Call a sequence X[1 · · n] of numbers oscillating if X[i] < X[i + 1] for all even i, and X[i] > X[i + 1] for all odd i. Describe an efficient algorithm to compute the length of the longest oscillating subsequence of an arbitrary array A of integers.

Let P(n) be the statement "the number 2^2n -1 is divisible by 3". Use mathematicalinduction to show that P(n) is true for all n >0

Describe an algorithm in pseudocode for finding the locations of the largest and smallest integers in a finite list of distinct integers.

The algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm 1 Euclid1(a,b)Require: a, b > 0Ensure: a = GCD(a, b) while a ̸= b do if a > b then a ← a − b else b ← b − a end if end whilereturn a We want to prove the correctness of this algorithm using the loop invariant technique. a. In a first attempt to prove the correctness of this algorithm, we propose the following loop invariant property: "GCD(a, b) is a factor of a and b". Explain why this property willnot help you as is. b. Use your previous observation to propose another loop invariant property that willhelp you to prove the correctness of the version of Euclid’s algorithm presented above. c. Show that your loop invariant property is true before executing the While loop forthe first time (i.e., initialization property)

We can sort n integers between 0 and 2n in O(n) time. True False Given a sorted array of n integers, we can find the median of them in O(1) time. True False If ƒ = 0(g), then g = 0(f). False True

Fermat's "Little" Theorem states that whenever n is prime and a is an integer, a^n−1≡1modn Then 281825≡ 263 mod827. c) If a=146 and n=331, then efficiently compute 146332≡ mod331Use The Extended Euclidean Algorithm to compute281^1≡362 mod827. Then 281^825≡???mod827. c) If a=146 and n=331, then efficiently compute146^332≡ ???mod331

= Trace the Winnow algorithm with 3: 1 for the following input. Suppose the domain is vectors of length n = 6 over {V1, V2, V3, V4, V5, V6} and the actual labels are with respect to formula v1/v3/v6; for example, the true label of (1, 0, 1, 0, 1, 1) is '1' and that of 1, 1, 1, 1, 1,0 is '0'. The input vectors are: x1 = (1, 0, 1, 1, 0, 1) x2 = (1,0,1,1,0,0) x3 = x4= X5 x6 x7 x8 = = = = x9 = (0, 1, 0, 1, 0, 1) 1, 1, 1, 0, 0, 1) (1, (0, 1, 1, 1, 1,0) (1, 1, 1, 1, 1, 1) (0, 1, 0, 1, 1, 1) (1, 1, 0, 1, 1, 1) (1, 0, 1, 0, 0, 1) You need to follow the same assumptions as in the example in Slide 4 of Module 11. Show your work and specify whether it is possible that the algorithm makes more mistakes after processing the above vectors; justify your answer.

Question 1. The input is an array a[1],. ,..., a[n] of size n containing natural numbers between 1 and n. Describe an O(n) algorithm that determines if the array contains 2 consecutive integers.

Given a Sorted Array of integers containing duplicates. Find the frequency of every unique element present in the array. Frequency is defined as the number of occurrence of any element in the array. Solve in Java

For any input array of n 3-digit numbers, prove that Radix Sort is guaranteed to correctly sort these n numbers, with the algorithm running in O(n) time.

Write a program that, given an array a[] of Ndouble values, finds a closest pair : two values whose difference is no greater than thethe difference of any other pair (in absolute value). The running time of your programshould be linearithmic in the worst case

given an array with repeating elements each element repeats thrice except one.find that one in O(1) space and O(n) time complexity in machine independent language.

Let A[1..n] be an array of n integers that each is larger than 1. give an O(n lg n)-time algorithm that decides if there are two integers x,y in A such that x=y2 (multiplication takes O(1) time)