Considering the search problem, we have a list of n integers A = (v1, v2, · · · Un). We want to design an algorithm to check whether an item v exists or not such that it should return either the index, i, if it was found or -1 otherwise, when not found. a) Implement the algorithm in Python b) Empirically, show the performance curve of the algorithm using time measurements. c) Using the basics of the theoretical analysis, write the complerity of its worst-case time. d) Prove your algorithm is correct–it terminates with the valid output in every case

Considering the search problem, we have a list of n integers A = (v1, v2, · · · Un). We want to design an algorithm to check whether an item v exists or not such that it should return either the index, i, if it was found or -1 otherwise, when not found. a) Implement the algorithm in Python b) Empirically, show the performance curve of the algorithm using time measurements. c) Using the basics of the theoretical analysis, write the complerity of its worst-case time. d) Prove your algorithm is correct–it terminates with the valid output in every case

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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Problem 2: Search an ArrayConsidering the search problem, we have a list of n integers A = ⟨v1, v2, · · · vn⟩. We wantto design an algorithm to check whether an item v exists or not such that it should returneither the index, i, if it was found or −1 otherwise, when not found.a) Implement the algorithm in Pythonb) Empirically, show the performance curve of the algorithm using time measurements.c) Using the basics of the theoretical analysis, write the complexity of its worst-casetime.d) Prove your algorithm is correct—it terminates with the valid output in every case
: In searching an element in an array, linear search can be used, even though simple to implement, but not efficient, with only O(n) time complexity. Assuming the array is already in sorted order, modify the search function below, using a better algorithm, so the average time complexity for the search function is O(log n). include <iostream> using namespace std; int search(int al), int s, int v) { 1/ Modify below codes. for (int i = 0; i <s; i++) { if (a[i] = v) return i; return -1; int main() { int intArray:10] = { 5, 7, 8, 9, 10, 12, 13, 15, 20, 34); // Search for element '12' in 10-elements integer array. cout << search(intArray, 10, 12); // '5' will be printed out. // Search for element '35' in 10-elements integer array. cout << search(intArray, 10, 35); // '-1' will be printed out. // Index '-l' means that the element is not found. return 0;
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
Consider the searching problem:Input: A sequence of n numbers A = [a1; a2; : : : ; an] and a value v.Output: An index i such that v = A[i] or the special value NILL if v does not appear in A.(i) Write pseudocode for linear search, which scans through the sequence, looking for v.(write the code)(ii) Using a loop invariant, prove that your algorithm is correct. proof by induction base case inductive step and proof of corretness
Problem 2: Finding an element.(i) Let n be an even positive integer. Let A be an array containing n not necessarydifferent integers. Elements in A are not necessary sorted. Describe an algorithmwhich in time O(n) tests whether there exists an element x in A that appears in Aat least n/2 times.(ii) Let S be a set containing n integers. An element x ∈ S is an approximate median ofS, if|{y ∈ S | y < x}| ≥ n/4and |{y ∈ S | y > x}| ≥ n/4Given an array A containing n different integers, not necessary sorted, describe analgorithm which in time O(n) finds all approximate medians of the set {A[i] | i =1, 2, . . . , n}.
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.
Problem 4: Divide and conquer.(i) Let A be an array containing n different integers. Elements in A are sorted inincreasing order. Given an integer x describe an algorithm which in time O(log n)finds an index i of x in A, that is, A[i] = x, where the length n of A is unknown(you may assume that the implementation of an array is such that for A[i], wherei > n, a special character ∞ is returned).(ii) Let A be an array containing n different integers. Elements in A are not necessarysorted. Given an integer x describe an algorithm which in time O(n log n) testswhether there exist two elements u and v in A such that their sum is equal to x, thatis, u + v = x. The length n of A is known.
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]
1. Consider the algorithm for the sorting problem that sorts an array by counting,for each of its elements, the number of smaller elements and then uses thisinformation to put the element in its appropriate position in the sorted array:ALGORITHMComparisonCountingSort(A[0..n − 1])//Sorts an array by comparison counting//Input: Array A[0..n//Output: Array S[0..n− 1] of orderable values− 1] of A’s elements sorted// in nondecreasing orderfor i ← 0 to nCount− 1 do[i]←0for i ← 0 to n − 2 dofor j ← i +1 to n − 1 doif A[i] < A[j ]Count[j ]← Count[j ] + 1else Count[i]← Count[i] + 1for i ←0 to n−1 doS[Count[i]]←A[i]return Sa. Apply this algorithm to sorting the list 60, 35, 81, 98, 14, 47.b. Is this algorithm stable?c. Is it in-place?
Question1 (b)Consider the followingalgorithmwhere Small is an array of ninteger values: AlgorithmQ1 //Input: An array Smallcontaining ninteger values b = ∞ for h = 1 to n-1 for k = h + 1 to n if (Small[h] –Small[k]) < b b = Small[h] –Small[k] return b i.Explain what does this algorithm compute. ii.Findthe asymptotic bound/complexity ofthis algorithm. iii.Producean improved algorithm to Question1 (b) above with a better asymptotic complexity/running time. iv.Findthe asymptotic bound/complexity of the improvedalgorithmin Question 1(b)iii. above.
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
For the 8-queens problem, define a heuristic function, design a Best First Search algorithm in which the search process is guided by f(n) = g(n) + h(n), where g(n) is the depth of node n and h(n) is the heuristic function you define, and give the pseudo code description.