Dynamic Programming Questions Medium
In the context of the Maximum Length of Pair Chain problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions of its subproblems.
The Maximum Length of Pair Chain problem involves finding the maximum number of pairs in a given set of pairs, where each pair (a, b) represents a relationship between two elements a and b. The goal is to find the longest chain of pairs such that each pair (a, b) in the chain satisfies the condition b < a.
To understand the optimal substructure, let's consider a set of pairs S = {(a1, b1), (a2, b2), ..., (an, bn)}. We can sort the pairs in ascending order of their second element (b) so that b1 ≤ b2 ≤ ... ≤ bn. Now, let's assume that the maximum length of the pair chain ending at pair (ai, bi) is denoted by L(i), where 1 ≤ i ≤ n.
To find the maximum length of the pair chain, we can iterate through the pairs in the sorted order and calculate L(i) for each pair. The optimal substructure property comes into play here. For each pair (ai, bi), we can calculate L(i) by considering all the previous pairs (aj, bj) where j < i and bj < ai. We can then choose the maximum L(j) and add 1 to it to obtain L(i). This is because if we include pair (ai, bi) in the chain, it should be preceded by a pair (aj, bj) where bj < ai, and the length of the chain ending at (ai, bi) would be 1 plus the length of the chain ending at (aj, bj).
By calculating L(i) for all pairs, we can determine the maximum length of the pair chain by finding the maximum value among all L(i). This approach utilizes the optimal substructure property, as the optimal solution to the problem is built upon the optimal solutions of its subproblems (i.e., the maximum length of the pair chain ending at each pair).
In summary, the concept of optimal substructure in the context of the Maximum Length of Pair Chain problem implies that the optimal solution to the problem can be obtained by considering the optimal solutions of its subproblems, which involves finding the maximum length of the pair chain ending at each pair.