove that Partition-Even- Odd is correct using a loop-invariant.