Question

An array a consisting of n elements - Integer k For each (1≤i≤n). Perform the following operation exactly one time: - Replace ai​ by ai​+x, where x∈[−k,k] which denotes x should lie in the range of −k and k, both inclusive. Determine the maximum length of the subsequence of array a, such that all numbers in that subsequence are equal after applying the given operation. Notes - A subsequence of array a is an array that can be derived from array a by removing some elements from it.

Answer 1

The maximum length of the subsequence is found by sorting the array and identifying the largest group of numbers that can be equalized using the operation where each element can be adjusted by a value in the range [-k, k].

Step-by-step explanation:

The maximum length of the subsequence of an array a, where each element ai can be replaced by ai + x, for x in the range [-k, k], refers to the largest subset of elements from the original array that can be made identical through the specified operation. To determine this length, one must look for the largest group of numbers within the array that can be equalized using the allowed changes to their values.

The technique involves initially sorting the array. After sorting, iterate through the array, and for each element ai, find the element aj (for i ≤ j) such that ai + k ≥ aj - k (the maximum value the i-th element can be increased to is at least the minimum value the j-th element can be decreased to). This pairing of elements indicates that the subsequence from ai to aj can be transformed into a sequence of equal numbers.

The maximum length of the subsequence will be the largest number of elements found between any such ai and aj. By conducting this operation throughout the array, one can find the longest subsequence of equal numbers post-operation.