Python 找到每个集合';s总和不';不要超过最大值
给定一个正整数数组,求最小子集数,其中:Python 找到每个集合';s总和不';不要超过最大值,python,algorithm,set,dynamic-programming,Python,Algorithm,Set,Dynamic Programming,给定一个正整数数组,求最小子集数,其中: 子集中每个元素的总和不超过值k 数组中的每个元素在任何子集中仅使用一次 数组中的所有值必须出现在任何子集中 基本上,一个“填充”算法,但需要最小化容器,并需要确保所有东西都被填充。我目前的想法是按降序排序,当总和超过k时开始创建集合,开始下一个集合,但不确定哪种方法更好 编辑: 例: 在输出集合中,使用所有1-5,并且在集合中仅使用一次。希望这能把事情弄清楚 也许有一种更聪明的方法可以找到最小数量的集合,但这里有一些代码使用Knuth的算法X来进行精确的
在输出集合中,使用所有1-5,并且在集合中仅使用一次。希望这能把事情弄清楚 也许有一种更聪明的方法可以找到最小数量的集合,但这里有一些代码使用Knuth的算法X来进行精确的覆盖运算,还有一个我去年编写的函数来生成总和小于给定值的子集。我的测试代码首先为问题中给出的数据找到一个解决方案,然后为一个更大的随机列表找到一个解决方案。它几乎可以立即找到[1,2,3,4,5]的最大和为10的解决方案,但在我的旧32位2GHz机器上,要解决更大的问题几乎需要20秒 这段代码只打印一个最小大小的解决方案,但修改它以打印所有最小大小的解决方案并不困难
""" Find the minimal number of subsets of a set of integers
which conform to these constraints:
The sum of each subset does not exceed a value, k.
Each element from the full set is only used once in any of the subsets.
All values from the full set must be present in some subset.
See https://stackoverflow.com/q/50066757/4014959
Uses Knuth's Algorithm X for the exact cover problem,
using dicts instead of doubly linked circular lists.
Written by Ali Assaf
From http://www.cs.mcgill.ca/~aassaf9/python/algorithm_x.html
and http://www.cs.mcgill.ca/~aassaf9/python/sudoku.txt
Written by PM 2Ring 2018.04.28
"""
from itertools import product
from random import seed, sample
from operator import itemgetter
#Algorithm X functions
def solve(X, Y, solution):
if X:
c = min(X, key=lambda c: len(X[c]))
for r in list(X[c]):
solution.append(r)
cols = select(X, Y, r)
yield from solve(X, Y, solution)
deselect(X, Y, r, cols)
solution.pop()
else:
yield list(solution)
def select(X, Y, r):
cols = []
for j in Y[r]:
for i in X[j]:
for k in Y[i]:
if k != j:
X[k].remove(i)
cols.append(X.pop(j))
return cols
def deselect(X, Y, r, cols):
for j in reversed(Y[r]):
X[j] = cols.pop()
for i in X[j]:
for k in Y[i]:
if k != j:
X[k].add(i)
#Invert subset collection
def exact_cover(X, Y):
newX = {j: set() for j in X}
for i, row in Y.items():
for j in row:
newX[j].add(i)
return newX
#----------------------------------------------------------------------
def subset_sums(seq, goal):
totkey = itemgetter(1)
# Store each subset as a (sequence, sum) tuple
subsets = [([], 0)]
for x in seq:
subgoal = goal - x
temp = []
for subseq, subtot in subsets:
if subtot <= subgoal:
temp.append((subseq + [x], subtot + x))
else:
break
subsets.extend(temp)
subsets.sort(key=totkey)
for subseq, _ in subsets:
yield tuple(subseq)
#----------------------------------------------------------------------
# Tests
nums = [1, 2, 3, 4, 5]
k = 10
print("Numbers:", nums, "k:", k)
Y = {u: u for u in subset_sums(nums, k)}
X = exact_cover(nums, Y)
minset = min(solve(X, Y, []), key=len)
print("Minimal:", minset, len(minset))
# Now test with a larger list of random data
seed(42)
hi = 20
k = 2 * hi
size = 10
nums = sorted(sample(range(1, hi+1), size))
print("\nNumbers:", nums, "k:", k)
Y = {u: u for u in subset_sums(nums, k)}
X = exact_cover(nums, Y)
minset = min(solve(X, Y, []), key=len)
print("Minimal:", minset, len(minset))
如果您共享一些示例输入和输出以及您编写的任何代码,这将非常有用。我不知道你所说的最小数量是什么意思-你是指唯一的集合吗?这里肯定有一个
itertools
解决方案。你关心算法的复杂性吗?请更详细一点。这是一种问题。一个普通的精确覆盖问题只需要条件2和3,但我认为添加条件1应该不会太难。这是(至少)np完全装箱问题。@AbhishekBansal感谢您提供这个问题的名称,但似乎并不是比反向排序和赋值更好的算法。谢谢你的帮助。
""" Find the minimal number of subsets of a set of integers
which conform to these constraints:
The sum of each subset does not exceed a value, k.
