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6-type-knapsack-problem.py
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#!/usr/bin/python3
# https://www.geeksforgeeks.org/0-1-knapsack-problem-dp-10/
# https://www.youtube.com/watch?v=iBnWHZmlIyY
# https://leetcode.com/problems/partition-equal-subset-sum
"""
# Dynamic Programming
# Knapsack variation Problems
1. Subset Sum Problem ~> https://www.geeksforgeeks.org/subset-sum-problem-dp-25/
2. Equal Subset Sum Partition ~> https://www.youtube.com/watch?v=UmMh7xp07kY
3. Count of Subset Sum
4. Minimum Subset Sum Difference
5. Number of Subset by a given Difference
6. Target Sum DP
"""
from typing import List
class SubsetSum:
def __init__(self) -> None:
self.memoization_table = None
def set_memoization_table(self, row: int, col: int) -> None:
# row ~> n = 6
# col ~> total = 30
# n x total
self.memoization_table = [[None for _ in range(col + 1)] for _ in range(row + 1)]
# Method 1: Recursion.
def is_subset_sum(self, nums: List[int], n: int, total: int) -> bool:
# Base Condition
if total == 0:
return True
if n == 0:
return False
# Memoization
if self.memoization_table[n][total] is not None:
return self.memoization_table[n][total]
# Choice Diagram
if nums[n - 1] <= total:
include = self.is_subset_sum(nums, n - 1, total - nums[n - 1])
exclude = self.is_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = include or exclude
return self.memoization_table[n][total]
else:
exclude = self.is_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = exclude
return self.memoization_table[n][total]
# Method 2: Top Down
def is_subset_sum_2(self, nums: List[int], n: int, total: int) -> bool:
# Initialize with Base Condition
matrix = [[True if c == 0 else False for c in range(total + 1)] for _ in range(n + 1)]
r = 1
while r <= n:
c = 1
while c <= total:
if nums[r - 1] <= c:
include = matrix[r - 1][c - nums[r - 1]]
exclude = matrix[r - 1][c]
matrix[r][c] = include or exclude
else:
matrix[r][c] = matrix[r - 1][c]
c += 1
r += 1
return matrix[n][total]
class EqualSubsetSumPartition:
def __init__(self) -> None:
self.memoization_table = None
def init_memoization_table(self, row: int, col: int) -> None:
# row ~> n = 6
# col ~> total = 30
# n x total
self.memoization_table = [[None for _ in range(col + 1)] for _ in range(row + 1)]
# Method 1: Recursion
def is_subset_sum(self, nums: List[int], n: int, total: int) -> bool:
# Base Condition
if total == 0:
return True
if n == 0:
return False
# Memoization
if self.memoization_table[n][total] is not None:
return self.memoization_table[n][total]
if nums[n - 1] <= total:
include = self.is_subset_sum(nums, n - 1, total - nums[n - 1])
exclude = self.is_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = include or exclude
return self.memoization_table[n][total]
else:
exclude = self.is_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = exclude
return self.memoization_table[n][total]
def is_eq_partition(self, nums: List[int]) -> bool:
total = sum(nums)
if total % 2 != 0:
return False
n = len(nums)
total //= 2
self.init_memoization_table(n, total)
return self.is_subset_sum(nums, n, total)
# Method 2: Top Down
def can_partition(self, nums: List[int]) -> bool:
total = sum(nums)
# Base Condition
if total % 2 != 0:
return False
n = len(nums)
total //= 2
# Initialize Matrix
matrix = [[True if c == 0 else False for c in range(total + 1)] for _ in range(n + 1)]
# Choice Diagram
r = 1
while r <= n:
c = 1
while c <= total:
if nums[r - 1] <= c:
include = matrix[r - 1][c - nums[r - 1]]
exclude = matrix[r - 1][c]
matrix[r][c] = include or exclude
else:
matrix[r][c] = matrix[r - 1][c]
c += 1
r += 1
return matrix[n][total]
class CountSubsetSum:
def __init__(self) -> None:
self.memoization_table = None
def set_memoization_table(self, row: int, col: int) -> None:
# row ~> n & col ~> total
self.memoization_table = [[None for _ in range(col + 1)] for _ in range(row + 1)]
