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EastonLee/find_additive_equations.py

Last active June 6, 2018 01:10
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find_additive_equations.py
#!/usr/bin/env python
# coding=utf-8
'''
This is not a difficult problem to solve, but the highlight of this problem-solving process is that
first I analyzed the vague problem very carefully and considered many edge cases, and
I solved it in 3 different ways from a naive way to a high-performance DP way and
I tested it with extensive test cases far beyond the scale of the problem asked to solve and
I gave the time and space complexity.
Problem:
You are given 5 numbers a,b,c,d,e, they are in range of [0, 100], find all possible additive equations.
For example, you are given 2,3,14,7,9, the expected answer is
2+7=9
2+3+9=14
Analysis:
1. Since 2+14=7+9 is not in the answer,
so the right hand side should be only one number
2. 1=1 should not be a valid additive equation
3. 0 and 100 should be in the range of input
4. 0,1,1,2,2 should be a valid input, duplicate numbers are allowed
5. 0+0=0, 0+0+0=0, 0+1=1, 0+0+1=1 should be valid
6. 0+0=0 and 0+0+0=0 should be considered as distinct equations
7. 1+2=3 and 2+1=3 should be the same
8. 1+1=2 and 1+1=2 should be considered as the same equation,
and they'd better be deduplicated
9. Result should be an unordered collection of equations
10. This is an NP problem
'''
import unittest
import cProfile
class Solution(object):
# method 1
# Python built-in `combinations`
# time complexity: O(n^2*2^n)
# space complexity: O(n^2*2^n)
def additive_equations1(self, numbers):
from itertools import combinations
N = len(numbers)
rst = set()
for i in range(N):
candidates = numbers[:i] + numbers[i+1:]
target = numbers[i]
subs = [c for n in range(1, N)
for c in combinations(candidates, n)
if sum(c) == target and len(c) > 1]
for s in subs:
# deduplicate
rst.add((tuple(sorted(s)), target))
return rst
# method 2
# recursive implementation
# time complexity: O(n^2*2^n), faster than method 1 due to early stop
# space complexity: O(n^2*2^n)
def additive_equations2(self, numbers):
numbers.sort(reverse=True)
rst = set()
def helper(target, partial, start):
for i, n in enumerate(numbers[start:]):
d = target - n
if d == 0 and len(partial) > 0:
rst.add((tuple(sorted(partial+[n])), sum(partial)+n))
if d < 0:
continue
if d >= 0:
helper(d, partial+[n], start+i+1)
for i, n in enumerate(numbers):
helper(n, [], i+1)
return rst
# method 3
# tabulation memoization, faster than the above 2 methods
# especially when len(numbers) is large, at the cost of space
# time complexity: O(n*M), where M is the value of max(numbers)
# note that this is a pseudo-polynomial algorithm
# space complexity: O(n*M*2^n)
def additive_equations3(self, numbers):
numbers.sort(reverse=True)
N = len(numbers)
M = max(numbers)
pre_tab = [[] for i in range(M+1)]
cur_tab = [[] for i in range(M+1)]
rst = set()
# fill this table
for i in range(N):
for j in range(M+1):
