forked from MTrajK/coding-problems
-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathfind_two_missing_numbers.py
More file actions
75 lines (57 loc) · 1.86 KB
/
find_two_missing_numbers.py
File metadata and controls
75 lines (57 loc) · 1.86 KB
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
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
'''
Find two missing numbers in a sequence
Find two missing numbers in a sequence,
all numbers are integers and they're smaller or equal to N+2 (N is length of the array).
Input: [2, 1, 4]
Output: [3, 5]
=========================================
Searching for 2 unknowns, math problem.
This solution is extension of 'find_one_missing_number.py'.
But in this case we also need the sum formula for the first squared N numbers (1^2 + 2^2 + ... + N^2).
And using those 2 formulas, we'll solve equations with 2 unknowns.
a + b = diff_sum (diff_sum = formula_sum + list_sum)
a^2 + b^2 = diff_squared_sum (diff_squared_sum = formula_squared_sum + list_squared_sum)
But in the end we'll need quadratic formula to find those values.
b1,2 = (diff_sum +- sqrt(2*diff_squared_sum - diff_sum^2)) / 2
Sum formula = N*(N+1)/2
Squared sum formula = N*(N+1)*(2*N+1)/6
Time Complexity: O(N)
Space Complexity: O(1)
Note: this idea also could be used when more than 2 numbers are missing,
but you'll need more computations/equations, because you'll have K unknowns.
'''
############
# Solution #
############
import math
def missing_numbers(nums):
# find sums from the array
s = 0
s_2 = 0
for i in nums:
s += i
s_2 += i * i
n = len(nums) + 2
# using formulas, compute the sums of the sequence
f_s = n * (n + 1) // 2
f_s_2 = n * (n + 1) * (2 * n + 1) // 6
# difference of sums
d = f_s - s
d_2 = f_s_2 - s_2
# using quadratic formula find the values
r = int(math.sqrt(2 * d_2 - d * d))
a = (d - r) // 2
b = (d + r) // 2
return [a, b]
###########
# Testing #
###########
# Test 1
# Correct result => [4, 5]
print(missing_numbers([2, 3, 1]))
# Test 2
# Correct result => [1, 2]
print(missing_numbers([5, 3, 4]))
# Test 3
# Correct result => [1, 5]
print(missing_numbers([2, 3, 4]))