# Method 1: Recursion
def get_subset_sum_count(self, nums: List[int], n: int, total: int) -> int:
# Base Condition
if total == 0:
return 1
if n == 0:
return 0
# Memoization
if self.memoization_table[n][total] is not None:
return self.memoization_table[n][total]
# Choice Diagram
if nums[n - 1] <= total:
include = self.get_subset_sum_count(nums, n - 1, total - nums[n - 1])
exclude = self.get_subset_sum_count(nums, n - 1, total)
self.memoization_table[n][total] = include + exclude
return self.memoization_table[n][total]
else:
exclude = self.get_subset_sum_count(nums, n - 1, total)
self.memoization_table[n][total] = exclude
return self.memoization_table[n][total]
# Method 2: Top Down
def count_subset_sum(self, nums: List[int], n: int, total: int) -> int:
# Initialize with Base Condition
matrix = [[1 if c == 0 else 0 for c in range(total + 1)] for _ in range(n + 1)]
r = 1
while r <= n:
c = 1
while c <= total:
if nums[r - 1] <= c:
include = matrix[r - 1][c - nums[r - 1]]
exclude = matrix[r - 1][c]
matrix[r][c] = include + exclude
else:
exclude = matrix[r - 1][c]
matrix[r][c] = exclude
c += 1
r += 1
return matrix[n][total]
class MinimumSubsetSumDifference:
def subset_sum(self, nums: List[int], total: int) -> List[bool]:
n = len(nums)
matrix = [[True if c == 0 else False for c in range(total + 1)] for _ in range(n + 1)]
r = 1
while r <= n:
c = 1
while c <= total:
if nums[r - 1] <= c:
include = matrix[r - 1][c - nums[r - 1]]
exclude = matrix[r - 1][c]
matrix[r][c] = include or exclude
else:
exclude = matrix[r - 1][c]
matrix[r][c] = exclude
c += 1
r += 1
return matrix[-1] # return last row
# Method 2: Top Down
def min_difference(self, nums: List[int]) -> int:
"""
nums = [1, 2, 7]
1. Find sum of nums as a the upper boundary i.e. Min diff can't go beyond this range
_range = 10
"""
_range = sum(nums)
"""
2. Consider lower half of the Range to calculate last row of subset sum matrix
lower_half_subset = {1, 2, 3, 4, 5}
upper_half_subset = {6, 7, 8, 9, 10}
Arguments ~> nums = [1, 2, 7], total = 5
small_subset_vector = [True, True, True, True, False, False]
"""
small_subset_vector = self.subset_sum(nums, _range // 2)
"""
3. Find the last index of True
Apply formula (Range - 2 * index) to find the minimum difference
[
(0, True), (1, True), (2, True), (3, True), (4, False), (5, False),
(6, False), (7, True), (8, True), (9, True), (10, True)
]
Min Diff is (7 - 3)
"""
for index, e in reversed(list(enumerate(small_subset_vector))):
if e:
return _range - 2 * index
class NumberOfSubsetByGivenSumDifference:
def __init__(self) -> None:
self.memoization_table = None
def set_memoization_table(self, row: int, col: int) -> None:
# row ~> n & col ~> total
self.memoization_table = [[None for _ in range(col + 1)] for _ in range(row + 1)]
# Method 1: Recursion
def count_subset_sum(self, nums: List[int], n: int, total: int) -> int:
# Base Condition
if total == 0:
return 1
if n == 0:
return 0
# Memoization
if self.memoization_table[n][total] is not None:
return self.memoization_table[n][total]
# Choice Diagram
if nums[n - 1] <= total:
include = self.count_subset_sum(nums, n - 1, total - nums[n - 1])
exclude = self.count_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = include + exclude
return self.memoization_table[n][total]
else:
exclude = self.count_subset_sum(nums, n - 1, total)
self.memoization_table[n][total] = exclude
return self.memoization_table[n][total]
def get_number_of_subset_from_diff(self, nums: List[int], diff: int) -> int:
n = len(nums)
total = (diff + sum(nums)) // 2
self.set_memoization_table(n, total)
return self.count_subset_sum(nums, n, total)
class TargetSum:
# Method 2: Top Down
def count_subset_sum(self, nums: List[int], n: int, total: int) -> int:
# Initialize
matrix = [[0 for _ in range(total + 1)] for _ in range(n + 1)]
matrix[0][0] = 1
for k, v in enumerate(nums):
if v == 0:
matrix[k + 1][0] = 2 * matrix[k][0]