# if numbers[i-1] is not included
cur_tab[j] = pre_tab[j][:] # a copy
# if numbers[i-1] is included
val = numbers[i]
# store index instead of value
if j-val == 0:
cur_tab[j].append([i])
if j-val >= 0:
# this number can't be used before
cur_tab[j].extend([k+[i] for k in pre_tab[j-val]
if i not in k])
pre_tab = cur_tab
# harvest the results
for i, n in enumerate(numbers):
# last line contains results
# this number itself should not be included
filtered = [j for j in cur_tab[n] if i not in j and len(j) > 1]
# deduplicate
for f in filtered:
# replace index with value now
f = [numbers[k] for k in f]
rst.add((tuple(sorted(f)), n))
return rst
def print_result(self, results):
if not results:
print('these are no available additive equations')
for result in results:
lhs = '+'.join(map(str, result[0]))
print(lhs + '=' + str(result[1]))
class Test(unittest.TestCase):
cases = [
[[2,3,7,9,14], {((2,3,9),14), ((2,7),9)}],
[[2,3,14,7,9], {((2,3,9),14), ((2,7),9)}],
[[50,50,50,50,100], {((50,50),100)}],
[[2,2,0,7,9], {((2,7),9), ((0,2,7),9), ((0,2),2)}],
[[0,0,0,0,0], {((0,0,0,0),0), ((0,0,0),0), ((0,0),0)}],
[[0,0,0,2,2], {((0,0),0), ((0,0,0,2),2), ((0,0,2),2), ((0,2),2)}],
[[1,1,2,2,2], {((1,1),2)}],
[[1,3,97,99,50], set()],
[[1,3,97,99,100], {((1,99),100), ((3,97),100)}],
[[1,3,97,99,0], set()],
]
all_cases = [
[[2,3,7,9,14], {((2,3,9),14), ((2,7),9)}],
[[2,3,14,7,9], {((2,3,9),14), ((2,7),9)}],
[[50,50,50,50,100], {((50,50),100)}],
[[0,1,1], {((0,1),1)}],
[[2,2,0,7,9], {((2,7),9), ((0,2,7),9), ((0,2),2)}],
[[0,0,0,0,0], {((0,0,0,0),0), ((0,0,0),0), ((0,0),0)}],
[[0,0,0,2,2], {((0,0),0), ((0,0,0,2),2), ((0,0,2),2), ((0,2),2)}],
[[1,1,2,2,2], {((1,1),2)}],
[[1,3,97,99,50], set()],
[[1,3,97,99,100], {((1,99),100), ((3,97),100)}],
[[1,3,97,99,0], set()],
# below are extensive tests, not required
[[1,2,3,4,5,6,7,8,9,45,90,100], set([((1, 2, 6), 9), ((4, 6, 90), 100), ((1, 2, 3, 4, 5, 6, 7, 8, 9), 45), ((1, 3, 4), 8), ((1, 2, 7, 90), 100), ((1, 6), 7), ((2, 8, 90), 100), ((1, 8), 9), ((2, 3), 5), ((1, 2, 3, 4, 5, 6, 7, 8, 9, 45), 90), ((1, 4, 5, 90), 100), ((1, 2, 3), 6), ((2, 3,4), 9), ((2, 6), 8), ((3, 6), 9), ((2, 5), 7), ((1, 2), 3), ((2, 4), 6), ((1, 5), 6), ((4, 5), 9), ((2, 3, 5, 90), 100), ((2, 7), 9), ((3, 7, 90), 100), ((3, 4), 7), ((3, 5), 8), ((1, 4), 5), ((1, 2, 5), 8), ((1, 3, 5), 9), ((1, 7), 8), ((1, 9, 90), 100),((1, 2, 3, 4, 90), 100), ((1, 2, 4), 7), ((1, 3, 6, 90), 100), ((1, 3), 4)])],
[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,45,90,100], set([((4, 5, 6, 7, 8, 10, 11, 12, 13, 14), 90), ((1, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15), 100), ((1, 2, 4, 10, 13, 15), 45), ((1, 2, 3, 5, 7, 12, 15), 45), ((3, 8, 10, 11, 13), 45), ((4, 5, 7, 8, 9, 12, 45), 90), ((1, 3, 4, 5, 7, 9, 11, 15, 45), 100), ((2,4, 5, 8, 9, 10, 11, 12, 14, 15), 90), ((1, 3, 4, 6, 7, 10, 14), 45), ((3, 5, 6, 8, 11, 12), 45), ((1, 7, 10, 12, 15), 45), ((1,7, 9, 13, 15), 45), ((3, 5, 10, 13, 14), 45), ((1, 3, 6, 7, 8, 10, 11, 12, 13, 14, 15), 100), ((4, 