This notebook was prepared by Donne Martin. Source and license info is on GitHub.

- Are we working with non-negative ints?
- Yes

- What is the range of the integers?
- Discuss the approach for 4 billion integers
- Implement for 32 integers

- Can we assume the inputs are valid?
- No

- None -> Exception
- [] -> Exception
- General case
- There is an int excluded from the input -> int
- There isn't an int excluded from the input -> None

The problem states to use a minimal amount of memory. We'll use a bit vector to keep track of the inputs.

Say we are given 4 billion integers, which is 2^32 integers. The number of non-negative integers would be 2^31. With a bit vector, we'll need 4 billion bits to map each integer to a bit. Say we had only 1 GB of memory or 2^32 bytes. This would leave us with 8 billion bits.

To simplify this exercise, we'll work with an input of up to 32 ints that we'll map to a bit vector of 32 bits.

input = [0, 1, 2, 3, 4...28, 29, 31] bytes [ 1 ] [ 2 ] [ 3 ] [ 4 ] index = 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 bit_vector = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 result = 30 * Loop through each item in the input, setting bit_vector[item] = True. * Loop through the bit_vector, return the first index where bit_vector[item] == False.

Complexity:

- Time: O(b), where b is the number of bits
- Space: O(b)

In [1]:

```
from bitstring import BitArray # Run pip install bitstring
class Bits(object):
def new_int(self, array, max_size):
if not array:
raise TypeError('array cannot be None or empty')
bit_vector = BitArray(max_size)
for item in array:
bit_vector[item] = True
for index, item in enumerate(bit_vector):
if not item:
return index
return None
```

In [2]:

```
%%writefile test_new_int.py
import unittest
class TestBits(unittest.TestCase):
def test_new_int(self):
bits = Bits()
max_size = 32
self.assertRaises(TypeError, bits.new_int, None, max_size)
self.assertRaises(TypeError, bits.new_int, [], max_size)
data = [item for item in range(30)]
data.append(31)
self.assertEqual(bits.new_int(data, max_size), 30)
data = [item for item in range(32)]
self.assertEqual(bits.new_int(data, max_size), None)
print('Success: test_find_int_excluded_from_input')
def main():
test = TestBits()
test.test_new_int()
if __name__ == '__main__':
main()
```

In [3]:

```
%run -i test_new_int.py
```