Optimal Substructure
Optimal Substructure is a fundamental concept in dynamic programming. It plays a crucial role in solving complex problems efficiently.
In dynamic programming, the optimal substructure property states that the optimal solution to a problem can be constructed from the optimal solutions of its subproblems. This allows us to break down a complex problem into smaller overlapping subproblems and solve each subproblem optimally. By combining the solutions to the subproblems, we can obtain the optimal solution to the original problem.
To illustrate this concept, let's consider the problem of finding the shortest path in a graph. The optimal substructure property tells us that the shortest path from a starting node to an ending node can be obtained by finding the shortest path from the starting node to each neighbor and selecting the minimum path among those options. By applying this approach recursively, we can find the shortest path from the starting node to the ending node.
Here's an example of a function that finds the shortest path in a graph using dynamic programming:
1// replace with ts logic relevant to content
2function shortestPath(graph, start, end) {
3 const dp = new Array(graph.length).fill(Infinity);
4 dp[start] = 0;
5
6 for (let i = 0; i < graph.length; i++) {
7 for (let j = 0; j < graph[i].length; j++) {
8 if (dp[i] + graph[i][j] < dp[j]) {
9 dp[j] = dp[i] + graph[i][j];
10 }
11 }
12 }
13
14 return dp[end];
15}
16
17const graph = [
18 [0, 4, 1, Infinity],
19 [4, 0, 2, 3],
20 [1, 2, 0, 5],
21 [Infinity, 3, 5, 0]
22];
23
24console.log(shortestPath(graph, 0, 3)); // Output: 5
25console.log(shortestPath(graph, 1, 2)); // Output: 2xxxxxxxxxxconsole.log(shortestPath(graph, 1, 2)); // Output: 2// Optimal Substructure// Optimal Substructure is a property often found in problems that can be solved using dynamic programming techniques. It refers to the idea that the optimal solution to a problem can be constructed from the optimal solutions of its subproblems.// In other words, if we break down a problem into smaller subproblems and solve each subproblem optimally, we can combine the solutions to the subproblems to obtain the optimal solution to the original problem.// This property is fundamental to dynamic programming and allows us to solve complex problems efficiently.// For example, let's consider the problem of finding the shortest path in a graph. The optimal substructure property states that the shortest path from a starting node to an ending node can be obtained by finding the shortest path from the starting node to each neighbor and selecting the minimum path among those options.// By applying this approach recursively, we can find the shortest path from the starting node to the ending node.// Here's an example of a function that finds the shortest path in a graph using dynamic programming:function shortestPath(graph, start, end) { const dp = new Array(graph.length).fill(Infinity); dp[start] = 0; for (let i = 0; i < graph.length; i++) { for (let j = 0; j < graph[i].length; j++) { if (dp[i] + graph[i][j] < dp[j]) { dp[j] = dp[i] + graph[i][j]; } } } return dp[end];}const graph = [ [0, 4, 1, Infinity], [4, 0, 2, 3],