dynamic programming knapsack problem python

the knapsack problem is a classic optimization problem where given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is maximized. dynamic programming is a popular technique to solve this problem efficiently.
in this tutorial, we'll walk through how to solve the knapsack problem using dynamic programming in python.
given a set of items, each with a weight and a value, and a knapsack with a maximum weight capacity, we need to determine the maximum value that can be obtained by selecting a subset of the items to put into the knapsack, without exceeding the maximum weight capacity.
dynamic programming is an algorithmic technique for solving optimization problems by breaking them down into simpler subproblems and solving each subproblem only once, storing the solutions to avoid redundant computations. the knapsack problem exhibits optimal substructure, making it suitable for dynamic programming.
identify the subproblems: in the knapsack problem, the subproblems involve determining the maximum value that can be obtained with a subset of items and a specific weight capacity.
formulate recurrence relations: define a recursive formula to express the solution of a subproblem in terms of solutions to smaller subproblems.
create a memoization table (or array): store the solutions to subproblems in a data structure (usually a table or array) to avoid redundant computations.
fill the memoization table bottom-up: compute the solutions to subproblems in a bottom-up manner, filling the table iteratively.
retrieve the final solution: once the table is filled, the solution to the original problem can be found by examining the entry corresponding to the maximum weight capacity of the knapsack.
let's implement the knapsack problem using dynamic programming in python.
this python function knapsack() takes three parameters:
dynamic programming off ...

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