The Problem: Given a set of items where each item contains a weight and value, determine the number of each to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.
Pseudo code for Knapsack Problem
for j from 0 to W do: m[0, j] := 0 for i from 1 to n do: for j from 0 to W do: if w[i] > j then: m[i, j] := m[i-1, j] else: m[i, j] := max(m[i-1, j], m[i-1, j-w[i]] + v[i])
A simple implementation of the above pseudo code using Python:
def knapSack(W, wt, val, n): K = [[0 for x in range(W+1)] for x in range(n+1)] for i in range(n+1): for w in range(W+1): if i==0 or w==0: K[i][w] = 0 elif wt[i-1] <= w: K[i][w] = max(val[i-1] + K[i-1][w-wt[i-1]], K[i-1][w]) else: K[i][w] = K[i-1][w] return K[n][W] val = [60, 100, 120] wt = [10, 20, 30] W = 50 n = len(val) print(knapSack(W, wt, val, n))
Running the code: Save this in a file named knapSack.py
$ python knapSack.py 220
Time Complexity of the above code:
O(nW) where n is the number of items and W is the capacity of knapsack.