The knapsack problem is a problem in combinatorial optimization:
Given a set of items, each with a weight and a value, determine the number of each item included 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. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively.
Types of knapsack problem:
- 0-1 Knapsack Problem
- for recursive solution time Complexity - O(2^n)
- using dynamic programming time complexity is O(nW) where "W" is capacity and "n" in no. of the item
- Unbounded Knapsack (Repetition of items allowed) Θ((W+1)*N)
- fractional knapsack problem O(nlogn)
Given knapsack
variables used
wt = [10,40,20,30]
val = [60,40,100,120]
capacity = 50
0-1 knapsack problem (Unbounded Knapsack (Repetition of items not allowed)
### 0-1 Knapsack Problem
## recursive solution time Complexity - O(2^n)
def zeroOneKnapsackRec(index,wt,val,capacity):
# base condition
if capacity == 0 or index == 0:
return 0
elif wt[index-1] > capacity:
return zeroOneKnapsackRec(index-1,wt,val,capacity)
else:
# if we take or leave
return max(val[index-1]
+ zeroOneKnapsackRec(index-1,wt,val,capacity-wt[index-1]), # taking element
zeroOneKnapsackRec(index-1,wt,val,capacity) # leaving element
)
zeroOneKnapsackRec(len(wt),wt,val,capacity)
Output: 220
now using dp on a recursive solution
Type of dynamic programming approach
- top-down approach( Memoization Technique (an extension of recursive approach) )
- bottom-up approach
# Memoization Technique ...
# Time Complexity: O(number of items*capacity).
# we have taken capacity and weigth because both variables are changing
dp = [[None for i in range(capacity + 1)] for j in range(len(wt) + 1)] ## array to store recursive call or for Memoization
def zeroOneKnapsackMemoization(index,wt,val,capacity):
# base condition
if capacity == 0 or index == 0:
return 0
if dp[index][capacity] is not None:
return dp[index][capacity]
else:
if wt[index-1] > capacity:
dp[index][capacity] = zeroOneKnapsackRec(index-1,wt,val,capacity)
return dp[index][capacity]
else:
dp[index][capacity] = max(val[index-1] \
+ zeroOneKnapsackRec(index-1,wt,val,capacity-wt[index-1]) ,\
zeroOneKnapsackRec(index-1,wt,val,capacity-wt[index-1]))
return dp[index][capacity]
zeroOneKnapsackMemoization(len(wt),wt,val,capacity)
Output: 220
def topDownKnapsack(wt,val,capacity):
dp = [[0 for i in range(capacity+1)] for j in range(len(wt)+1)]
for i in range(len(wt)+1):
for j in range(capacity+1):
if i == 0 or j == 0:
dp[i][j] = 0
elif wt[i-1] <= j:
dp[i][j] = max(
val[i - 1] + dp[i - 1][j - wt[i - 1]],
dp[i - 1][j])
else:
dp[i][j] = dp[i - 1][j]
return dp
print("ans :",topDownKnapsack(wt,val,capacity)[len(wt)][capacity])
# to see dp table
from pandas import *
matrix = topDownKnapsack(wt,val,capacity)
df = DataFrame(matrix)
print(df)
Output: ans : 220
Table formed :
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
2 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 100
3 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 100 100 100 100 100 100 100 100 100 100 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160
4 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 100 100 100 100 100 100 100 100 100 100 160 160 160 160 160 160 160 160 160 160 180 180 180 180 180 180 180 180 180 180 220
Unbounded Knapsack (Repetition of items allowed)
def recursiveUnboundedKnapsack(val, wt, W, n):
"""
W = capacity,
n = index
"""
if n == 0 or W == 0:
return 0
elif wt[n - 1] > W:
return recursiveUnboundedKnapsack(val, wt, W, n-1)
else:
return max(
val[n - 1] + recursiveUnboundedKnapsack(val, wt, W - wt[n - 1], n),
recursiveUnboundedKnapsack(val, wt, W, n-1),
)
print(recursiveUnboundedKnapsack(val, wt, capacity, len(wt)))
Output: 300
def unboundedKnapsackDp(val, wt, W, n):
"""
W = capacity,
n = index
"""
dp = [[0 for _ in range(W + 1)] for _ in range(n + 1)]
for i in range(n + 1):
for j in range(W + 1):
if i == 0 or j == 0:
dp[i][j] = 0
elif wt[i - 1] > j:
dp[i][j] = dp[i - 1][j]
else:
dp[i][j] = max(val[i - 1] + dp[i][j - wt[i - 1]], dp[i - 1][j])
return dp
print("ans :",unboundedKnapsackDp(val, wt, capacity, len(wt))[len(wt)][capacity])
# to see dp table
from pandas import *
matrix = unboundedKnapsackDp(val, wt, capacity, len(wt))
df = DataFrame(matrix)
print(df)
Output: ans : 300
Table formed :
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 120 120 120 120 120 120 120 120 120 120 180 180 180 180 180 180 180 180 180 180 240 240 240 240 240 240 240 240 240 240 300
2 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 120 120 120 120 120 120 120 120 120 120 180 180 180 180 180 180 180 180 180 180 240 240 240 240 240 240 240 240 240 240 300
3 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 120 120 120 120 120 120 120 120 120 120 180 180 180 180 180 180 180 180 180 180 240 240 240 240 240 240 240 240 240 240 300
4 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 60 60 120 120 120 120 120 120 120 120 120 120 180 180 180 180 180 180 180 180 180 180 240 240 240 240 240 240 240 240 240 240 300
## fractional knapsack problem (greedy approach)
class item(object):
""" class to store item """
def __init__(self,wt,val,index):
self.wt = wt
self.val = val
self.index = index
self.cost = val // wt
def __lt__(self, other):
return self.cost < other.cost
class build_item_list:
""" class to build item list """
def __init__(self,wt_list,val_list):
self.wt_list = wt_list
self.val_list = val_list
def build(self):
item_list = []
for index in range(len(self.wt_list)):
item_list.append(item(self.wt_list[index],self.val_list[index],index))
return item_list
class FractionalKnapSack(object):
""" knapsack sol class """
def __init__(self,wt_list,val_list,capacity):
# Builder design pattern
self.item_list = build_item_list(wt_list,val_list).build()
self.capacity = capacity
self.totalValue = 0
self.totalValue = self.getMaxValue()
def getMaxValue(self):
""" function return max val a knapsack can hold"""
self.item_list.sort(reverse=True)
for i in self.item_list:
curWt = int(i.wt)
curVal = int(i.val)
if self.capacity - curWt >= 0:
self.capacity -= curWt
self.totalValue += curVal
else:
fraction = self.capacity / curWt
self.totalValue += curVal * fraction
self.capacity = int(self.capacity - (curWt * fraction))
break
return self.totalValue
def __repr__(self):
return f" Total value of knapsack is {self.totalValue}"
def __str__(self):
return f" Total value of knapsack is {self.totalValue}"
print(FractionalKnapSack(wt,val,capacity))
Output: Total value of knapsack is 240.0
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