Knapsack Problem

Examples

• 1

  Knapsack Problem Basics
  1

  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
  

  Given:

  1. Values(array v)
  2. Weights(array w)
  3. Number of distinct items(n)
  4. Capacity(W)

  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.

  edited Jan 8 at 2:22

  

  

  

  

  
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• 0

  Solution Implemented in C#
  0

  public class KnapsackProblem
  {
      private static int Knapsack(int w, int[] weight, int[] value, int n)
      {
          int i;
          int[,] k = new int[n + 1, w + 1];
          for (i = 0; i <= n; i++)
          {
              int b;
              for (b = 0; b <= w; b++)
              {
                  if (i==0 || b==0)
                  {
                      k[i, b] = 0;
                  }
                  else if (weight[i - 1] <= b)
                  {
                      k[i, b] = Math.Max(value[i - 1] + k[i - 1, b - weight[i - 1]], k[i - 1, b]);
                  }
                  else
                  {
                      k[i, b] = k[i - 1, b];
                  }
              }
          }
          return k[n, w];
      }
  
      public static int Main(int nItems, int[] weights, int[] values)
      {
          int n = values.Length;
          return Knapsack(nItems, weights, values, n);
      }
  }
  

  created Oct 12 '16 at 21:36

  
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