In this problem you will explore what happens if in the integer multiplication algorithm we saw in class we dothe splitting of the numbers differently. But before we get to the final problem, let's start off with this one:Part (a): Define the recurrence relationT(n)≤≤ {{ST5T ()+cnif n < 3otherwise.Argue that T(n) is O(nlog, 5).We now come to the actual algorithmic problem:⚫ Part (b): Either by modifying the algorithm we saw in class or otherwise, present an algorithm thatcan multiply two n bit numbers a = (an-1,..., ao) and b = (b-1,..., bo) in time O(nlog, 5),which is O(n147) (and is better than the runtime of the algorithm we saw in class).HintIt might be useful to divide up the numbers into three parts each with roughly n/3 bits andthen argue that one only needs to perform 5 smaller multiplications of n/3-bit numbers insteadof the trivial 9 such multiplications. Further this support page might also be useful ingeneralizing the O(nlog₂ 3) algorithm to one with a better runtime as required by this problem.)

Here are some examples of algorithms that achieve O(n log⁵ base 2) (which is equivalent to O(n^1.47)…

Answered: In this problem you will explore what…

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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We have already had a recurrence relation of an algorithm, which is T (n) = 4T (n/2) + n log n. We know T (1) ≤c.(a) express it as T (n) = O(f (n)), by using the iteration method.(b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T (n) at the iteration step t, and prove 3 that this form is correct for any iteration step t by using mathematical induction. Then by finding out the eventual number of t and substituting it into your general form of T (n), you get the O(·) notation of T (n).]
1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.
Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The problem is computing the sum of n numbers. This algorithm divides the problem into two instances of the same problem: to compute the sum of the first ⌊n/2⌋ numbers and compute the sum of the remaining ⌊n/2⌋ numbers. Once each of these two sums is computed by applying the same method recursively, we can add their values to get the sum in question. A-)T(n)=T(n/2)+1 B-)T(n)=T(n/2)+2 C-)T(n)=2T(n/2)+1 D-)T(n)=2T(n/2)+2
Consider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.
For each of the following recurrences, give an expression for the runtime T(n) if the recurrence can be solved with the Master Theorem. Otherwise, indicate that the Master Theorem does not apply. For all cases, we have T(x) = 1 when x ≤ 100 (base of recursion). Ex.) T(n) = 3T(n/3) + √n We have nlog, a T(n) = O(n). a) T(n) = 5T(n/3) +2023n¹.6 b) T(n) = 9T(n/3) + 1984n² = n. Since f(n) = O(n¹-) (for any € < 1/2), we are at case 1 and n³ c) T(n) = 8T(n/2) + log n 4 d) T(n) = 16T(n/2) + n² log³ n
7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).
Let G: (0,1}" {0,1)"+l be defined as follows: G(x....x) = (X1, X Ox2, X O x2 O X3, ..., X1 O X2 O ... O x X). Prove or G is a PRG. disprove
Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The problem is computing the sum of n numbers. This algorithm divides the problem into two instances of the same problem: to compute the sum of the first Ln/2J numbers and compute the sum of the remaining Ln/2J numbers. Once each of these two sums is computed by applying the same method recursively, we can add their values to get the sum in question. T(n) = T(n/2) + 1 T(n) = 2T(n/2) + 2 T(n) = 2T(n/2) + 1 T(n) = T(n/2) + 2
Given an integer array nums, return all the triplets [nums[i], nums[j], nums[k]] such that i != j, i != k, and j != k, and nums[i] + nums[j] + nums[k] == 0. This time, your problem has additional details: Constraints: The solution set must not contain duplicate triplets. The order of the output and the order of the triplets does not matter. 3 <= nums.length <= 3000 -105 <= nums[i] <= 105 Function definition for Java: public List<List<Integer>> threeSum(int[] nums) { // Your code here } Function definition for Python: def threeSum(self, nums: List[int]) -> List[List[int]]: #Your code here Announced Test Cases: Input: nums = [0,1,1] Output: [] Explanation: The only possible triplet does not sum up to 0.Input: nums = [-5,0,5,10,-10,0] Output: [[-10,0,10],[-5,0,5]] Explanation: There are two possible combinations of triplets that satisfy: (-5,0,5) and (-10,0,10). Hint: There are 3 well-known ways to solve this problem!
2. Expand the following recurrence to help you find a closed-form solution, and then use induction to prove your answer is correct. T(n) = T(n-1) + 5 for n> 0; T(0) = 8. 3. Give a recursive algorithm for the sequential search and explain its running time.
To have random-access lookup, a grid should have a scheme for numbering the tiles.For example, a square grid has rows and columns, which give a natural numberingfor the tiles. Devise schemes for triangular and hexagonal grids. Use the numberingscheme to define a rule for determining the neighbourhood (i.e. adjacent tiles) of agiven tile in the grid. For example, if we have a four-connected square grid, wherethe indices are i for rows and j for columns, the neighbourhood of tile i, j can bedefined asneighbourhood(i, j) = {i ± 1, j,i, j ± 1}
In the friends level of abstracting recursion, you can give your friend any legal instance that is smaller than yours according to some measure as long as you solve in your own any instance that is sufficiently small. For which of these algorithms has this been done? If so, what is your measure of the size of the instance? On input instance (n, m), either bound the depth to which the algorithm recurses as a function of n and m, or prove that there is at least one path down the recursion tree that is infinite. algorithm R, (n, m) (pre-cond): n & mints. (post-cond): Say Hi begin if(n ≤ 0) else Print("Hi") R₂ (n - 1,2m) end if end algorithm algorithm R, (n, m) (pre-cond): n & m ints. (post-cond): Say Hi begin if(n ≤ 0) else Print("Hi") Rb(n-1, m) R₂(n, m-1) end if end algorithm

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