Given a strictly increasing array A of positive integers where,
Input: A = [1, 3, 7, 11, 12, 14, 18]
Explanation: The longest subsequence that is Fibonacci-like: [1, 11, 12]
Other possible subsequences are [3, 11, 14] or [7, 11, 18].
Input: A = [1, 2, 3, 4, 5, 6, 7, 8]
Explanation: The longest subsequence that is Fibonacci-like: [1, 2, 3, 5, 8].
A Fibonacci-like sequence is such that it has each two adjacent terms that determines the next expected term. For example, with 1, 1, we expect that the sequence must continue 2, 3, 5, 8, 13, … and so on.
We will use Set or Map to determine quickly whether the next term of Fibonacci sequence is present in the array A or not. Because of the exponential growth of these terms, there will be not more than log(M) searches to get next element on each iteration.
For each starting pair A[i], A[j], we maintain the next expected value y = A[i] + A[j] and the previously seen largest value x = A[j]. If y is in the array, then we can then update these values (x, y) -> (y, x+y) otherwise we stop immediately.
Below is the implementation of above approach:
Time Complexity: O(N2*log(M)), where N is the length of array and M is max(A).
- Length of longest strict bitonic subsequence
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- Length of the longest increasing subsequence such that no two adjacent elements are coprime
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- Maximum length subsequence such that adjacent elements in the subsequence have a common factor
- Longest subsequence such that every element in the subsequence is formed by multiplying previous element with a prime
- Longest subsequence with a given AND value | O(N)
- Longest Zig-Zag Subsequence
- Longest subsequence with a given AND value
- Longest subsequence with no 0 after 1
- Longest subsequence whose average is less than K
- Longest Increasing Subsequence using BIT
- Longest Consecutive Subsequence
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