5, 6, 7, 10, 13), 45), ((3, 8, 9, 10, 12, 13, 45), 100), ((1, 4, 5, 9, 12, 14, 45), 90), ((2, 3, 6, 9, 11, 14, 45), 90), ((2, 4, 9), 15), ((1, 2, 3, 4, 5, 7, 8, 10, 15, 45), 100), ((1, 6, 9, 14, 15), 45), ((1, 2, 3, 4, 6, 9, 11, 12, 13, 14, 15), 90), ((4, 7, 9, 11, 14), 45), ((3, 4, 9, 11, 13, 15, 45), 100), ((1, 2, 4, 5, 6, 7, 11, 12, 13, 14, 15), 90), ((1, 3, 4, 5, 9, 10, 13), 45), ((1, 3, 11), 15), ((1, 2,6, 10, 12, 14), 45), ((1, 3, 5, 7, 8, 9, 10, 12, 45), 100), ((2, 3, 7, 8, 9, 11, 15, 45), 100), ((1, 5, 6, 9, 11, 13), 45), ((3, 5, 9, 13, 15, 45), 90), ((4, 5, 6, 7, 8, 15), 45), ((1, 4, 7, 8, 10, 15, 45), 90), ((2, 3, 4, 5, 6, 7, 13, 15, 45), 100), ((2,3, 4, 5, 8, 10, 13, 45), 90), ((1, 4, 6, 7, 10, 13, 14, 45), 100), ((5, 8, 9, 10, 13), 45), ((2, 4, 5, 9, 10, 15, 45), 90), ((2, 3, 5, 8, 12, 15, 45), 90), ((3, 4, 5, 7, 9, 13, 14, 45), 100), ((1, 2, 5, 6, 8, 11, 12, 45), 90), ((6, 10, 11, 13, 15, 45), 100), ((3, 4, 5, 9, 10, 14), 45), ((2, 5), 7), ((5, 7, 8, 12, 13), 45), ((3, 7, 10, 11, 14, 45), 90), ((1, 2, 3, 6, 7, 10, 12, 14,45), 100), ((1, 4, 6, 7, 8, 10, 12, 13, 14, 15), 90), ((3, 4, 11, 12, 15, 45), 90), ((1, 2, 3, 6, 8, 11, 14), 45), ((1, 3, 8, 10, 11, 12, 45), 90), ((1, 2, 7, 9, 11, 15), 45), ((2, 3, 4, 9, 10, 12, 15, 45), 100), ((4, 9), 13), ((2, 4, 5, 8, 11, 15, 45), 90), ((1, 5, 7, 9, 11, 12), 45), ((1, 4, 5, 8, 10, 13, 14, 45), 100), ((4, 7), 11), ((1, 2, 3, 4, 5, 7, 8, 15), 45), ((1, 3, 9), 13), ((1, 2, 3, 4, 5, 8, 9, 11, 12, 45), 100), ((2, 6, 8, 9, 10, 11, 12, 13, 14, 15), 100), ((1, 3, 5, 6, 7, 11, 12, 45), 90), ((1, 2, 3, 5, 7, 8, 14, 15, 45), 100), ((2, 4, 5, 7, 13, 14, 45), 90), ((3, 7, 9, 11, 12, 13, 45), 100), ((1, 4, 5, 10, 12, 13, 45), 90), ((4, 6, 7, 9, 14, 15, 45), 100), ((1, 4, 7, 9, 10, 11, 13, 45), 100), ((1, 2, 3, 5, 6, 7, 8, 10, 13, 45), 100), ((6, 7,9, 11, 12, 45), 90), ((2, 3, 4, 8, 11, 12, 15, 45), 100), ((3, 6, 7, 14, 15), 45), ((1, 2, 3, 4, 8, 10, 12, 15, 45), 100), ((3,5, 6, 7, 11, 13, 45), 90), ((1, 2, 4, 6, 8, 9, 11, 14, 45), 100), ((1, 3, 7, 8, 10, 11, 15, 45), 100), ((1, 2, 6, 7, 8, 10, 11,45), 90), ((5, 6, 8, 9, 12, 15, 45), 100), ((2, 3, 5, 8, 13, 14), 45), ((1, 2, 3, 4, 5, 6, 11, 13), 45), ((3, 10, 13, 14, 15, 45), 100), ((1, 2, 4, 11, 12, 15, 45), 90), ((1, 6, 9, 12, 13, 14, 45), 100), ((1, 3, 10, 12, 14, 15, 45), 100), ((1, 3, 4, 6, 7,11, 13), 45), ((2, 3, 4, 5, 6, 11, 14, 45), 90), ((2, 6, 9, 13, 15, 45), 90), ((1, 8, 9, 12, 15), 45), ((5, 7, 9, 10, 11, 13, 45), 100), ((3, 4, 7, 9, 10, 12), 45), ((3, 6, 7, 8, 10, 11), 45), ((2, 4, 6, 7, 10, 11, 15, 45), 100), ((2, 11, 13, 14, 15, 45),100), ((1, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15), 90), ((2, 5, 7, 8, 9, 14), 45), ((3, 4, 6, 8, 10, 11, 13, 45), 100), ((2, 3, 4, 6, 7, 11, 12), 45), ((1, 2, 5, 6, 7, 9, 11, 14, 45), 100), ((2, 4, 6, 8, 10, 11, 14, 45), 100), ((1, 4, 7, 8, 12, 13), 45), ((1, 4, 6, 7, 9, 10, 11, 13, 14, 15), 90), ((1, 3, 5, 7, 10, 14, 15, 45), 100), ((3, 4, 5, 7, 10, 12, 14, 45), 100), ((1, 3, 4, 6, 7,8, 