0/1 DYNAMIC PROGRAMMING KNAPSACK PROBLEM

0/1 DYNAMIC PROGRAMMING KNAPSACK PROBLEM

• 2.
  DESIGN TECHNIQUE Example /Problem Algorithm Design Technique Dynamic Programing 0/1 Dynamic Programming Knapsack Problem
• 3.
  Dynamic Programming Technique  DynamicProgramming is a technique in computer programming that helps to efficiently solve a class of problems that have overlapping subproblems and optimal substructure property.  If any problem can be divided into subproblems, which in turn are divided into smaller subproblems, and if there are overlapping among these subproblems, then the solutions to these subproblems can be saved for future reference. In this way, efficiency of the CPU can be enhanced. This method of solving a solution is referred to as dynamic programming.
• 4.
  0/1 DP KNAPSACKPROBLEM • Knapsack is basically means bag. A bag of given capacity. We want to pack n items in your luggage. • Input: Knapsack of capacity List (Array) of weight and their corresponding value. • Output: To maximize profit and minimize weight in capacity. The knapsack problem where we have to pack the knapsack with maximum value in such a manner that the total weight of the items should not be greater than the capacity of the knapsack.
• 5.
  0/1 DP KNAPSACKPROBLEM Weights • Items – Profit Bag = Weight = ? Kg. W 1) 0/1 Problem 0 or 1 Suppose, 2 Kg 1Kg 2kg DP Method 2) Fractional Problem Suppose, 3 Kg 2 Kg 1 Kg Greedy Method
• 6.
  Example of 0/1DP Knapsack Problem • Consider the following instance of knapsack problem N = 4 , Weights = { 3, 4, 6, 5} W = 8 , Profits (Values) = { 2, 3, 1, 4} Find out the optimal solution for the given instance. Given: Weights = { 3, 4, 6, 5}, Profits (Values) = { 2, 3, 1, 4}, W = 8 , N = 4 .
• 7.
  Solution: N = 4, WEIGHTS = { 3, 4, 6, 5} W = 8 , PROFITS (VALUES) = { 2, 3, 1, 4} W I Pi Wi 2 3 3 4 4 5 1 6 0 1 2 3 4 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 3 4 4 5 5 5 6 5 6 M
• 8.
  Algorithm of KnapsackProblem Dynamic-0-1-knapsack (v, w, n, W) { for w = 0 to W do c[0, w] = 0 for i = 1 to n do c[i, 0] = 0 for w = 1 to W do if wi ≤ w then if vi + c[i-1, w-wi] then c[i, w] = vi + c[i-1, w-wi] else c[i, w] = c[i-1, w] else c[i, w] = c[i-1, w] }
• 9.
  Python Program for0/1 DP Knapsack Problem
• 10.
  Time Complexity of0/1 DP Knapsack Problem Complexity Analysis Time Complexity: O(N*W). where ‘N’ is the number of weight element and ‘W’ is capacity. • As for every weight element we traverse through all weight capacities 1<=w<=W. • Auxiliary Space: O(N*W). The use of 2-D array of size ‘N*W’.
• 11.
  Thank You