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 8, Problem 4P
(a)
Program Plan Intro
To describe a deterministic
(b)
Program Plan Intro
To prove the lower bound is
(c)
Program Plan Intro
To explain a randomized algorithm whose expected number of the comparison is
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A cryptarithm is a mathematical puzzle for which the goal is to find the correspondence between letters and digits, such that the given arithmetic equation consisting of letters holds true when the letters are converted to digits.
You have an array of strings crypt, the cryptarithm, and an an array containing the mapping of letters and digits, solution. The array crypt will contain three non-empty strings that follow the structure: [word1, word2, word3], which should be interpreted as the word1 + word2 = word3 cryptarithm.
If crypt, when it is decoded by replacing all of the letters in the cryptarithm with digits using the mapping in solution, becomes a valid arithmetic equation containing no numbers with leading zeroes, the answer is true. If it does not become a valid arithmetic solution, the answer is false.
Note that number 0 doesn't contain leading zeroes (while for example 00 or 0123 do).
Computer Science
There is an n × n grid of squares. Each square is either special, or has a positive integer costassigned to it. No square on the border of the grid is special.A set of squares S is said to be good if it does not contain any special squares and, starting fromany special square, you cannot reach a square on the border of the grid by performing up, down,left and right moves without entering a cell belonging to S.
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- Design an algorithm which receives an arbitrary n × n grid, runs in time poly-nomial in n and determines a good set of squares with minimum total cost.
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…
Chapter 8 Solutions
Introduction to Algorithms
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- 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…arrow_forwardImplement the sieve of Eratosthenes: a function for computing prime numbers, known to the ancient Greeks. Choose an integer n. This function will compute all prime numbers up to, and including, n. First insert all numbers from 1 to n into a set. Then erase all multiples of 2 (except 2); that is, 4, 6, 8, 10, 12, ... . Erase all multiples of 3, that is, 6, 9, 12, 15, ... . Go up to ?⎯⎯√n. The remaining numbers are all primes. Make sure that you use functions in your solution, including a main function. Define def sieveOfEratosthenes(n) that receives the maximum value and returns a list of all prime numbers less than or equal to this value in main() print the list of prime numbers.arrow_forwardTennis is a racket sport that is played by two opposing players on S sets. Each set consists of at least Kgames. A set is won by a player if that player wins at least K games and at least 2 games more than theopponent. Once a set is won, the set is ended and the match continues to a new set (if any) where bothplayers start from 0 game won for that new set.For example, let K = 6, then a set can be ended with any of the following.• P1 (Player 1) wins 6 games while P2 (Player 2) wins 3 games → P1 wins the set.• P1 wins 7 games while P2 wins 9 games → P2 wins the set.On the other hand, a set cannot be ended with any of the following.• P1 wins 6 games while P2 wins 5 games → no player wins at least 2 games more than the opponent.• P1 wins 0 game while P2 wins 5 games → no player wins K = 6 games.• P1 wins 7 games while P2 wins 0 games → the set is already ended when P1 won the first 6 games.• P1 wins 8 games while P2 wins 5 games → the set must already be ended before it reaches this…arrow_forward
- A magic square is a square array of non-negative integers where the sums of the numbers on each row,each column, and both main diagonals are the same. An N × N occult square is a magic square with Nrows and N columns with additional constraints:• The integers in the square are between 0 and N, inclusive.• For all 1 ≤ i ≤ N, the number i appears at most i times in the square.• There are at least two distinct positive integers in the square.For example, the following is a 5 × 5 occult square, where the sums of the numbers on each row, eachcolumn, and both main diagonals are 7:0 0 0 3 42 4 0 0 10 0 3 4 05 0 0 0 20 3 4 0 0For a given prime number P, you are asked to construct a P × P occult square, or determine whether nosuch occult square exists.InputInput contains a prime number: P (2 ≤ P ≤ 1000) representing the number of rows and columns in theoccult square.OutputIf there is no P × P occult square, simply output -1 in a line. Otherwise, output P lines, each contains Pintegers…arrow_forwardPLZ help with the following in Java try abd keep it simple so i can understand You have a long flowerbed in which some of the plots are planted, and some are not. However, flowers cannot be planted inadjacent plots.Given an integer array flowerbed containing 0's and 1's, where 0 means empty and 1 means not empty, and an integer n, return ifn new flowers can be planted in the flowerbed without violating the no-adjacent-flowers rule. Example 1: Input: flowerbed = [1,0,0,0,1], n = 1 Output: true Example 2: Input: flowerbed = [1,0,0,0,1], n = 2 Output: false Constraints: 1 <= flowerbed.length <= 2 * 104 • flowered [i] is 0 or 1. • There are no two adjacent flowers in flowerbed. 0 <= n <= flowerbed.lengtharrow_forwardA general state space search algorithm is implemented in search.py. Go through this and try to understand it. An example of how to implement a particular search problem using the state space code is shown in pitcher.py. Note in particular how the PitcherState class works in conjunction with the Search class. Using the PitcherState class as a model, write a Simpsons class that implements a solution to the following slightly modified problem from an episode of the Simpsons called Gone Maggie Gone from 2009. Maybe you remember it. Homer Simpson has to move his daughter Maggie, his dog Santa's Little Helper, and a jar of medical marijuana that looks like candy across a river. He can only take one item in his boat at a time. He can't leave Maggie alone with the medical marijuana (or she will eat it) and he can't leave Santa's Little Helper alone with Maggie (because the dog will pester the girl). Formulate the actions for this problem and implement them. Be careful to ensure that illegal…arrow_forward
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