Each element from the full set is only used once in any of the subsets.
All values from the full set must be present in some subset.
See https://stackoverflow.com/q/50066757/4014959
Uses Knuth's Algorithm X for the exact cover problem,
using dicts instead of doubly linked circular lists.
Written by Ali Assaf
From http://www.cs.mcgill.ca/~aassaf9/python/algorithm_x.html
and http://www.cs.mcgill.ca/~aassaf9/python/sudoku.txt
Written by PM 2Ring 2018.04.28
"""
from itertools import product
from random import seed, sample
from operator import itemgetter
#Algorithm X functions
def solve(X, Y, solution):
if X:
c = min(X, key=lambda c: len(X[c]))
for r in list(X[c]):
solution.append(r)
cols = select(X, Y, r)
yield from solve(X, Y, solution)
deselect(X, Y, r, cols)
solution.pop()
else:
yield list(solution)
def select(X, Y, r):
cols = []
for j in Y[r]:
for i in X[j]:
for k in Y[i]:
if k != j:
X[k].remove(i)
cols.append(X.pop(j))
return cols
def deselect(X, Y, r, cols):
for j in reversed(Y[r]):
X[j] = cols.pop()
for i in X[j]:
for k in Y[i]:
if k != j:
X[k].add(i)
#Invert subset collection
def exact_cover(X, Y):
newX = {j: set() for j in X}
for i, row in Y.items():
for j in row:
newX[j].add(i)
return newX
#----------------------------------------------------------------------
def subset_sums(seq, goal):
totkey = itemgetter(1)
# Store each subset as a (sequence, sum) tuple
subsets = [([], 0)]
for x in seq:
subgoal = goal - x
temp = []
for subseq, subtot in subsets:
if subtot <= subgoal:
temp.append((subseq + [x], subtot + x))
else:
break
subsets.extend(temp)
subsets.sort(key=totkey)
for subseq, _ in subsets:
yield tuple(subseq)
#----------------------------------------------------------------------
# Tests
nums = [1, 2, 3, 4, 5]
k = 10
print("Numbers:", nums, "k:", k)
Y = {u: u for u in subset_sums(nums, k)}
X = exact_cover(nums, Y)
minset = min(solve(X, Y, []), key=len)
print("Minimal:", minset, len(minset))
# Now test with a larger list of random data
seed(42)
hi = 20
k = 2 * hi
size = 10
nums = sorted(sample(range(1, hi+1), size))
print("\nNumbers:", nums, "k:", k)
Y = {u: u for u in subset_sums(nums, k)}
X = exact_cover(nums, Y)
minset = min(solve(X, Y, []), key=len)
print("Minimal:", minset, len(minset))
Numbers: [1, 2, 3, 4, 5] k: 10
Minimal: [(2, 3, 5), (1, 4)] 2
Numbers: [1, 2, 3, 4, 8, 9, 11, 12, 17, 18] k: 40
Minimal: [(1, 8, 9, 18), (4, 11, 17), (2, 3, 12)] 3