Dynamic Programming Questions Medium
The time complexity of the Dynamic Programming approach for solving the Longest Increasing Subsequence in an Array problem is O(n^2), where n is the length of the input array.
In this approach, we use a dynamic programming table to store the lengths of the longest increasing subsequences ending at each index of the array. We initialize the table with all values set to 1, as the minimum length of any subsequence is 1.
Then, we iterate through the array from left to right, for each index i, we compare the value at index i with all the previous indices j (0 <= j < i). If the value at index i is greater than the value at index j, we update the length of the longest increasing subsequence ending at index i as the maximum of its current length and the length of the longest increasing subsequence ending at index j plus 1.
Finally, we find the maximum value in the dynamic programming table, which represents the length of the longest increasing subsequence in the array.
The time complexity of this approach is O(n^2) because we have nested loops. The outer loop iterates through each index of the array, and the inner loop iterates through all the previous indices. Therefore, the total number of iterations is approximately n*(n+1)/2, which simplifies to O(n^2).
Note that there is also an optimized version of the Longest Increasing Subsequence problem that can be solved in O(n log n) time complexity using binary search and dynamic programming.