9, 10, 11, 12, 14, 15), 100), ((3, 4, 6, 8, 10, 14), 45), ((1, 2, 4, 5, 7, 8, 9, 12, 13, 14, 15), 90), ((1, 3, 6, 8, 13, 14,45), 90), ((1, 3, 4, 5, 8, 10, 14, 45), 90), ((2, 3, 4, 6, 7, 8, 10, 15, 45), 100), ((2, 3, 4, 5, 8, 11, 12), 45), ((1, 2, 4, 7, 8, 9, 14, 45), 90), ((4, 5, 6, 12, 13, 15, 45), 100), ((4, 5, 7, 14, 15), 45), ((2, 7, 10, 11, 15), 45), ((1, 3, 4, 5, 6, 12, 14), 45), ((2, 4, 6, 8, 11, 14), 45), ((3, 5, 7, 8, 10, 12, 45), 90), ((1, 5, 7, 9, 10, 13), 45), ((1, 3, 4, 5, 13, 14, 15, 45),100), ((1, 6, 7, 8, 9, 11, 13, 45), 100), ((4, 5, 9, 13, 14, 45), 90), ((1, 2, 4, 6, 9, 10, 13, 45), 90), ((4, 5, 6, 7, 11, 12,45), 90), ((2, 4, 6, 7, 8, 13, 15, 45), 100), ((10, 90), 100), ((2, 3, 4, 5, 7, 8, 11, 15, 45), 100), ((1, 2, 3, 4, 5, 6, 7, 9,11, 13, 14, 15), 90), ((2, 3, 5, 6, 8, 9, 10, 12, 45), 100), ((4, 12, 14, 15), 45), ((1, 3, 4, 10, 11, 12, 14, 45), 100), ((1, 2, 4, 5, 6, 12, 15), 45), ((2, 4, 6, 9, 10, 14), 45), ((4, 6, 10, 11, 14), 45), ((5, 6, 7, 8, 14, 15, 45), 100), ((3, 5, 6, 7, 8, 11, 15, 45), 100), ((4, 6, 9, 12, 14), 45), ((3, 6, 7, 10, 14, 15, 45), 100), ((4, 6, 7, 9, 10, 12, 13, 14, 15), 90), ((4, 5, 7, 8, 9, 10, 12, 45), 100), ((3, 4, 6, 7, 9, 12, 14, 45), 100), ((1, 2, 3, 5, 6, 7, 9, 12, 45), 90), ((1, 2, 7, 8, 13, 14, 45), 90), ((2, 3, 4, 8, 9, 14, 15, 45), 100), ((2, 3, 5, 6, 14, 15), 45), ((1, 6), 7), ((2, 3, 5, 9, 12, 14), 45), ((1, 4, 5, 8, 9, 10, 11, 13, 14, 15), 90), ((1, 7, 8, 9, 10, 11, 12, 13, 14, 15), 100), ((1, 2, 6, 7, 11, 13, 15, 45), 100), ((1, 2, 5, 10, 11, 12, 14, 45), 100), ((2, 3, 4, 7, 8, 9, 10, 12, 45), 100), ((2, 6, 9, 10, 13, 15, 45), 100), ((4, 5, 6), 15), ((3, 8, 9, 11, 14), 45), ((1, 2, 4, 7, 8, 10, 13), 45), ((3, 5, 6, 12, 14, 15, 45), 100), ((4, 5, 7, 12, 13, 14, 45), 100), ((2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 15), 90), ((1, 2, 3, 6, 8, 10, 12, 13, 45), 100), ((1, 3, 7, 8, 11, 15, 45), 90), ((1, 2, 4, 8, 10, 11, 12, 13, 14, 15), 90), ((1, 5, 7, 8, 11, 13), 45), ((2, 3, 7, 8, 10, 15, 45), 90), ((1, 4, 5, 10, 11, 14), 45), ((1, 2, 3, 8, 9, 10, 12), 45),((2, 4, 11, 13, 15), 45), ((1, 6, 11, 12, 15), 45), ((2, 3, 5, 8, 12, 15), 45), ((3, 13, 14, 15, 45), 90), ((1, 5, 7, 8, 10, 11, 13, 45), 100), ((5, 6, 8, 12, 14, 45), 90), ((1, 5, 7, 9, 10, 11, 12, 45), 100), ((1, 2, 3, 8), 14), ((1, 3, 4, 6, 7, 11, 13, 45), 90), ((1, 3, 5, 9, 11, 12, 14, 45), 100), ((3, 4, 5, 6, 8, 9, 10, 45), 90), ((1, 3, 4, 5, 6, 7, 14, 15, 45), 100), ((2, 3, 4, 5, 6, 9, 12, 14, 45), 100), ((3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15), 100), ((4, 11, 12, 13, 15, 45), 100), ((1, 2, 3, 8, 12, 14, 15, 45), 100), ((1, 2, 4, 9, 11, 13, 15, 45), 100), ((2, 13), 15), ((1, 2, 3, 5, 6, 13, 15), 45), ((3, 13, 14, 15), 45), ((1, 2, 3, 5), 11), ((2, 5, 6, 8, 10, 11, 13, 45), 100), ((2, 5, 8), 15), ((2, 4, 5, 6, 7, 8, 9, 14, 45), 100), ((5, 7), 12), ((2, 3, 5, 9, 11, 15, 45), 90), ((1, 2, 4, 5, 6, 7, 8, 12, 45), 90), ((2, 5, 6, 13, 14, 15, 45), 100), ((4, 7, 8, 9, 13, 14, 45), 100), ((2, 3, 4, 7, 