Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 14.1, Problem 8E
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You are standing in front of an infinitely long straight fence; that is, the fence extends infinitely to your left and to your right. The fence has a single gate in it but you do not know where it is. Your goal is to minimize the distance you need to walk in order to find the gate. If n (which is unknown) is the distance to the gate in yards, design and an analyze an efficient algorithm for finding the gate in terms of n.
Let n E N be a natural number with a divisor d E N. If the sum of all of the positive divisors of n is equal to 2n + d, then we call n a near-
perfect number.
For example, if we have n = 12, then the divisors of n are 1, 2, 3, 4, 6 and 12. Therefore, the sum of the divisors of n is equal to
1+2+3+4+6+12=2*12+4, and 4|12. Hence, 12 is a near-perfect number.
Write a function, near_perfect_number, which accepts any integer n as input, and returns True if n is a near-perfect number, and False
otherwise.
You are given a collection of n bolts of different widths and n corresponding nuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However, there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to each nut.
Design an algorithm for this problem with Θ(n log n) average-case complexity (in terms of nut-bolt comparisons).
Explain why your algorithm has Θ(n log n) average case complexity.
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- implement Running time algorithm Careful(n) pre-cond: n is an integer. post-cond: Q(n) “Hi”s are printed for some odd function Qarrow_forwardYou are given a collection of n bolts of different widths and n corresponding nuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However, there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to each nut. Design an algorithm for this problem with Θ(n log n) average-case complexity (in terms of nut-bolt comparisons). Explain why your algorithm has Θ(n log n) average-case complexity. You may use any result without explicitly solving recurrence equations.arrow_forwardsuppose that n is not 2i for any integer i. How would we change the algorithm so that it handles the case when n is odd? I have two solutions: one that modifies the recursive algorithm directly, and one that combines the iterative algorithm and the recursive algorithm. You only need to do one of the two (as long as it works and does not increase the BigOh of the running time.)arrow_forward
- Given is a strictly increasing function, f(x). Strictly increasing meaning: f(x)< f(x+1). (Refer to the example graph of functions for a visualization.) Now, define an algorithm that finds the smallest positive integer, n, at which the function, f(n), becomes positive. The things left to do is to: Describe the algorithm you came up with and make it O(log n).arrow_forwardThe coordinates of a polygon can be represented as a list of tuples: [(x1, y1), (x2, y2), .., (xn, yn)], where (x1, y1), ., (xn, yn) are points of the polygon in a counterclockwise order. Find the area of such a polygon using using shoelace formula. It is no longer required to take absolute value. п-1 п—1 A = > xi+1Yi ) – x1Yn i=1 i=1 1 x142 + x2Y3 + + xn-1Yn + xnY1 - x2y1 – X3Y2 – ··- xn Yn-1 – x1Yn| >>> area ([(0,0), (1,0), (0,1)]) 0.5 >>> area ([(0,0), (1,0),(1,1),(0,1)]) 1.0 >> area([(0,0),(2,0), (2,2),(1,2), (0,1)]) 3.5 Write the function area(C) to find the area of a triangle with vertices at c = [(x1, y1), (x2, y2),..., (xn, yn)].arrow_forwardGiven n> 3 points P1 = (X1, y1), ..., Pn = (Xn, Yn) in the coordinate plane, design an algorithm to check whether all the points lie within a triangle with its vertices at three of the points given. You can either design an algorithm from scratch or transform the problem to another one with a known algorithm. %3Darrow_forward
- In Python, generate a random matrix A with 100 entries each of which is an independent numberpseudo-randomly drawn (uniformly) from the interval (0, 1). Let B = A + AT . Use Python to calculatethe diagonal matrix D with the same eigenvalues as B. Then use the QR method to block-diagonalize Bover R. Do at least 100 steps of the QR switching. You should end up with a matrix E which is differentthan D. Answer the question of why.arrow_forwardJacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges. Further details of the method can be found at Jacobi Method with a formal algorithm and examples of solving a system of 2/3 linear equations. This also includes a python implementation of the algorithm using numpy. But, in this assignment, you are required to write python implementation using list/nested list data structure without using any of the built-in python libraries for matrix operations. Write a python function Jacobi(A, b, x_init) where A is a diagonal dominant matrix, b is list of constants (right-hand side of the equations), and x_init (a list) is an initial guess to the solution. The function returns the solution as a list (each value of the list is rounded off to two decimal places). >>> Jacobi([[2, 1, 1], [0, 2, 1],…arrow_forwardGiven an n-element sequence of integers, an algorithm executes an O(n)-time computation for each even number in the sequence, and an O(logn)-time computation for each odd number in the sequence. What are the best-case and worst-case running times of this algorithm? Why? Show with proper notations.arrow_forward
- Carry out all the steps in the the Euclidean algorithm for 78 and 64 as we did in class. This means find the greatest common divisor according to this algorithm. If you get a zero at some point do not leove it out. Each line should be a complete mothematical equation. Use same format ond conventions for each equation as shown in the previous problem.arrow_forwardwith n=6 and A=(3,5,4,1,3,2). Draw the corresponding walkthrough as shownarrow_forwardSuppose, T(n) = 12+22+32+…..+n2. Show that, T(n) is in Θ(n3 ).arrow_forward
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