Dynamic Programming Questions Medium
In the context of the Longest Increasing Subsequence Size problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.
In the Longest Increasing Subsequence Size problem, we are given a sequence of numbers and we need to find the length of the longest increasing subsequence within that sequence. An increasing subsequence is a sequence of numbers where each number is greater than the previous number.
To understand the concept of optimal substructure, let's consider an example. Suppose we have a sequence of numbers: 2, 4, 3, 5, 1, 7, 6. The longest increasing subsequence in this sequence is 2, 3, 5, 7, which has a length of 4.
Now, let's break down the problem into subproblems. We can start by considering the longest increasing subsequence ending at each position in the sequence. For example, the longest increasing subsequence ending at position 1 is just the number 2, which has a length of 1. Similarly, the longest increasing subsequence ending at position 2 is 2, 4, which has a length of 2.
To find the longest increasing subsequence ending at each position, we can use dynamic programming. We can maintain an array dp[] where dp[i] represents the length of the longest increasing subsequence ending at position i. We can initialize dp[i] to 1 for all positions.
Now, for each position i, we can iterate through all previous positions j (from 0 to i-1) and check if the number at position i is greater than the number at position j. If it is, then we can update dp[i] as dp[j] + 1, indicating that we can extend the longest increasing subsequence ending at position j by including the number at position i.
By finding the maximum value in the dp[] array, we can determine the length of the longest increasing subsequence in the given sequence.
The optimal substructure property comes into play when we update the dp[] array. At each position i, we consider all previous positions j and update dp[i] based on the optimal solution to the subproblem at position j. This ensures that the overall solution is optimal.
In summary, the concept of optimal substructure in the context of the Longest Increasing Subsequence Size problem means that the optimal solution to the problem can be constructed from optimal solutions to its subproblems. By breaking down the problem into smaller subproblems and using dynamic programming, we can efficiently find the length of the longest increasing subsequence in a given sequence.