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- “Rectangles” A rectangle can be specified in a 2-dimensional plane using the top left (north west) point and the bottom right (south east) point. Given n rectangles (using 2 points each), give an O(n log n) algorithm that tells if any two rectangles from the list overlap. Two rectangles are said to overlap, if there is a common point in both of them. [If a rectangle is entirely contained in another, they are still said to “overlap”, even though their lines don’t cross.]python The number of possible binary tree topologies (given all possible heights) with n nodes is the Catalan Number (in closed form) Cn = 1/(n+1)*(2n choose n). Recursively, this can also be written as: question ) Write a function cn_fast(n) that computes the n-th catalan number using the closed form formulation, and Conduct doubling experiments (just n = 1, 2, 4, and 8) to compute runtimes. Plot the result on a log-log plot.Count consecutive summers def count_consecutive_summers(n): Like a majestic wild horse waiting for the rugged hero to tame it, positive integers can be broken down as sums of consecutive positive integers in various ways. For example, the integer 42 often used as placeholder in this kind of discussions can be broken down into such a sum in four different ways: (a) 3 + 4 + 5 + 6 + 7 + 8 + 9, (b) 9 + 10 + 11 + 12, (c) 13 + 14 + 15 and (d) 42. As the last solution (d) shows, any positive integer can always be trivially expressed as a singleton sum that consists of that integer alone. Given a positive integer n, determine how many different ways it can be expressed as a sum of consecutive positive integers, and return that count. The number of ways that a positive integer n can be represented as a sum of consecutive integers is called its politeness, and can also be computed by tallying up the number of odd divisors of that number. However, note that the linked Wikipedia de0inition…
- Count consecutive summers def count_consecutive_summers(n): Like a majestic wild horse waiting for someone to come and tame it, positive integers can be broken down as sums of consecutive positive integers in various ways. For example, the integer 42 often used as placeholder in this kind of discussions can be broken down into such a sum in four different ways: (a) 3 + 4 + 5 + 6 + 7 + 8 + 9, (b) 9 + 10 + 11 + 12, (c) 13 + 14 + 15 and (d) 42. As the last solution (d) shows, any positive integer can always be trivially expressed as a singleton sum that consists of that integer alone. Given a positive integer n, determine how many different ways it can be expressed as a sum of consecutive positive integers, and return that count. The count of how many different ways a positive integer n can be represented as a sum of consecutive integers is also called its politeness, and can be alternatively computed by counting how many odd divisors that number has. However, note that the linked…5. Let F(n) denote the Fibonacci numbers. Prove the following property for all n ≥ 1 (Hint: Induction)F(1) + F(2) + · · · + F(n) = F(n + 2) − 1 7. The binary search algorithm is used with the following sorted list, with parameter x having the value“Chicago”. Name the elements against which x is compared.Boston, Charlotte, Indianapolis, New Orleans, Philadelphia, San Antonio, Yakima 8. The binary search algorithm is used with the following sorted list, with parameter x having the value“flour”. Name the elements against which x is compared.butter, chocolate, eggs, flour, shortening, sugarin this code: def insertSort(A): for i in range(1,len(A)): j = i-1 while A[j]>A[j+1] and j>=0: A[j],A[j+1] = A[j+1],A[j] j -= 1 A = [3,1,11,9,10,2] insertSort(A) print(A) 1) What is the Hypothesis? 2) What is Your Induction
- Baby Names: Each year, the government releases a list of the 10,000 most common baby namesand their frequencies (the number of babies with that name). The only problem with this is thatsome names have multiple spellings. For example, "John" and ''.Jon" are essentially the same namebut would be listed separately in the list. Given two lists, one of names/frequencies and the otherof pairs of equivalent names, write an algorithm to print a new list of the true frequency of eachname. Note that if John and Jon are synonyms, and Jon and Johnny are synonyms, then John andJohnny are synonyms. (It is both transitive and symmetric.) In the final list, any name can be usedas the "real" name.EXAMPLEInput:Names: John (15), Jon (12), Chris (13), Kris (4), Christopher (19)Synonyms: (Jon, John), (John, Johnny), (Chris, Kris), (Chris, Christopher)Output: John (27), Kris (36)Algortimhs Recursive Thinking) Carefully read the comment of the following function, then complete the function so that it correctly solves the problem. Verify your answer by writing down the three questions rule and answer the questions for your completed algorithm. /* Return true if there exist duplicate values in sublist a[first...last], i.e., there are a[i]==a[j] for some i,j where i is not equal to j @param a: the vector/list @param first, last: two integer specifying the starting and ending indices of list@return return true if there are duplicate values in a[first...last], otherwise return false */ bool ContainDuplicate (vector<int> a, int first, int last) { //1. what should be the base case? What should be returned? // general case if (ContainDuplicate (a, first+1, last)==true) _______________; //Finish the rest based upon comment:// if a[first] does not equal to ANY value in a[first+1...last], // then return false;// else return true } (Recursive Thinking) For the…Implement the following two sorting algorithms in a program called p3.py. Write two separate functions for these algorithms. Both functions must take a list of integers as the input parameter.1) Bogosort: first shuffle the list argument (i.e., randomize the positions of every element) and then check to see if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.2) Bozosort: choose two elements in the list at random, swap them, and then check if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.Write a main() function and call both sorting functions using the same list as their arguments. The list can be of any size (try a small list first). Does any of your algorithms terminate? If yes, count the number of iterations it uses to sort the list. Does it always use the same number of repetitions? If…
- Question 14 papa .Using Python construct the following code: Implement "reverse" as a method of SinglyLinkedList, called as L.reverse(), that reverses the SinglyLinkedList instance L using only a constant amount of additional space (i.e., as an in-place algorithm) and not using any recursion. Full explain this question and text typing work only We should answer our question within 2 hours takes more time then we will reduce Rating Dont ignore this line* Assignment: Longest increasing subsequence * * Sequences are a natural source of computational problems. One such * family of problems involves finding subsequences with specified * properties in a given sequence. This exercise asks you to write * a program that, given a sequence * * s(0), s(1), ..., s(n-1) * * of integers as input, finds a ___longest increasing subsequence___ * of the sequence s. * * For example, suppose we are given as input the sequence * * 72, 16, 51, 17, 6, 21, 92, 59, 54, 78, 41, 33, 94, * 85, 83, 56, 2, 46, 57, 44, 73, 6, 47, 47, 0. * * In this sequence, a longest increasing subsequence has length 7. * One example of such an increasing subsequence is * * 16 < 17 < 21 < 54 < 56 < 57 < 73. * * More generally, your program must be such that * given a sequence * * s(0), s(1), ..., s(n-1) * * of integers as input, the program returns a subsequence * * s(i_1), s(i_2), ..., s(i_k) * * that meets all…This code is for python Searching for Data In this lab we'll compare the time it takes to search for an item in a list vs searching for it in a binary search tree. Part 1: Populating and Searching a List To get started create a function, populate, that takes as a parameter a positive integer n, populates a list of length n with random integers in the range of [0, n] and returns that list. Next create a function that takes as a parameter a list and an integer and returns True if that integer is found in the list and False if it is not. Finally, in your main script call your function for a value of n to get a list of length n and then go through that list and look for each element in that list within the list and print out that count. Needless to say if you did everything right you should print out n Part 2: Populating and Searching a Binary Search Tree Download the provided BST.py module. This module is a stripped down implementation of a Binary Search Tree and associated Node…