8, 10, 11, 45), 90), ((1, 2, 5, 7, 8, 10, 12), 45), ((1, 6, 7, 8, 10, 13, 45), 90), ((2, 3, 4, 5, 6, 10, 11, 14, 45), 100), ((1, 4, 6, 8, 10, 12, 14, 45), 100), ((3, 4, 6, 13, 14, 15, 45), 100), ((3, 5, 6, 9, 10, 12), 45), ((6, 7, 8, 10, 11, 13, 45), 100), ((2, 5, 6, 7, 9, 10, 11, 12, 13, 15), 90), ((2, 3, 4, 5, 6, 11, 14), 45), ((3, 4, 5, 6, 7, 8, 10, 12, 45), 100), ((3, 9, 10, 11, 12, 45), 90), ((2, 6, 9, 11, 13, 14, 45), 100), ((1, 2, 3, 4, 9, 10, 11, 15, 45), 100), ((1, 3, 4, 5, 8, 11, 13), 45), ((1, 6, 7), 14), ((1, 2, 3, 4, 6, 7, 9, 11, 12, 45), 100), ((1, 2, 5, 7, 8, 9, 10, 13, 45), 100), ((1, 2, 3, 5, 6, 7, 8, 9, 14, 45), 100), ((3, 7, 8, 10, 12, 15, 45), 100), ((2, 3, 4, 11, 12, 13), 45), ((7, 10, 11, 12, 15, 45), 100), ((1, 3, 7, 9, 11, 14), 45), ((2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13), 90), ((2, 3, 4, 6, 7, 8, 12, 13, 45), 100), ((5, 7, 8, 12, 13, 45), 90), ((1, 2, 7, 10, 12, 13), 45), ((4, 5, 6, 7, 9, 14, 45), 90), ((1, 2, 4, 5, 9, 11, 13), 45), ((1, 3, 6, 8, 9, 13, 15, 45), 100), ((1, 5, 6, 8, 12, 13), 45), ((2, 5, 8, 9, 10, 11), 45), ((2, 7, 9, 10, 13, 14, 45), 100), ((1, 4, 7, 9, 11, 13, 45), 90), ((1, 4, 6, 8, 9, 12, 15, 45), 100), ((1, 3, 5, 10, 11, 12, 13, 45), 100), ((2, 3, 4, 10, 11, 12, 13, 45), 100), ((1, 7, 9, 11, 13, 14, 45), 100), ((2, 8, 9, 12, 14, 45), 90), ((3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15), 100), ((3, 4, 5, 7, 8, 9, 12, 13, 14, 15), 90), ((4, 7, 9, 10, 12, 13, 45), 100), ((2, 5, 11, 12, 15, 45), 90), ((2, 3, 7, 8, 10, 11, 14, 45), 100), ((1, 2, 3, 7, 9, 10, 13), 45), ((1, 3, 5, 6, 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45), ((1, 2, 3, 4, 10, 12, 13, 45), 90), ((1, 2, 3, 7, 9, 10, 11, 12, 45), 100), ((1, 2, 5, 10, 13, 14, 45), 90), ((1, 2, 3, 5, 9, 10, 12, 13, 45), 100), ((2, 3, 7, 8, 11, 14, 45), 90), ((6, 8, 12, 14, 15, 45), 100), ((2, 5, 6, 9, 10, 13), 45), ((2, 6, 7, 10, 11, 12, 13, 14, 15), 90), ((4, 5, 6, 8, 9, 11, 12, 45), 100), ((1, 4, 5, 8, 13, 14, 45), 90), ((3, 4, 5, 9, 10, 11, 13, 45), 100), ((5, 11, 12, 13, 14, 45), 100), ((2, 3, 10, 11, 14, 15, 45), 100), ((4, 8, 10, 11, 12), 45), ((3, 5, 6, 7, 10, 11, 13, 45), 100), ((2, 3, 4, 7, 10, 14, 15, 45), 100), ((2, 3, 8, 9, 10, 13, 45), 90), ((1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15), 90), ((2, 6, 7, 9, 10, 11), 45), ((1, 3, 4, 5, 7, 10, 15), 45), ((1, 2, 4, 5, 9, 11, 13, 45), 90), ((1, 5, 6, 8, 11, 14, 45), 90), ((1, 2, 3, 4, 9, 12, 14), 45), ((4, 5, 7, 14, 15, 45), 90), ((3, 6, 10, 11, 15), 45), ((1, 3, 4, 11, 12, 14, 45), 90), ((2, 4, 5, 8, 10, 11, 15, 45), 100), ((1, 2, 3, 9), 15), ((6, 7), 13), ((2, 3, 4, 6, 7, 9, 14, 45), 90), ((2, 7), 9), ((1, 3, 7, 9, 12, 13), 45), ((3, 4, 8, 11, 14, 15, 45), 100), ((1, 5, 8, 9, 10, 12), 45),((7, 11, 12, 15), 45), ((1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15), 90), ((2, 3, 6, 7, 8, 10, 12, 13, 14, 15), 90), ((1, 4, 5, 6,7, 9, 10, 13, 45), 100), ((1, 4, 5, 7, 8, 9, 11), 45), ((1, 7, 10, 13, 14), 45), ((1, 4, 5, 7, 13, 15, 