There are two types of professional wrestlers: "babyfaces" ("good guys") and "heels" ("bad guys"). Between any pair of professional wrestlers, there may or may not be a rivalry. Suppose we have n professional wrestlers and we have a list of r pairs of wrestlers for which there are rivalries. Give an O(n +r)-time algo- rithm that determines whether it is possible to designate some of the wrestlers as babyfaces and the remainder as heels such that each rivalry is between a babyface and a heel. If it is possible to perform such a designation, your algorithm should produce it.

There are two types of professional wrestlers: "babyfaces" ("good guys") and "heels" ("bad guys"). Between any pair of professional wrestlers, there may or may not be a rivalry. Suppose we have n professional wrestlers and we have a list of r pairs of wrestlers for which there are rivalries. Give an O(n +r)-time algo- rithm that determines whether it is possible to designate some of the wrestlers as babyfaces and the remainder as heels such that each rivalry is between a babyface and a heel. If it is possible to perform such a designation, your algorithm should produce it.

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 18SA

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There are two types of professional wrestlers: “babyfaces” (“good guys”) and“heels” (“bad guys”). Between any pair of professional wrestlers, there may ormay not be a rivalry. Suppose we have n professional wrestlers and we have a listof r pairs of wrestlers for which there are rivalries. Give an O.n C r/-time algorithm that determines whether it is possible to designate some of the wrestlers asbabyfaces and the remainder as heels such that each rivalry is between a babyfaceand a heel. If it is possible to perform such a designation, your algorithm shouldproduce it.
PLEASE HELP ME. kindly show all your work 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}. 3. Let n ∈ N \ {0}. Describe the largest set of values n for which you think 2n < n!. Use induction toprove that your description is correct.Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! ∈ O(n log2 n). Thank you. But please show all work and all steps
Given a non-empty list items of positive integers in strictly ascending order, find and return thelongest arithmetic progression whose all values exist somewhere in that sequence. Return the answer as a tuple (start, stride, n) of the values that define the progression. To ensure unique results to facilitate automated testing, if there exist several progressions of the same length, this function should return the one with the lowest start. If several progressions of equal length emanate from the lowest start, return the progression with the smallest stride. INTRUCTIONS: Input a comment or a solution on how you solve the problem, write a psuodocode and flow chart about th problem
In this exercise we will develop a dynamic programming algorithm for finding the maximum sum of consecutive terms of a sequence of real numbers. Thatis, given a sequence of real numbers a1, a2,… , an,the algorithm computes the maximum sum ∑k(top) i=j(bottom) a_iwhere 1 ≤ j ≤ k ≤ n. b) Let M(k) be the maximum of the sums of consecutiveterms of the sequence ending at ak. That is, M(k) =max1≤j≤k∑ki=j ai. Explain why the recurrence relationM(k) = max(M(k − 1) + ak, ak) holds for k = 2, ..., n.c) Use part (b) to develop a dynamic programming algorithm for solving this problem.d) Show each step your algorithm from part (c) uses tofind the maximum sum of consecutive terms of thesequence 2,−3, 4, 1,−2, 3.e) Show that the worst-case complexity in terms of thenumber of additions and comparisons of your algorithm from part (c) is linear.
If we choose to represent each genotype as a partial permutation, then provide an algebraic expression, in terms of n, for the size of the genetic search space (the number of possible valid genotypes), in terms of n. Assume that n ≥ 4.
Suppose we have the following sets: P, Q, and R. Prove or disprove that if for all x, (x ∈ P) → ((x ∈ Q) → (x ∈ R)), then P ⋂ Q ⊆ R
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Given is a list of K distinct coin denominations (V1,...,Vk) and the total sum S>0. Find the minimum number of coins whose sum is equal to S (we can use as many coins of one type as we want), or report that it’s not possible to select coins in such a way that they sum up to S. Justify your explanation!
Consider GCET’s central library where the librarian Mario is required to sort thebooks before closing each day. The library has long shelves, and the books arenumbered through 1 to N, the librarian wishes to arrange the books so that they appearin the sequence 1,2, …, N.He intends to do this by a sequence of moves. In each move Mario can pick up anybook and insert it at a different place. Let he has 5 books and they are initially arrangedin the order: 1 4 5 3 2Mario will rearrange this in ascending order by first moving book 2 to the beginningof the shelf to get : 1 2 4 5 3Then moving book 3 to position 3, he gets 1 2 3 4 5You are required to write a program for Mario that determines the minimum numberof moves that are necessary to sort the bookshelf. C programming, please solve with full answer and solve with Flowchart and Discription
Consider GCET’s central library where the librarian Mario is required to sort thebooks before closing each day. The library has long shelves, and the books arenumbered through 1 to N, the librarian wishes to arrange the books so that they appearin the sequence 1,2, …, N.He intends to do this by a sequence of moves. In each move Mario can pick up anybook and insert it at a different place. Let he has 5 books and they are initially arrangedin the order: 1 4 5 3 2Mario will rearrange this in ascending order by first moving book 2 to the beginningof the shelf to get : 1 2 4 5 3Then moving book 3 to position 3, he gets 1 2 3 4 5You are required to write a program for Mario that determines the minimum numberof moves that are necessary to sort the bookshelf. please solve with Flowchart and description
Implement the algorithm for an optimal parenthesization of a matrix chain product as dis-cussed in the class.Use the following recursive function as part of your program to print the outcome, assumethe matrixes are namedA1, A2, ..., An.PRINT-OPTIMAL-PARENS(s, i, j){if (i=j) thenprint “A”i else{print “(”PRINT-OPTIMAL-PARENS(s,i,s[i, j])PRINT-OPTIMAL-PARENS(s, s[i, j] + 1, j)print “)”} }a- Test your algorithm for the following cases:1. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,3, X,12,5,50, Y,6>.2. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,50,6, X,15,40,18, Y,30,15, Z,3,12,5>. 3. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<50,6, X,15,40,18, Y,5,10,3,12,5, Z,40,10,30,5>. X=10 Y=20 Z=30
In the maximum coverage problem, we have a set of n elements E, a set of m sets (whichare subsets of E), S1, S2, .., Sm. We are also given an integer m ≥ k ≥ 1. Our goal is to pick k sets thatcover the maximum number of elements. We say an element is covered if we pick at least one set fromS1, S2, .., Sm in our solution that contains that element. Devise a greedy algorithm that is a (1 −1/e) approximation. That is if there is an optimum solution thatcan cover OPT number of elements, the greedy algorithm is guaranteed to cover at least (1 −1/e) *OPT elements. Here e is the base of natural logarithm. Use the following relation (1-1/k)k ~1/e.