Dynamic programming algorithm

Dynamic programming is a commonly used algorithmic technique when it comes to solving optimization problems with overlapping subproblems. It decomposes the problem into a series of overlapping sub-problems, and uses recursive or iterative methods to solve these sub-problems, and finally obtains the optimal solution of the problem.

The core idea of dynamic programming is to decompose the original problem into smaller sub-problems, and construct the solution of the original problem by solving these sub-problems. When solving subproblems, dynamic programming saves the solutions of subproblems so that they can be reused when needed, thereby avoiding double computation.

General steps of dynamic programming

To implement the dynamic programming algorithm, you can follow the steps below:

Determine the state of the problem: First, the state of the problem needs to be determined, and these states should be able to uniquely represent the sub-problems of the problem. The state can be a combination of one or more variables, a number, an array, a matrix, etc., depending on the nature of the problem.

Define the state transition equation : According to the definition and nature of the problem, determine the transition relationship between the states of the problem, that is, how to transfer from one state to another. This equation is usually defined based on recurrence relations or optimal substructure properties.
Determine initial conditions : Determine the solution to the smallest subproblem, that is, the value of the initial state. These initial conditions are the boundary conditions of the problem and are used to start the recursive calculation.
Determine the calculation order : Determine the order in which to calculate the solution of the sub-problems, usually starting from the smallest sub-problem, and gradually calculate the larger sub-problems until the solution of the original problem is calculated. This order can be either top-down recursive or bottom-up iterative.
Calculate the optimal solution : Calculate the optimal solution of the original problem according to the state transition equation and the initial conditions. Calculations can be performed recursively or iteratively.
Construct the optimal solution : construct the optimal solution of the original problem based on the calculated optimal solution and the saved intermediate results. This step is usually performed by backtracking or tracking intermediate results.

It should be noted that the implementation of dynamic programming algorithm can use recursive or iterative way, depending on the nature of the problem and the requirement of computational efficiency. During implementation, data structures such as arrays, matrices, or hash tables can be used to store intermediate results so that they can be searched and used when needed.

Special DP - Backpack

The knapsack problem is a classic optimization problem that can be solved by a dynamic programming algorithm. In the knapsack problem, there is a knapsack and a set of items, each of which has its own weight and value. The goal is to select some items to put into the knapsack so that the total weight of the items put into the knapsack does not exceed the capacity of the knapsack while maximizing the total value of the items put into the knapsack.

Knapsack problems can be divided into two types: 0-1 knapsack problem and infinite knapsack problem.

0-1 knapsack problem

Each item can only be selected to be put in the backpack once or not put in. That is, item selection is a binary decision. In this case, the state of dynamic programming can be defined as "the maximum value of knapsack capacity j among the first i items". The state transition equation can be expressed as: dp[i][j] = max(dp[i-1][j], dp[i-1][jw[i]] + v[i]) Among them, dp[i ][j] represents the maximum value of the first i items when the knapsack capacity is j, w[i] represents the weight of the i-th item, and v[i] represents the value of the i-th item.

complete knapsack problem

Each item can be selected to be placed in the backpack multiple times, that is, the selection of the item is a non-negative integer. In this case, the state of dynamic programming can be defined as "the maximum value of knapsack capacity j among the first i items". The state transition equation can be expressed as: dp[i][j] = max(dp[i-1][j], dp[i][jw[i]] + v[i]) Among them, dp[i][ j] represents the maximum value of the first i items when the knapsack capacity is j, w[i] represents the weight of the i-th item, and v[i] represents the value of the i-th item.

The implementation steps of the dynamic programming algorithm are as follows:

Define the state of the problem: Determine the definition of the state, that is, the meaning and dimensions of the dp array.
Initialization: According to the definition of the problem, initialize the initial value of the dp array.
State transition: According to the state transition equation, use a loop to traverse the items and backpack capacity, and update the value of the dp array.
Return result: According to the definition of the problem, get the value of the optimal solution from the dp array.
Optional step: If you need to construct a specific combination of items for the optimal solution, you can use an additional data structure (such as a two-dimensional array or a hash table) to store the selected information, and then construct the optimal solution based on this information.

Through the above steps, the knapsack problem can be solved using the dynamic programming algorithm, and the optimal item selection scheme and total value can be obtained.

Summarize

To sum up, the key to realizing the dynamic programming algorithm is to determine the state of the problem and the state transition equation, and perform recursive or iterative calculations according to the calculation sequence, and finally obtain the optimal solution of the original problem.

DP - Dynamic Programming