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

# Solution Notebook¶

## Constraints¶

• Is the array sorted?
• Yes
• Are the elements in the array distinct?
• No
• Does a magic index always exist?
• No
• If there is no magic index, do we just return -1?
• Yes
• Are negative values allowed in the array?
• Yes
• If there are multiple magic values, what do we return?
• Return the left-most one
• Can we assume this fits memory?
• Yes

## Test Cases¶

• None input -> -1
• Empty array -> -1
a[i]  -4 -2  2  6  6  6  6 10
i    0  1  2  3  4  5  6  7


Result: 2

a[i]  -4 -2  1  6  6  6  6 10
i    0  1  2  3  4  5  6  7


Result: 6

a[i]  -4 -2  1  6  6  6  7 10
i    0  1  2  3  4  5  6  7


Result: -1

## Algorithm¶

We'll use a binary search to split the search space in half on each iteration. To obtain more efficiency, we can do a little better than a naive left and half split.

In the example below, we see that i == 5 cannot be the magic index, otherwise a[5] would have to equal 5 (note a[4] == 6).

a[i]  -4 -2  2  6  6  6  6 10
i    0  1  1  3  4  5  6  7
mid


Similarly, in the example below we can further trim the left search space.

a[i]  -4 -2  2  2  2  6  6 10
i    0  1  2  3  4  5  6  7
mid

• Calculate mid
• If mid == array[mid], return mid
• Recurse on the left side of the array
• start: 0
• end: min(mid-1, array[mid]
• Recurse on the right side of the array
• start: max(mid+1, array[mid]
• end: end

Complexity:

• Time: O(log(n))
• Space: O(log(n))

## Code¶

In [1]:
from __future__ import division

class MagicIndex(object):

def find_magic_index(self, array):
if array is None or not array:
return -1
return self._find_magic_index(array, 0, len(array) - 1)

def _find_magic_index(self, array, start, end):
if end < start or start < 0 or end >= len(array):
return -1
mid = (start + end) // 2
if mid == array[mid]:
return mid
left_end = min(mid - 1, array[mid])
left_result = self._find_magic_index(array, start, end=left_end)
if left_result != -1:
return left_result
right_start = max(mid + 1, array[mid])
right_result = self._find_magic_index(array, start=right_start, end=end)
if right_result != -1:
return right_result
return -1


## Unit Test¶

In [2]:
%%writefile test_find_magic_index.py
import unittest

class TestFindMagicIndex(unittest.TestCase):

def test_find_magic_index(self):
magic_index = MagicIndex()
self.assertEqual(magic_index.find_magic_index(None), -1)
self.assertEqual(magic_index.find_magic_index([]), -1)
array = [-4, -2, 2, 6, 6, 6, 6, 10]
self.assertEqual(magic_index.find_magic_index(array), 2)
array = [-4, -2, 1, 6, 6, 6, 6, 10]
self.assertEqual(magic_index.find_magic_index(array), 6)
array = [-4, -2, 1, 6, 6, 6, 7, 10]
self.assertEqual(magic_index.find_magic_index(array), -1)
print('Success: test_find_magic')

def main():
test = TestFindMagicIndex()
test.test_find_magic_index()

if __name__ == '__main__':
main()

Overwriting test_find_magic_index.py

In [3]:
%run -i test_find_magic_index.py

Success: test_find_magic