45), 90), ((3, 5, 9, 11,12, 15, 45), 100), ((2, 5, 6, 7, 11, 14, 45), 90), ((5, 7, 10, 11, 12, 45), 90), ((1, 3, 6, 9, 12, 14), 45), ((1, 2, 3, 4, 8, 9, 10, 11, 13, 14, 15), 90), ((1, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15), 100), ((1, 3, 5, 6, 7, 9, 11, 13, 45), 100), ((3, 6, 8, 9, 14, 15, 45), 100), ((1, 3, 5, 7, 8, 10, 11, 45), 90), ((2, 3, 4), 9), ((3, 4, 5, 6, 13, 14, 45), 90), ((2, 3, 5, 6, 8, 9, 12,45), 90), ((3, 7, 8, 10, 13, 14, 45), 100), ((1, 2, 5, 9, 10, 13, 15, 45), 100), ((4, 5), 9), ((3, 5, 10, 11, 12, 14, 45), 100), ((2, 3, 6, 9, 10, 12, 13, 45), 100), ((1, 2, 3, 5, 8, 9, 13, 14, 45), 100), ((3, 7, 8, 9, 13, 15, 45), 100), ((1, 2, 3, 7, 8, 10, 11, 13, 45), 100), ((2, 8), 10), ((1, 2, 3, 9, 10, 11, 12, 13, 14, 15), 90), ((1, 3, 4, 6, 8, 9, 14), 45), ((1, 2, 3, 6, 7, 12, 14, 45), 90), ((1, 3, 4, 9, 13, 15, 45), 90), ((4, 8, 9, 10, 14, 45), 90), ((2, 3, 6, 7, 13, 14), 45), ((3, 4, 6, 7, 10, 11,14, 45), 100), ((1, 4, 11, 14, 15), 45), ((1, 3, 4, 6, 7, 8, 9, 10, 13, 14, 15), 90), ((3, 4, 6, 9, 10, 11, 12, 45), 100), ((1,3, 4, 5, 7, 11, 14), 45), ((3, 4, 6, 7, 12, 13), 45), ((1, 5, 6, 9, 11, 13, 45), 90), ((4, 8), 12), ((2, 5, 8, 11, 14, 15, 45),100), ((1, 2, 4, 5, 6, 10, 13, 14, 45), 100), ((2, 3, 5, 6, 7, 9, 11, 12, 45), 100), ((1, 2, 3, 5, 8, 9, 10, 11, 12, 14, 15), 90), ((3, 4, 5, 8, 10, 15), 45), ((2, 3, 4, 5, 7, 10, 11, 13, 45), 100), ((1, 7, 8, 10, 14, 15, 45), 100), ((1, 3, 5, 6, 8, 10, 12, 45), 90), ((1, 2, 3, 8, 9, 10, 12, 45), 90), ((4, 5, 9, 11, 12, 14, 45), 100), ((1, 2, 3, 5, 6, 7, 10, 11), 45), ((1, 4, 6, 9, 12, 13, 45), 90), ((3, 6), 9), ((2, 8, 10, 12, 13, 45), 90), ((1, 2, 3, 4, 8, 12, 15, 45), 90), ((4, 6, 8, 10, 13, 14, 45), 100), ((1, 3, 6, 7, 9, 10, 12, 13, 14, 15), 90), ((1, 2, 3, 5, 7, 13, 14, 45), 90), ((1, 4, 5, 6, 8, 10, 11), 45), ((1, 2, 6, 7, 9, 11, 12, 13, 14, 15), 90), ((1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14), 90), ((1, 8, 9, 13, 14, 45), 90), ((3, 4, 6, 9, 10, 13, 45), 90), ((1, 2, 4, 8, 9, 10, 11), 45), ((2, 4, 12, 13, 14, 45), 90), ((1, 2, 3, 4, 6, 7, 9, 13), 45), ((1, 2, 4, 7, 9, 10, 12), 45), ((1, 2, 3, 4, 90), 100), ((2, 5, 6, 7, 8, 10, 11, 12, 14, 15), 90), ((1, 4, 6, 9, 11, 14), 45), ((5, 6, 9, 11, 14, 45), 90), ((2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14), 90), ((1, 12, 13, 14, 15, 45), 100), ((3, 7, 8, 13, 14, 45), 90), ((1, 3, 7, 8, 12,14), 45), ((1, 2, 8, 9, 12, 13, 45), 90), ((2, 3, 4, 6, 8, 10, 12), 45), ((4, 10, 12, 14, 15, 45), 100), ((4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15), 100), ((2, 3, 5, 8, 10, 12, 15, 45), 100), ((4, 5, 9, 12, 15, 45), 90), ((2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15), 90), ((1, 4, 7, 9, 11, 13), 45), ((2, 8, 9, 11, 12, 13, 45), 100), ((3, 4, 6, 9, 11, 12), 45), ((2, 4, 6, 7, 9, 10, 11, 12, 14, 15), 90), ((2, 6, 7, 9, 10, 11, 45), 90), ((1, 5, 7, 9, 10, 13, 45), 90), ((1, 4), 5), ((2, 3, 4, 6, 9, 10, 11), 45), ((3, 4,5, 10, 11, 12, 45), 90), ((5, 7, 8, 9, 11, 15, 45), 100), ((1, 2, 