### Problem

Given a set of items, each with a weight and a value. Determine the maximum value of items to include in a collection so that the total weight is less than or equal to a given limit

### Example

Given 3 items with weights = {10, 20 , 30} and values = {60, 100, 120} respectively, knapsack weight capacity is 50. The maximum value of items to include in the knapsack is 220

### Dynamic programming approach

Have 2 options at each collecting step

• Including the `i` item if not exceeding the weight limit
• Excluding the `i` item if exceeding the weight limit

### Implementation

Fill up a cache matrix `int[][] cache = new int[N+1][W+1]` with

• 0 to `N` items as columns
• 0 to `W` weight-limit as rows
• each matrix cell `cache[i][w]` is the maximum value can be attained with weight less than or equal to `w` using `i` items
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• `i` is the current item
• `w` is the current weight limit
• `weights[i-1]` is weight of current item as `int[] weights` is 0 based index
• `values[i-1]` is value of current item as `int[] values` is 0 based index
• `cache[i][w]` is maximum value of current item as `int[][] cache` is 1 based index
• `weights[i-1] > w` weight of the current item is more than the weight limit
• `cache[i][w] = cache[i-1][w]` excludes the current item
• `cache[i-1][w-weights[i-1]] + values[i-1]` includes the current item