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 5.3, Problem 2E
Program Plan Intro
To decide whether PERMUTE-WITHOUT-IDENTITY
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Each decimal number has a base k same. The singular digits of a base k number are called k-its. How about we characterize the k-itwise XOR of two k-its an and b as (a+b)modk.
The k-itwise XOR of two base k numbers is equivalent to the new number framed by taking the k-itwise XOR of their relating k-its. The k-itwise XOR of two decimal numbers an and b is signified by a⊕kb and is equivalent to the decimal portrayal of the k-itwise XOR of the base k portrayals of an and b. All further numbers utilized in the assertion underneath are in decimal except if indicated. When k=2 (it is in every case valid in this form), the k-itwise XOR is as old as bitwise XOR.
You have hacked the criminal data set of Rockport Police Department (RPD), otherwise called the Rap Sheet. However, to get to it, you require a secret word. You don't have any acquaintance with it, yet you are very certain that it lies somewhere in the range of 0 and n−1…
Use two ways to count the number of r-combinations of[n]={1,2,…,n}that contains 1 or 2 or 3. First, separate cases with Case 1 counting all r-combinations of [n] that contains 1 , Case 2 counting all r-combination of [n] that contains 2 but not 1 , and Case 3 counting allrcombination of [n] that contains 3 but not 1 or 2 . Second, count all r-combinations of [n] not containing any of1,2,3and use the subtraction rule.
Prove the following argument is valid:
Væ [p(x) → (q(x) V r(x)]
Væ (r(x) → q(x)]
-3x q(x)
3x p(x)
..
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- Give regular expressions for the i (a) {w : w contains at most two 1s} (b) {w : w contains at least two 1s} (c) {w: every odd position in w is 1} (d) {w : w contains an even number of 0s, orarrow_forwardHW: Find Z{-n u(-n))}arrow_forward6 7 8 9 10 ======= 11 12 13 14 15 16 17 18 19 20 21 22 XO la b 1 m Periodicity U 50 33 97 D XI E F G H I J L M N Use linear congruential generator approach to find Xi, what is the periodicity? Rename cells C4, C5, C6 and C7 as X0, a, b and m, respectively. RUDIMENTS OF RANDOM NUMBER GENERATION (1 OF 2) LINEAR CONGRUENTIAL GENERATORS (BYLEHMER 1951 A sequence of integers defined by the following recursive formula X = (ax +b) MOD m, 1 = 0, 1.... which means divide (ax, + b) by m, let X... be the remainder of this division a multiplier b increment m the modulus x0 seed or initial point 0 Parrow_forward
- The Legendre Polynomials are a sequence of polynomials with applications in numerical analysis. They can be defined by the following recurrence relation: for any natural number n > 1. Po(x) = 1, P₁(x) = x, Pn(x) = − ((2n − 1)x Pn-1(x) — (n − 1) Pn-2(x)), n Write a function P(n,x) that returns the value of the nth Legendre polynomial evaluated at the point x. Hint: It may be helpful to define P(n,x) recursively.arrow_forwardWhat does it mean to say that f(n) is in O(g(n))?arrow_forwardFind a regular expression for the set {a"bm: n > 3,m is even}.arrow_forward
- Given the function T(n) = n3 + 20n + 5, show that T(n) is O(n3)arrow_forwardQ4// Simplify and then implement the Boolean function F(A,B,C,D) = N (1,2,3,5,6,7,9,11,13,15) using POS karnough map , draw the implementation. %3Darrow_forward7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).arrow_forward
- Solve following recurrence with iterative substitution: T(n)= 2T(n/2)+c *O(1)arrow_forwardUse substitution to show T(n) = 4T(n/4)+n is T(n) is both in O(n²) and O(n log n)- note for O(n?) will need to subtract off a lower-order term (e.g. n² is a lower-order term to n³) to make it work. %3D Explain why it can be in both.arrow_forwardConsider the following procedure with a parameter n, and n is a positive integer: int Q(int n) { if (n==1) else return 1; return Q (n-2) + 2*n ; Set up a recurrence relation for the number of multiplications performed by the procedure. M(n) = M( )+ for n>1 M(0) =arrow_forward
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