DynamischeProgrammierungsAnsatz

Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is a technique used in computer science and mathematics to optimize solutions by remembering the results of subproblems and avoiding recomputation. The core idea is to solve each subproblem only once and store its solution, typically in a table or array. When the same subproblem is encountered again, its stored solution is retrieved instead of being recalculated. This approach significantly reduces the time complexity of algorithms, especially for problems exhibiting overlapping subproblems and optimal substructure.

A problem exhibits optimal substructure if an optimal solution to the problem contains optimal solutions to its subproblems. Overlapping subproblems means that a recursive approach to solving the problem would repeatedly solve the same subproblems. Dynamic programming tackles these characteristics by using either a top-down approach with memoization or a bottom-up approach with tabulation. Memoization involves storing the results of function calls and returning the cached result when the same inputs occur again. Tabulation, on the other hand, systematically builds up solutions to subproblems from the smallest ones to the larger ones, filling a table in a specific order. Famous examples of problems solved efficiently using dynamic programming include the Fibonacci sequence, the shortest path problem, and the knapsack problem.