Design a dynamic programming algorithm for the Coin-change problem described below: Input: An amount of money C and a set of n possible coin values with an unlimited supply of each kind of coin. Output: The smallest number of coins that add up to C exactly, or output that no such set exists. The algorithm must take O(n C) time. You must also write and explain the recurrence. Example 1: Input: C24, Coin values = = [1, 5, 10, 25, 50] Output: 6 (since 24 = 10+ 10+1+1 +1 + 1) Example 2: Input: C = 86, Coin values = [1, 5, 6, 23, 35, 46, 50] Output: 2 (since 86 = 46+35+5)

Design a dynamic programming algorithm for the Longest Common Subsequence problem de- scribed below Input: Two strings x = x1x2 xm and y = Y1Y2... Yn Output: The length of the longest subsequence that is common to both x and y. . The algorithm must take O(m n) time. You must also write and explain the recurrence. (I want the largest k such that there are 1 ≤ i₁ < ... < ik ≤ m and 1 ≤ j₁ < ... < jk ≤ n such that Xi₁ Xi2 Xik = Yj1Yj2 ··· Yjk) Example 1: Input: x = 'abcdefghijklmnopqrst' and y = 'ygrhnodsh ftw' Output: 6 ('ghnost' is the longest common subsequence to both strings) Example 2: Input: x = 'ahshku' and y = ‘asu' Output: 3 ('asu' is the longest common subsequence to both strings)

Design a dynamic programming algorithm for the problem described below Input: A list of numbers A = = [a1,..., an]. Output: A contiguous subsequence of numbers with the maximum sum. The algorithm must take O(n) time. You must also write and explain the recurrence. (I am looking for an i ≥ 1 and k ≥ 0 such that a + ai+1 + ···ai+k has the largest possible sum among all possible values for i and k.) Example 1: Input: A[5, 15, -30, 10, -5, 40, 10]. Output: [10, 5, 40, 10] Example 2: Input: A = [7, 5, 7, 4, -20, 6, 9, 3, -4, -8, 4] Output: [6,9,3]

Design a dynamic programming algorithm for the Longest Increasing Subsequence problem described below: Input: A sequence of n integers Output: The length of the longest increasing subsequence among these integers. The algorithm must take O(n²) time. You must also write and explain the recurrence. Example 1: Input: [5, 3, 6, 8, 4, 6, 2, 7, 9, 5] Output: 5 ([3, 4, 6, 7, 9]) Example 2: Input: [12, 42, 66, 73, 234, 7, 543, 16] Output: 6 ([42, 66, 73, 234, 543])

Design a dynamic programming algorithm for the Subset Sum problem described below: Input: A set of n integers A and an integer s Output: A subset of A whose numbers add up to s, or that no such set exists. The algorithm must take O(n·s) time. You must also write and explain the recurrence. Example 1: Input: A = {4, 7, 5, 2, 3}, s = 12 Output: {7,2,3} Example 2: Input: A{4, 7, 5,3}, s = 6 Output: 'no such subset'

TECNOLOGIE DEL WEB 2023/2023 (VER 1.1) Prof. Alfonso Pierantonio 1. Project Requirements The project consists in designing and implementing a Web application according to the methodology and the technologies illustrated and developed during the course. This document describe cross-cutting requirements the application must satisfy. The application must be realized with a combination of the following technologies: PHP MySQL HTML/CSS JavaScript, jQuery, etc templating The requirements are 2. Project size The application must have at least 18 SQL tables The number of SQL tables refers to the overall number of tables (including relation normalizations). 3. Methodology The application must be realized by adopting separation of logics, session management, and generic user management (authentication/permissions). Missing one of the above might correspond to a non sufficient score for the project. More in details: 3.1 Separation of Logics The separation of logics has to be realizse by using…

Write a C program to calculate the function sin(x) or cos(x) using a Taylor series expansion around the point 0. In other words, you will program the sine or cosine function yourself, without using any existing solution. You can enter the angles in degrees or radians. The program must work for any input, e.g. -4500° or +8649°. The function will have two arguments: float sinus(float radians, float epsilon); For your own implementation, use one of the following relations (you only need to program either sine or cosine, you don't need both): Tip 1:  Of course, you cannot calculate the sum of an infinite series indefinitely. You can see (if not, look in the program) that the terms keep getting smaller, so there will definitely be a situation where adding another term will not change the result in any way (see problem 1.3 – machine epsilon). However, you can end the calculation even earlier – when the result changes by less than epsilon (a pre-specified, sufficiently small number, e.g.…

Write a C program that counts the number of ones (set bits) in the binary representation of a given number. Example:Input: 13 (binary 1101)Output: 3 units

I need help to resolve or draw the diagrams. thank you