Longest Bitonic Sequence
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Given an array arr[0 … n-1] containing n positive integers, a subsequence of arr[] is called Bitonic if it is first increasing, then decreasing. Write a function that takes an array as argument and returns the length of the longest bitonic subsequence. A sequence, sorted in increasing order is considered Bitonic with the decreasing part as empty. Similarly, decreasing order sequence is considered Bitonic with the increasing part as empty.
The first part (where the value of the items increases) of the Bitonic Subsequence is nothing but similar to finding Longest Increasing Subsequence.
Similarly, the second part, where the value of the items decreases, is similar to finding Longest Decreasing Subsequence. Longest Decreasing Subsequence is similar to Longest Increasing Subsequence, but instead of the values of the items increasing, in case of Longest Decreasing Subsequence the value of the items decreases from left to right.
Lets nums[] be the given array:
Create longestIncreasingSubsequence[] array based off of nums[i] from left to right where longestIncreasingSubsequence[i] gives length of longest increasing subsequence ending with nums[i], including.
Create longestDecreasingSubsequence[] array based off of nums[i] from right to left where longestDecreasingSubsequence[i] gives length of longest decreasing subsequence starting with nums[i].
For all i from 0 to (n - 1), where n = length of nums[] array, return max(longestIncreasingSubsequence[i] + longestDecreasingSubsequence[i] - 1
The '-1' is because when we find the longest increasing and decreasing subsequences, the eventual peak of the mountain will be shared by both which needs to be subtracted.
This problem is the same as calculating the bitonic sequence with an added condition that a strictly increasing or strictly decreasing sequence is not considered valid. (It has to be increasing and then decreasing)