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Sum of Good Subsequences Approach to the Problem We are tasked with finding the sum of all possible good subsequences in a given array nums where a good subsequence satisfies the condition that the absolute difference between any two consecutive elements is exactly 1. Here’s how we can systematically solve this problem.
Understand the Problem
Subsequences: A subsequence is a sequence derived from an array by deleting some or no elements without changing the order of the remaining elements. Example: [1, 2, 1] has subsequences: [1], [2], [1], [1, 2], [2, 1], [1, 2, 1]. Good Subsequences: A subsequence is good if the absolute difference between consecutive elements is exactly 1. Goal: Compute the sum of all good subsequences modulo 10⁹+7. Brute-Force Approach Generating All the Subsequences: Use recursion or bitmasking to generate all possible subsequences of the array. Check if each subsequence is good by ensuring the condition ∣a[i]−a[i−1]∣ = 1 is satisfied for all consecutive elements. Sum up all elements of the good subsequences. Time Complexity: Generating all subsequences takes O(2^n), where n is the length of the array. Verifying each subsequence for the good property takes O(n) in the worst case. Overall: O(n⋅2^n). This is infeasible for large n (up to 10⁵). Optimal Approach To handle the constraints efficiently, we use a dynamic programming approach with optimization based on frequencies of numbers in the array. Algorithm We Use:-
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