Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 30, Problem 5P
Program Plan Intro
To prove that
Program Plan Intro
To prove that
Program Plan Intro
To prove that
Program Plan Intro
To write an
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Consider the function
f(n) = 2n if n is evenf(n) = 3n^2 if n is odd From the definitions:i) Prove or disprove that f(n) is O( n^2) (`Big-Oh of n squared’).ii) Prove or disprove that f(n) is Ω( n^2) (`Big-Omega of n squared’).iii) Prove or disprove that f(n) is o( n2^2) (`little-oh of n squared’) I have proven (i) and disproven (iii), I am having trouble with (ii), becuase if i choose c = 1 and n0 = 1, i get this inequality 4 >= 4 which is true, but for any other value of n it is false, but is this correct since it's only there exists c and n0 such that f(n) >= c g(n) or do other n values, for example n0 = 2, 3 ,4 , need to be true as well for it to be proven?Thank you very much
1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.
Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but c is relatively prime to both a and b . Let n = s × a + t × b be some linear combination of a and b, where s and t are integers. Prove that n cannot be a divisor of c.
Follow the definition of relative primes, and use contradiction.
Knowledge Booster
Similar questions
- 2. Prove from the definition that 2n2 + 100n log n + 1000 = O(n2) 3. For what values of n is 50 n lg n greater than 0.5 n2? Why do we say that 0.5 n2 is asymptotically larger, if 50 n lg n is larger for many values? (Hint: you may need to graph the functions or play around with your calculator.)arrow_forwardApply a suitable approach to compare the asymptotic order of growth forthe following pair of functions. Prove your answer and conclude by telling if f(n) ?Ѳ(g(n)), f(n) ?O(g(n)) or f(n) ?Ω(g(n)). f(n) = 100n2+ 20 AND g(n) = n + log narrow_forwardWe mentioned that if we want to prove P = NP, we only need to pick up any one NPC problem and design a polynomial-time algorithm for the problem. If you want to prove P = NP, select one NPC problem based on your preference and describe your idea of a polynomial-time algorithm that solves the problem. It does not have to be a formal algorithm or pseudo-code, a description of your idea of designing such an algorithm would be fine.arrow_forward
- How can I implement the complex step into this code such that xk+1 = xk - h*(f(xk)/Im(f(xk + ih)), where k = 0,1,2,... and Im = Imaginary number. Hence, I need to modify the code so that the default value of the derivative df has to be the new approximation Im(f(x0 + ih))/h.arrow_forwardPLEASE 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 stepsarrow_forwardYou 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.arrow_forward
- Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.arrow_forwardSmall witnesses are easy to findThe definition of NP requires that the number of bits of the witness is at most polynomialin the number of bits of the input, i.e., |w| = poly(n). Suppose a decision problemA ∈ NP has the property that witnesses, when they exist, are at most logarithmic size,i.e., |w| = O(log n). Show that this implies A ∈ P.arrow_forward5. Practice with RSA algorithma. Pick two prime numbers p, q, for example,• const int P=23;• const int Q=17;• int PQ=P*Q; b. Find a e that is relatively prime with (p-1)(q-1)Call RelativelyPrime () c. Calculate the inverse modulo (p-1)(q-1) of e to be your dUse inverse () c++arrow_forward
- Please written by computer source Given two strings x1…xn, y1…ym find the length of their longest common subsequence, that is, the largest k for which there exist indices i1<…<ik and j1<=<jk such that xi1…xik…=yj1=yjk. Show how to do this in time O(nm) and find its space complexity.arrow_forwardProve that P ⇒ (Q ∨ R) is equivalent to (P ∧ (∼ Q)) ⇒ R.arrow_forwardQuestion 3 The Extended Euclidean Algorithm allows us to efficiently compute inverses in Z∗n (and also in Galois fields). If gcd(n,a)=1, and EEA gives 1=sn+ta, then a−1≡tmodn. Use this method to compute the following inverses. Please give the answer in the natural range from 0 to n−1 where n is the modulus. Modulo 674, 103−1≡ Modulo 982, 483−1≡ Modulo 220, 149−1≡ Modulo 571, 443−1≡ Full explain this question and text typing work only thanksarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- C++ Programming: From Problem Analysis to Program...Computer ScienceISBN:9781337102087Author:D. S. MalikPublisher:Cengage Learning
C++ Programming: From Problem Analysis to Program...
Computer Science
ISBN:9781337102087
Author:D. S. Malik
Publisher:Cengage Learning