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Dynamic programming is a computational technique that is used to solve complex optimization problems by breaking them down into smaller subproblems. It is based on the principle of optimality, which states that the optimal solution to a problem can be found by solving its subproblems and combining their solutions.
One of the classic examples of dynamic programming is the problem of finding the shortest path between two points in a network. In this problem, we are given a network of roads connecting a number of cities. The goal is to find the shortest path between two specific cities.
To solve this problem using dynamic programming, we first identify the subproblems. In this case, the subproblems are the shortest paths between pairs of cities. We then solve these subproblems, storing the results in a tableau or memo. Once all of the subproblems have been solved, we can use the stored results to quickly find the shortest path between the two specific cities.
Here is an example of how we can use dynamic programming to solve the shortest path problem. Suppose we are given the following network of roads:
To find the shortest path between cities A and C, we first need to identify the subproblems. In this case, the subproblems are the shortest paths between pairs of cities. There are five pairs of cities in this network: (A,B), (A,D), (A,E), (B,C), and (D,E).
We can then solve each of these subproblems, storing the results in a tableau. Here is what the tableau would look like after solving all of the subproblems:
The entries in the tableau represent the shortest distances between pairs of cities. For example, the entry in the second row and third column (highlighted in bold) indicates that the shortest distance between cities B and C is 2.
Once we have solved all of the subproblems and filled in the tableau, we can use the stored results to quickly find the shortest path between the two specific cities. In this case, we want to find the shortest path between cities A and C. We can do this by following the entries in the tableau, starting at city A and moving to the neighboring city with the smallest distance. In this case, the smallest distance is 1, and it leads us to city B. We then move to the neighboring city with the smallest distance from city B, which is 2 and leads us to city C.
Thus, the shortest path between cities A and C is A-B-C, and the total distance is 3 (1 + 2). We can verify that this is indeed the shortest path by checking the entries in the tableau.
In summary, dynamic programming is a powerful technique for solving complex optimization problems by breaking them down into smaller subproblems. By using the principle of optimality, we can avoid solving the same subproblem multiple times and quickly find the optimal solution to the problem. The shortest path problem is just one example of how dynamic programming can be used to solve complex problems efficiently.