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

Solution Notebook

Problem: Find the magic index in an array, where array[i] = i.

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