3, 4, 7, 13, 15), 45), ((2, 3, 6, 7, 8, 9, 10, 45), 90), ((1,5, 7, 8, 11, 13, 45), 90), ((1, 2, 3, 4, 5, 6, 10, 14), 45), ((1, 2, 3, 5, 8, 12, 14, 45), 90), ((1, 2, 6), 9), ((1, 3, 7, 8, 9, 10, 11, 12, 14, 15), 90), ((1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 15), 90), ((2, 5, 6, 8, 9, 11, 14, 45), 100), ((2, 3, 4, 6, 7, 8,11, 14, 45), 100), ((2, 5, 7, 8, 10, 13), 45), ((2, 5, 6, 9, 10, 11, 12, 45), 100), ((1, 2, 3, 4, 8, 12, 15), 45), ((8, 9, 11, 12, 15, 45), 100), ((1, 6, 7, 8, 11, 12, 45), 90), ((2, 4, 6, 8, 10, 15), 45), ((1, 2, 4, 5, 6, 12, 15, 45), 90), ((2, 4, 5, 6, 8, 11, 12, 13, 14, 15), 90), ((2, 3, 7, 9, 10, 14, 45), 90), ((1, 3, 12, 14, 15), 45), ((1, 8, 9, 13, 14), 45), ((2, 3, 5, 7, 9,10, 12, 13, 14, 15), 90), ((1, 2, 4, 5, 10, 11, 12, 45), 90), ((1, 2, 3, 4, 5, 7, 10, 13, 45), 90), ((1, 3, 6, 10, 12, 13), 45), ((1, 3, 4, 10, 12, 15), 45), ((2, 6, 11, 12, 14, 45), 90), ((1, 2, 4, 6, 7, 12, 13), 45), ((1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 14, 15), 90), ((1, 5, 9, 12, 13, 15, 45), 100), ((3, 10), 13), ((1, 8, 10, 12, 14, 45), 90), ((1, 2, 13, 14, 15, 45), 90), ((3, 6,7, 8, 9, 10, 12, 45), 100), ((1, 3, 4, 6, 8, 9, 10, 14, 45), 100), ((1, 2, 5, 7, 8, 9, 13), 45), ((1, 3, 5, 9, 10, 12, 15, 45),100), ((2, 7, 9, 13, 14), 45), ((1, 2, 5, 7, 12, 13, 15, 45), 100), ((2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15), 90), ((2, 3, 4, 5),14), ((8, 9, 13, 15, 45), 90), ((1, 2, 3, 7, 8, 9, 15, 45), 90), ((2, 6, 7, 8, 9, 13, 45), 90), ((1, 9), 10), ((1, 3, 4, 7, 8, 9, 10, 13, 45), 100), ((1, 2, 3, 4, 5, 7, 11, 12), 45), ((1, 5, 6, 7, 8, 9, 12, 13, 14, 15), 90), ((4, 5, 9, 13, 14), 45), ((4, 5, 10, 12, 14), 45), ((1, 2, 3, 4, 5, 6, 7, 8, 9, 45), 90), ((1, 2, 7, 8, 13, 14), 45), ((4, 5, 7, 8, 9, 12), 45), ((4, 9, 13, 14, 15, 45), 100), ((3, 5, 9, 10, 13, 15, 45), 100), ((4, 5, 6, 8, 10, 12, 45), 90), ((1, 3, 4, 5, 7, 8, 12, 15, 45), 100), ((1, 2, 4, 6, 8, 11, 13), 45), ((2, 5, 6, 8, 10, 14), 45), ((1, 5, 10, 12, 13, 14, 45), 100), ((2, 3, 4, 5, 7, 11, 13), 45), ((2, 3, 4, 6, 7, 11, 12, 45), 90), ((3, 5, 6, 7, 10, 14), 45), ((1, 2, 3, 10, 11, 13, 15, 45), 100), ((3, 4, 5, 6, 7, 9, 10, 11, 45), 100), ((1, 3, 5, 6, 9, 10, 11, 45), 90), ((1, 4, 5, 6, 7, 8, 14), 45), ((2, 3, 5, 7, 9, 14, 15, 45), 100), ((2, 3, 8, 9, 10, 13), 45), ((3, 5, 6, 8, 9, 11, 13, 45), 100), ((2, 9, 10, 11, 13, 45), 90), ((2, 3, 5, 10, 12, 13), 45), ((6, 12, 13, 14, 45), 90), ((1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15), 90), ((1, 2, 3, 4, 5, 9, 10, 11, 45), 90), ((1, 4, 5, 6, 14, 15), 45), ((2, 8, 9, 12, 14), 45), ((1, 2, 5, 7, 11, 14, 15, 45), 100), ((1, 6, 7, 8, 10, 13), 45), ((2, 3, 6), 11), ((3, 4, 5, 8, 12, 13, 45), 90), ((1, 2,4, 6, 7, 10, 15, 45), 90), ((1, 5, 11, 13, 15), 45), ((3, 7, 9, 10, 11, 15, 45), 100), ((1, 4, 6, 7, 13, 14), 45), ((2, 3, 5, 6, 9, 11, 12, 13, 14, 15), 90), ((3, 5, 6, 7, 8, 9, 11, 12, 14, 15), 90), ((3, 5, 7, 9, 10, 11), 45), ((1, 9, 10, 12, 13, 45), 90), ((1, 3, 5, 6, 7, 8, 15), 45), ((1, 