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 4.6, Problem 3E
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To show that the case 3 of the master theorem is overstated.
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Justify (argue, prove, ...) that the following formula is valid
Vx A(x) V Vx B(x) → Vx (A(x) V B(x))
3. Using the pumping lemma and contradiction, prove that L = {ab n
1. Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71).
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Chapter 4 Solutions
Introduction to Algorithms
Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Prob. 5E
Ch. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Prob. 9ECh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Prob. 3ECh. 4 - Prob. 1PCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Prob. 6P
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- It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.arrow_forward1. Using the pumping lemma and contradiction, prove that L = (ab² n>=0} is not regular. 2. Using the pumping lemma and contradiction, prove that L = {a"bm n> m} is not regular. 3. Using the pumping lemma and contradiction, prove that L = (aºb narrow_forwardIf a, b, m ∈ Z+, ab mod m = [(a mod m)(b mod m)] mod m Show the equality of this expression.arrow_forwardFor f(a, b) = (a | b) | b (a) Simplify f(a, b). (b) Find DNF for f(a, b). (c) Is f(a, b) satisfiable?arrow_forwardIn each case below, show using the pumping lemma that the givenlanguage is not a CFL.e. L = {x ∈ {a, b, c}∗ | na(x) = max {nb(x), nc(x)}}arrow_forwardQuestion 2 : Consider the following functions: log2 x • x.log₂x (²9) ² X log₂ x • 2x • √x x² • (log₂ x)² (3) ² The aforementioned functions represent the number of operations carried out by a set of algo- rithms that solve the same computational problem in different ways. Your task it to order the functions from slowest to fastest asymptotic growth rate (i.e., sort the algorithms from best to worst in terms of runtime complexity). Explain your reasoning for each consecutively ordered function pair by using the limit method For example, given a set of three functions {f(n), g(n), h(n)}, if you claim that asymptotically f(n) < g(n) < h(n), then you must prove that limn→∞(n) = 0 and limn→∞ g(n) = 0.arrow_forwardConsider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?arrow_forwardProve that if M is any λ-NF A such that λ /∈ L(M), then there exists an ordinary NFA N, with exactly one final state, such that L(M) = L(N). (Hint: Make a λ-NFA with the desired properties. Then use the Theorem from this section, taking full advantage of the fine print.)arrow_forwardQuestion 4 satisfying gi = N(92), 92 = 2(93), .. Partition your list in equivalence classes such that functions f(n) and g(n) are in the same class if and only if f(n) = 0(g(n)). Rank the following functions by order of growth; that is, find an arrangement g1, 92, ... of the functions n2 n! (3/2)" n3 lg n lg(n!) 22" In In n 1 Inn VIgn en (n + 1)! nlgn 22"+1arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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