0/1 DYNAMIC PROGRAMMING KNAPSACK PROBLEM

DESIGN TECHNIQUE
Example / Problem
Algorithm Design
Technique
Dynamic Programing
0/1 Dynamic Programming
Knapsack Problem

Dynamic Programming
Technique
 Dynamic Programming 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.

0/1 DP KNAPSACK PROBLEM
• 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.

0/1 DP KNAPSACK PROBLEM
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

Example of 0/1 DP 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 .

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

Algorithm of Knapsack Problem
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]
}

Python Program for 0/1 DP Knapsack Problem

Time Complexity of 0/1 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’.

0/1 DYNAMIC PROGRAMMING KNAPSACK PROBLEM

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