Dynamic programming - definition
Dynamic programming - definition
Dynamic programming is a method of designing algorithms that involves solving optimisation problems by decomposing them into smaller sub-problems, the results of which are stored and reused in further calculations. This approach eliminates the need to recalculate the same values multiple times, which significantly reduces computational complexity compared to recursive solutions without memorisation (so-called memoryless naive recursion).
The basis of dynamic programming is the existence of a structure of optimal sub-solutions, which means that the optimal solution of the main problem can be obtained by combining the solutions of its sub-problems. An additional condition is the occurrence of the so-called overlap of subproblems, i.e. their multiple occurrence in different stages of computation. The technique is based on storing the results of partial calculations in data structures such as arrays or dictionaries, thus avoiding redundant operations.
Dynamic programming comes in two main varieties: a top-down approach with memoisation of results (top-down with memoization) and a bottom-up approach with table building (bottom-up). Both versions aim to optimise the number of function calls and computational operations. This is particularly important in problems with exponential time complexity, where the use of this method achieves a solution in polynomial time.
Applications of dynamic programming cover a wide range of computer and mathematical issues, such as string analysis, scheduling, graph search, resource decomposition, data compression or routing. The method is also relevant in control theory, computational economics and molecular biology, where high-performance optimisation algorithms are required. Thanks to its versatility, it is one of the basic tools in the arsenal of computer algorithmics.
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