else:
matrix[k + 1][0] = matrix[k][0]
r = 1
while r <= n:
c = 1
while c <= total:
if nums[r - 1] <= c:
include = matrix[r - 1][c - nums[r - 1]]
exclude = matrix[r - 1][c]
matrix[r][c] = include + exclude
else:
matrix[r][c] = matrix[r - 1][c]
c += 1
r += 1
return matrix[n][total]
def find_target_sum_ways(self, nums: List[int], s: int) -> int:
n = len(nums)
if n == 0:
return 0
sum_of_nums = sum(nums)
if sum_of_nums < s or (sum_of_nums + s) % 2 != 0:
return 0
total = (s + sum_of_nums) // 2
return self.count_subset_sum(nums, n, total)
if __name__ == "__main__":
print('\n:~: Subset Sum :~:\n')
subset_sum_inputs = (
{
'nums': [3, 34, 4, 12, 5, 2],
'total': 30,
'expected': False,
},
{
'nums': [3, 34, 4, 12, 5, 2],
'total': 9,
'expected': True
},
)
for i in subset_sum_inputs:
_n = len(i['nums'])
ss_rc = SubsetSum()
ss_rc.set_memoization_table(_n, i['total'])
ss_rc_o = ss_rc.is_subset_sum(i['nums'], _n, i['total'])
print(f'Method 1: Recursion.\nExpected Output - {i["expected"]}\nOriginal Output - {ss_rc_o}', end='\n\n')
ss_td = SubsetSum()
ss_td_o = ss_td.is_subset_sum_2(i['nums'], _n, i['total'])
print(f'Method 2: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {ss_td_o}',
end='\n-----------------------\n')
print('\n:~: Equal Subset Sum Partition :~:\n')
equal_subset_sum_partition_inputs = (
{
'nums': [1, 5, 11, 5],
'expected': True,
},
{
'nums': [5, 2, 1, 3],
'expected': False
},
)
for i in equal_subset_sum_partition_inputs:
essp_rc = EqualSubsetSumPartition()
essp_rc_o = essp_rc.is_eq_partition(i['nums'])
print(f'Method 1: Recursion.\nExpected Output - {i["expected"]}\nOriginal Output - {essp_rc_o}', end='\n\n')
essp_td = EqualSubsetSumPartition()
essp_td_o = essp_td.can_partition(i['nums'])
print(f'Method 2: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {essp_td_o}',
end='\n-----------------------\n')
print('\n:~: Count Subset Sum :~:\n')
count_subset_sum_inputs = (
{
'nums': [10, 2, 5, 3, 8, 6],
'total': 10,
'expected': 3,
},
)
for i in count_subset_sum_inputs:
_n = len(i['nums'])
css_rc = CountSubsetSum()
css_rc.set_memoization_table(_n, i['total'])
css_rc_o = css_rc.get_subset_sum_count(i['nums'], _n, i['total'])
print(f'Method 1: Recursion.\nExpected Output - {i["expected"]}\nOriginal Output - {css_rc_o}', end='\n\n')
css_td = CountSubsetSum()
css_td_o = css_rc.count_subset_sum(i['nums'], _n, i['total'])
print(f'Method 2: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {css_td_o}',
end='\n-----------------------\n')
print('\n:~: Minimum Subset Sum Difference :~:\n')
min_subset_sum_diff_inputs = (
{
'nums': [1, 6, 11, 5],
'expected': 1,
},
{
'nums': [1, 2, 7],
'expected': 4,
},
)
for i in min_subset_sum_diff_inputs:
mssd_td = MinimumSubsetSumDifference()
mssd_td_o = mssd_td.min_difference(i['nums'])
print(f'Method 2: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {mssd_td_o}',
end='\n-----------------------\n')
print('\n:~: Number of Subset by a given Difference :~:\n')
number_of_subset_by_given_diff_inputs = (
{
'nums': [1, 1, 2, 3],
'diff': 1,
'expected': 3,
},
)
for i in number_of_subset_by_given_diff_inputs:
nssd_rc = NumberOfSubsetByGivenSumDifference()
nssd_rc_o = nssd_rc.get_number_of_subset_from_diff(i['nums'], i['diff'])
print(f'Method 1: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {nssd_rc_o}',
end='\n-----------------------\n')
print('\n:~: Target Sum :~:')
print(':~: TargetSum == NumberOfSubsetByGivenSumDifference :~:\n')
target_sum_inputs = (
{
'nums': [1, 1, 2, 3],
'target': 1,
'expected': 3,
},
{
'nums': [1, 1, 1, 1, 1],
'target': 3,
'expected': 5
},
{
'nums': [1],
'target': 2,
'expected': 0
},
{
'nums': [0, 0, 1],
'target': 1,
'expected': 4
},
)
for i in target_sum_inputs:
ts_rc = TargetSum()
ts_rc_o = ts_rc.find_target_sum_ways(i['nums'], i['target'])
print(f'Method 1: Top Down.\nExpected Output - {i["expected"]}\nOriginal Output - {ts_rc_o}',
end='\n-----------------------\n')