Given an unsorted array of integers, write a function in Python to find the length of the longest increasing subsequence (LIS) in the array. For example, given the array [10, 9, 2, 5, 3,7, 101, 18], the LIS is [2, 3, 7, 101], which has a length of 4.Your solution should have a time complexity of O(n log n), where n is the length of the input array.Here's some code to get you started:def longest increasing_subsequence(arr):# TODO: implement functionpass# example usagearr = [10, 9, 2, 5, 3, 7, 101, 18]print(longest_increasing_subsequence(arr)) # should print 4

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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.
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In Python: Code two versions of Quicksort algorithms, one using the Lomuto method and one using the Hoare method. Clearly document your code so that it is clear how and why your algorithms are correct. For each of the two algorithms the output must be the following three items: the total numberof comparisons, the total number of swaps, and the total running time. Demonstrate your two Quicksort algorithms on the input array A= [2, 8, 7, 1, 3, 5, 6, 4].
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