Prove that the size of an instance will always decrease at least by a factor of 2 after two successive iterations of Euclid's algorithm.
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- 5. Let R = {1, 3, π, 4, 1, 9, 10}, S = {{1}, 3, 9, 10}, T = {1, 3, π}, and U = {{1, 3, π}, 1}. Which of thefollowing are true? For those that are not, why not? (d) 1 ⊆ U(e) {1} ⊆ T(f) {1} ⊆ S(g) {1} ∈ SArtificial Intelligence - Adversarial Search 1. Consider the following three variants of minimax search: the simple version, alpha-beta search, anddepth-limited search, and consider the games of tic-tac-toe and chess. For the chess game, supposethat the Threefold Repetition Rule and the Fifty-Move Rule, and the similar rules if any, are notconsidered, i.e., the game will not terminate if the same position occurs multiple times. For eachcombination of minimax variant and game, answer the following question: can that minimax variantpossibly never terminate, in computing the best next move? Justify your answer.Problem 8: Divide and Conquer Recurrences Consider the recurrence that naturally arises in some divide-and-conquer algorithms: T(1) = 1, T(n) = 4T(n/2) + n for n > 1 [a] Explain what the initial condition and the various parts of this recurrence mean in the context of the analysis of algorithms. [b] Use the Master Theorem to determine the order of magnitude of T(n). [c] Using the methods of our course, derive a closed-form formula for T(n). [d] For the case n = 2, test your formula for T(n) against the original recurrence.
- [Introduction to the Design and Analysis of Algorithms, 3rd Edition] Maxima search. A point (xi, yi) in the Cartesian plane is said to be dominated by point (xj , yj ) if xi ≤ xj and yi ≤ yj with at least one of the two inequalities being strict. Given a set of n points, one of them is said to be a maximum of the set if it is not dominated by any other point in the set. For example, in the figure below, all the maximum points of the set are circled. Design an efficient algorithm for finding all the maximum points of a given set of n points in the Cartesian plane. What is the time efficiency class of your algorithm?2 (**) Rubik's Cube Consider the task of solving Rubik's Cube. Please formulate it as a state-space search problem. Specifically, please define the state space and the successor function. For simplicity, let us consider a 2 × 2 × 2 cube.1.Consider the following Boolean Satisfiability problem:Given a Boolean expression consisting of variables, does the expression have an assignment for the set of variables where the expression evaluates to TRUE. (i)Outline, in pseudocode, an exact algorithm for the problem. This should guarantee a solution if one exists. (Please state clearly the algorithm used) (ii)For the exact method used in part (1) (i), estimate the time required for the algorithm to find an exact solution in the big O-notation.
- Prove that the following problem is NP: We're provided S and t. Is there a subset of S whose members add to t? Algorithm and Data Structure IssuesUsing c++ Apply both breadth-first search and best-first search to a modified version of MC problem. In the modified MC, a state can contain any number of M’s and any number of C’s on either side of the river. Assume the goal is always to move all the persons on the left side to the right side. The Initial state should be a parameter given to the program at beginning of execution. As in the original problem, boat capacity =2, the boat cannot move by itself, and on either side C’s should not outnumber M’s. For best-first search, you need to come up with an appropriate heuristic. In addition to solving the problem, your grade will also be based on th effectiveness of the heuristic. As an example, the program should execute as follows. Initial state… Enter number of M’s on left side of the river: 3 Enter number of C’s on left side of the river: 1 Enter number of M’s on right side of the river: 0 Enter number of C’s on right side of the river: 0 Enter location of the boat: L The output…Correct answer will be upvoted else downvoted. Computer science. There is an exhibit a1,a2,… ,an of n positive integers. You should isolate it into a negligible number of ceaseless portions, to such an extent that in each section there are no two numbers (on various positions), whose item is an ideal square. Also, it is permitted to do all things considered k such activities before the division: pick a number in the exhibit and change its worth to any certain integer. However, in this form k=0, so it isn't significant. What is the base number of consistent fragments you should utilize in the event that you will make changes ideally? Input The primary line contains a solitary integer t (1≤t≤1000) — the number of experiments. The principal line of each experiment contains two integers n, k (1≤n≤2⋅105, k=0). The second line of each experiment contains n integers a1,a2,… ,an (1≤ai≤107). It's surefire that the amount of n over all experiments doesn't surpass 2⋅105.…
- 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 caseThe function below has the recursive relation T(x,y) = 2T(y/2)+O(1) when x,y>0Knowing these information, find the following: 1) The worst-case time complexity of the function2) The worst-case auxiliary space complexity of the function3) Explanation as to how the time complexity of the function can be improved without using multiplication operations.Let A ={a1,...,an} and B ={b1,...,bm} be two sets of numbers. Consider the problem of finding their intersection, i.e., the set C of all the numbers that are in both A and B. a. Design a brute-force algorithm for solving this problem and determine its efficiency class. b. Design a presorting-based algorithm for solving this problem and determine its efficiency cla