Dynamic programming (Dynamic Programming) is an optimization method for solving multi-stage decision-making problems. It decomposes the problem into several overlapping sub-problems, and uses the solutions of the sub-problems to construct the solution of the original problem.
A dynamic programming algorithm usually includes the following steps:
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Define the state: Divide the original problem into several sub-problems, and determine the state variables that need to be stored. State variables are the key factors to describe the problem, and their values can represent certain characteristics or properties of the problem.
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Determine the state transition equation: By observing the characteristics and constraints of the problem, find out the recursive relationship between different states, that is, determine the transition relationship between sub-problems. This equation describes how to move from one state to the next, and is usually obtained by optimal choice or some optimality criterion.
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Initialize boundary conditions: For the initial subproblem or base case, initial values or boundary conditions need to be determined. These conditions serve as a starting point in the problem-solving process to ensure that the algorithm proceeds correctly.
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Recursive solution: According to the state transition equation, the solutions of all sub-problems are calculated by recursion. Usually, a bottom-up method is adopted to calculate small-scale sub-problems first, and gradually expand the scale until the original problem is solved.
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Calculate the optimal solution: According to the solutions of the sub-problems that have been calculated, select the optimal solution in a certain way, and construct the optimal solution of the original problem.
The core idea of the dynamic programming algorithm is to use the solutions of the sub-problems that have been solved to avoid repeated calculations, thereby improving the efficiency of the algorithm. It is suitable for those problems with overlapping sub-problems and optimal sub-structure properties, which can greatly simplify the problem-solving process.
It should be noted that the dynamic programming algorithm may need to reasonably divide and store the state space in practical applications, as well as to make a trade-off between time and space complexity. In addition, dynamic programming algorithms usually require the problem to satisfy no aftereffect, that is, the solution of the subproblem will not change with the change of other parts of the problem.
I hope the above introduction will help you understand the principle of dynamic programming algorithm.