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 C.5, Problem 4E
Program Plan Intro
Toprove the given equationif 0< k < np, where 0 < p < 1 and q = 1 − p.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Suppose the following PDA P = ({q, r}, {0,1}, {Z0, X}, δ, q, Z0, ∅) is given:
0, Z0/XZ0 1, Z0/ϵ
0, X /XX 1, X /XX 1, X /X ϵ, X/ϵ
Convert P to a PDA P′ with L(P′) = N(P).
Show that p ↔ q is logically equivalent to (p ∧ q) ∨ (¬p ∧¬q).
Show that 100n 3 + 50n 2 + 75 is in 0 (20n 3) by finding a positive K that satisfiesthe equation (100n 3 + 50n2 + 75) / 20n 3 $ K.
Chapter C Solutions
Introduction to Algorithms
Ch. C.1 - Prob. 1ECh. C.1 - Prob. 2ECh. C.1 - Prob. 3ECh. C.1 - Prob. 4ECh. C.1 - Prob. 5ECh. C.1 - Prob. 6ECh. C.1 - Prob. 7ECh. C.1 - Prob. 8ECh. C.1 - Prob. 9ECh. C.1 - Prob. 10E
Ch. C.1 - Prob. 11ECh. C.1 - Prob. 12ECh. C.1 - Prob. 13ECh. C.1 - Prob. 14ECh. C.1 - Prob. 15ECh. C.2 - Prob. 1ECh. C.2 - Prob. 2ECh. C.2 - Prob. 3ECh. C.2 - Prob. 4ECh. C.2 - Prob. 5ECh. C.2 - Prob. 6ECh. C.2 - Prob. 7ECh. C.2 - Prob. 8ECh. C.2 - Prob. 9ECh. C.2 - Prob. 10ECh. C.3 - Prob. 1ECh. C.3 - Prob. 2ECh. C.3 - Prob. 3ECh. C.3 - Prob. 4ECh. C.3 - Prob. 5ECh. C.3 - Prob. 6ECh. C.3 - Prob. 7ECh. C.3 - Prob. 8ECh. C.3 - Prob. 9ECh. C.3 - Prob. 10ECh. C.4 - Prob. 1ECh. C.4 - Prob. 2ECh. C.4 - Prob. 3ECh. C.4 - Prob. 4ECh. C.4 - Prob. 5ECh. C.4 - Prob. 6ECh. C.4 - Prob. 7ECh. C.4 - Prob. 8ECh. C.4 - Prob. 9ECh. C.5 - Prob. 1ECh. C.5 - Prob. 2ECh. C.5 - Prob. 3ECh. C.5 - Prob. 4ECh. C.5 - Prob. 5ECh. C.5 - Prob. 6ECh. C.5 - Prob. 7ECh. C - Prob. 1P
Knowledge Booster
Similar questions
- Consider f(n) = 3n^2+2n-1, mathematically show that f(n) is O(n²), (n²), and O(n²).arrow_forwardConsider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…arrow_forwardConsider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.arrow_forward
- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.arrow_forwardA country has coins with k denominations 1 = d1 < d2 < ... < dk, and you want to make change for n cents using the smallest number of coins. For example, in the United States we have d1 = 1, d2 = 5, d3 = 10, d4 = 25, and the change for 37 cents with the smallest number of coins is 1 quarter, 1 dime, and 2 pennies, which are a total of 4 coins. To solve for the general case (change for n cents with k denominations d1 ... dk), we refer to dynamic programming to design an algorithm. 1. We will come up with sub-problems and recursive relationship for you. Let be the minimum number of coins needed to make change for n cents, then we have: Explain why the above recursive relationship is correct. [Formal proof is not required] 2. Use the relationship above to design a dynamic programming algorithm, where the inputs include the k denominations d1 ... dk and the number of cents n to make changes for, and the output is the minimum number of coins needed to make change for n. Provide…arrow_forwardProve that P ⇒ (Q ∨ R) is equivalent to (P ∧ (∼ Q)) ⇒ R.arrow_forward
- Subject : calculas Show that: ¬q 1) p→¬q 2) (p∧r)∨s 3) s→(t∨u) 4) ¬t∧¬u where ¬ is denied.arrow_forwardLet P2(x) be the least squares interpolating polynomial for f(x) := sin(πx) on the interval [0,1] (with weight function w(x) = 1). Determine nodes (x0,x1,x2) for the second-order Lagrange interpolating polynomial Pˆ2(x) so that P2 = Pˆ2. You are welcome to proceed theoretically or numerically using Python.arrow_forwardGiven 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!arrow_forward
- Prove that (p →(q →r)) → ((p →q) →(p →r)) ≡ T for all propositions p, q, r.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_forwardDe ning T(0) = a; T(1) = b; T(2) = c, and T(n + 3) = (1=3)(T(n + 2) + T(n + 1) + T(n)) for n >= 0, solve for the limit T(n) as n -> ∞. Show your work.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole