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 21.3, Problem 3E
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To give a sequence of m MAKE-SET, UNION and FIND-SET operations, n of which are MAKE-SET operations that take
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For σarthm := (0, S, +, ·). Prove PA ⊨ ∀x (x ≠ 0 ⟹ ∃y(x = S(y))). In English, this say that every element other than 0 is a successor.
Prove that the amortized complexity of the union operation in the solution is O(log n). Consider the number of times a given element may change its set.
Order the following functions by asymptotics with respect to Ω. That is, find an orderingf1 , f2 , · · ·, of the following functions such that f1 = Ω(f2), f2 = Ω(f3) and so on.n2, (√2)logn , n!, log(n)!, (3/2)n , n3, log2n, loglogn, 4logn, 2n, nlogn, 2logn , 2√2logn , log(n!)
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- List the elements of set A = {x:x є N, 2<x<9} where N= {1, 2, 3, ...}arrow_forwardGive the uppor-bound for the recurrence T(n)=2π(n/2)+n∧2, using the Recursion Tree method. You must show at least 3 levels of the tree, and give the explicit log base when using the tree height!arrow_forwardComputing the union of two sets using Boolean values is simple and efficient. Since the union of two sets is a combination of the members of bothsets, we can build a new union set by Oring the corresponding elements ofthe two BitArrays. In other words, a member is added to the new set if thevalue in the corresponding position of either BitArray is True.Computing the intersection of two sets is similar to computing the union;only for this operation we use the And operator instead of the Or operator.Similarly, the difference of two sets is found by executing the And operatorwith a member from the first set and the negation of the corresponding member of the second set. We can determine if one set is a subset of another set byusing the same formula we used for finding the difference. For example, if:setA(index) && !(setB(index))evaluates to False then setA is not a subset of setB.The BitArray Set Implementation Write The code for a CSet class based on a BitArray.arrow_forward
- Suppose that we implement a union-find structure by representing each set using a balanced search tree. Describe and analyze algorithms for each of the methods for a union-find structure so that every operation runs in at most O(logn) time in worst case.arrow_forwardCorrect answers are nodes B and A, please explain with proper working that how we can come up with these answers.arrow_forwardPlease don't copy Please show a detailed explanation of how to do each step. Show how to make the intersection of two NFA. How can there be a bijection? 1) FOR NFA NOT DFA: Give examples of two distinct countably infinite sets A and B such that there is a bijection between A⋃ B and A∩B.arrow_forward
- Implement Algorithm Using Schreier Generators in Schreier-Sims methodInput : a set S of generators of a group G;Output : a base B for G; a strong generating set S of G relative to B;arrow_forwardLet S be a set of sets with the following recursive definition ∅ ∈ S. If X ⊆ S, then X ∈ S. (a) List three different elements of S. (b) Explain why S has infinitely many elementsarrow_forwardLet C(n) denote the number of comparisons performed by quicksort in sorting an array of size n in the worst case. Write the recurrence relation for C(n) and solve.arrow_forward
- prove the assertion. An failed or successful binary search in an ordered array of N keys needs no more than lg N 1 comparisons.arrow_forwardMake K-map as followsarrow_forwardFor an array A of size N and A[0] > A[N-1] (0 indexed), devise an efficient algorithm to find a pair of adjacent elements A[il and A[i+1] such that Ali] > A[i+1]. Can you always find such an adjacent pair and why? Justify your conclusion with proof.arrow_forward
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