algorithms
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For relatively small values of n, algorithms with larger orders can be more efficient than algorithms with smaller orders. Use a graphing calculator or computer to answer this question.
For what values of n is an
[50 log2(n)]
operations? (Assume n is an integer such that n > 1.)
n?
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- 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.Consider the problem of counting, in a given text, the number of substrings that start with an A and end with a B. For example, there are four such substrings in CABAAXBYA.a. Design a brute-force algorithm for this problem and determine its efficiency class.b. Design a more efficient algorithm for this problem with complexity O (n)Consider the “recursion tree” and “subproblem graph” for our two algorithms. The case n = 4 is illustrated below. For the case n = 4, the recursion tree has 16 vertices and 15 edges, while the subproblem graph has 5 vertices and 10 edges. For the case n = 10, determine the number of vertices and edges in the recursion tree, as well as the number of vertices and edges in the subproblem graph. Clearly justify your answers.
- Develop a brute-force algorithm that finds the longest common sequential elements of two given lists with the Java programming language. For example, the following two lists are given: S1 = [B,C,D,A,A,C,D] S2 = [A,C,D,B,A,C] Common lists are: [B,C], [C,D,A,C], [D,A,C], [A,A,C], [A,C], [C,D], … Note that although both lists have a common [C,D,A,C] (sub)list, both list items are skipped and the left-to-right order is intact. Apply the brute force algorithm. Explain Time Complexity?The travel time function of an algorithm has the form: f(n) = 3n^2+4n+8.a. Determine the values of c and n0, so that big-oh O(n^2) satisfies the rule f(n) <= cg(n); n≥n0,b. Show (picture) the graphs of f(n) and g(n).For each, draw the recursion tree, find the height of the tree, the running time of each layer, and the sum of running times. Then use this info to find the explicit answer for T(n). a. T(n) = 2T(n/4) + √ n (n is a power of 4 (n = 4^k) for some positive integer k) b. T(n) = 9T(n/3) + n^2 (n is a power of 3 (n = 3^k) for some positive integer k) c. T(n) = T(n/2) + 1 (n is a power of 2 (n = 2^k) for some positive integer k)
- The rooted Fibonacci trees Tn are defined recursively in the following way. T1 and T2 are both the rooted trees consisting of a single vertex, and for n = 3, 4, …, the rooted tree Tn is constructed from a root with Tn-1 as its left subtree and Tn-2 as its right subtree. 1. How many vertices, leaves, and internal vertices does the rooted Fibonacci tree Tn have, where n is a positive integer? 2. What is its height?In this question you will explore Graph Colouring algorithms. Given a graph G, we say that G is k-colourable if every vertex of G can be assigned one of k colours so that for every pair u, v of adjacent vertices, u and v are assigned different colours. The chromatic number of a graph G, denoted by χ(G), is the smallest integer k for which graph G is k-colorable. To show that χ(G) = k, you must show that the graph is k-colourable and that the graph is not (k − 1)-colourable. Question: It is NP-complete to determine whether an arbitrary graph has chromatic number k, where k ≥ 3. However, determining whether an arbitrary graph has chromatic number 2 is in P. Given a graph G on n vertices, create an algorithm that will return TRUE if χ(G) = 2 and FALSE if χ(G) 6= 2. Clearly explain how your algorithm works, why it guarantees the correct output, and determine the running time of your algorithm.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.
- Conceptual Question: How do you find the Big Oh, Big Omega, and Big Theta of an Algorithm? What do each of them mean? Can you provide an example for solving for each with some algorithm? Like... (Java) for (i = 1; i < n; i++) { j = i; while ((j > 0) && (s[j] < s[j-1])) { temp = s[j]; s[j] = s[j-1]; s[j-1] = temp; } j--; } And what about if Algorithm is recurssive like... public int runBinarySearchRecursively(int[] sortedArray, int key, int low, int high) { int middle = (low + high) / 2; if (high < low) { return -1; } if (key == sortedArray[middle]) { return middle; } else if (key < sortedArray[middle]) { return runBinarySearchRecursively(sortedArray, key, low, middle - 1); } else { return runBinarySearchRecursively(sortedArray, key, middle + 1, high); } }Modeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…READ AND SOLVE VERY CAREFULLY WITH PYTHONthanks in advance Given a jungle matrix N*M:jungle = [[1, 0, 0, 0],[1, 1, 0, 1],[0, 1, 0, 0],[1, 1, 1, 1,]]Where 0 means the block is dead end and 1 means the block can be used in the path from source to destination. Task: (Use Python)Starting at position (0, 0), the goal is to reach position (N-1, M-1).Your program needs to build and output the solution matrix, a 4*4 matrix with 1’s in positions used to get from the starting position (0,0) to the ending position (N-1, M-1) with the following constraints:- You can only move one space at a time.- You can only move in two directions: forward and down.- You can only pass thru spaces on the jungle matrix marked ‘1’If you cannot reach the ending position, print a message that you’re trapped in the jungle Algorithm:If destination is reached print the solution matrixElse Mark current cell in the solution matrix Move forward horizontally and recursively check if this leads to…