5, 6, 9, 10, 14), 45), ((1, 2, 3, 5, 9, 10, 15, 45), 90), ((1, 4, 5, 10, 11, 14, 45), 90),((2, 3, 7, 8, 11, 14), 45), ((2, 5, 6, 7, 8, 9, 11, 13, 14, 15), 90), ((1, 6, 10, 11, 13, 14, 45), 100), ((3, 6, 8, 11, 13, 14,45), 100), ((2, 7, 11, 12, 13, 45), 90), ((1, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14), 90), ((2, 4, 6), 12), ((2, 4, 6, 8, 10, 12, 13, 45), 100), ((1, 2, 4, 5, 6, 8, 10, 12, 13, 14, 15), 90), ((2, 4, 5, 7, 9, 13, 15, 45), 100), ((1, 4, 5, 8, 12, 15), 45), ((4,5, 8, 10, 13, 15, 45), 100), ((1, 3, 4, 5, 7, 10, 15, 45), 90), ((2, 5, 6, 8, 9, 10, 15, 45), 100), ((5, 6, 7, 10, 12, 15, 45),100), ((4, 5, 6, 7, 8, 9, 10, 12, 14, 15), 90), ((1, 2, 3, 4, 9, 11, 15, 45), 90), ((1, 2, 5, 9, 11, 13, 14, 45), 100), ((1, 3,6, 7, 8, 9, 11), 45), ((4, 6), 10), ((5, 6, 8, 9, 10, 11, 12, 14, 15), 90), ((6, 9), 15), ((1, 8, 9, 10, 13, 14, 45), 100), ((1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 15), 90), ((1, 3, 4, 6, 7, 9, 15, 45), 90), ((1, 3, 4, 5, 6, 9, 12, 15, 45), 100), ((5, 6, 7, 12, 15), 45), ((1, 2, 6, 11, 12, 13), 45), ((2, 6, 10, 13, 14), 45), ((1, 5, 6, 8, 9, 11, 15, 45), 100), ((3, 4, 8), 15)])],
# method 1: 40s, method 2: 1.00s, method 3: 1.18s
# [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,100], set()],
# # method 1: forever, method 2: 1.7s, method 3: 2.3s
# [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,100], set()],
# # method 1: forever, method 2: 0.016s, method 3: 0.008s
# [list(range(1,31)), set()],
# # method 1: forever, method 2: 3.5s, method 3: 4.6s
# [list(range(1,31)) + [100], set()],
# # method 1: forever, method 2: 8.0s, method 3: 7.2s
# [list(range(1,51)) + [100], set()],
# # method 1: forever, method 2: 12.6s, method 3: 13.0s
# [list(range(41)) + [100], set()],
# # method 1: forever, method 2: 17.1s, method 3: 14.6s
# [list(range(61)) + [100], set()],
# # method 1: forever, method 2: 23.9s, method 3: 17.1s
# [list(range(71)) + [100], set()],
# # method 1: forever, method 2: 45.2s, method 3: 21.9s
# [list(range(81)) + [100], set()],
# # method 1: forever, method 2: 99.6s, method 3: 31.5s
# [list(range(91)) + [100], set()],
# # method 1: forever, method 2: 311.8s, method 3: 52.2s
# [list(range(101)), set()],
]
def test(self):
for ci, co in self.all_cases:
print('input numbers are: ' + str(ci))
co1, co2, co3 = Solution().additive_equations1(ci), \
Solution().additive_equations2(ci), \
Solution().additive_equations3(ci)
assert co1 == co2 == co3 == co
# Solution().print_result(Solution().additive_equations3(ci))
# cProfile.runctx('Solution().additive_equations3(ci)', \
# globals(), locals(), sort='cumtime')
if __name__ == '__main__':
for ci, co in Test.cases:
print('input numbers are: ' + str(ci))
Solution().print_result(Solution().additive_equations3(ci))
# running all test cases without printing needs about 160s
# for ci, co in Test.all_cases:
# print('input numbers are: ' + str(ci))
# Solution().additive_equations3(ci)
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