MissingNumber

Problem

https://leetcode.com/problems/missing-number/

Given an array nums containing n distinct numbers in the range [0, n], return the only number in the range that is missing from the array.

Example 1:

Input: nums = [3,0,1]

Output: 2

Explanation:

n = 3 since there are 3 numbers, so all numbers are in the range [0,3]. 2 is the missing number in the range since it does not appear in nums.

Example 2:

Input: nums = [0,1]

Output: 2

Explanation:

n = 2 since there are 2 numbers, so all numbers are in the range [0,2]. 2 is the missing number in the range since it does not appear in nums.

Example 3:

Input: nums = [9,6,4,2,3,5,7,0,1]

Output: 8

Explanation:

n = 9 since there are 9 numbers, so all numbers are in the range [0,9]. 8 is the missing number in the range since it does not appear in nums.

Constraints:

n == nums.length
1 <= n <= 10:sup:`4`
0 <= nums[i] <= n
All the numbers of nums are unique.

Follow up: Could you implement a solution using only O(1) extra space complexity and O(n) runtime complexity?

Solution

Let \(m \in \{1, ..., n\}\) be the missing number. Since nums \(= \{1, ..., n\} - \{m\}\),

\[\begin{split}\begin{align} \DeclareMathOperator{\sumof}{sumof} \sumof(nums) &= \left( \sum_{i=1}^{n} i \right) - m \\ m &= \sum_{i=1}^{n} i - \sumof(nums) \end{align}\end{split}\]

Use the formula

\[\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\]

to calculate the the sum from 1 to \(n\) quickly. Note that \(n\) is one more than the number of elements in nums.

Pattern

Array, Hash Table, Math, Binary Search, Bit Manipulation, Sorting

Code

from typing import List


def missingNumber(nums: List[int]) -> int:
    """Find the single missing number in a list of integers.
    """
    n = len(nums)
    s = sum(nums)
    return n * (n + 1) // 2 - s

Test

>>> from MissingNumber import missingNumber
>>> list1 = list(range(4))
>>> list1.remove(2)
>>> missingNumber(list1)
2
>>> list2 = list(range(100))
>>> list2.remove(47)
>>> missingNumber(list2)
47

MissingNumber.missingNumber(nums: List[int]) → int: Find the single missing number in a list of integers.