Problem Description

The problem presents us with a simple task: Given an array of integers called nums, we are to find the highest value integer that appears exactly once in the array. If all integers appear more than once or if the array is empty, the function should return -1.

For example, if the input array is [5, 7, 3, 9, 4, 9, 8, 3, 1], the function should return 8, as it is the largest integer that appears only once.

This is essentially a frequency problem where we need to count how many times each number appears, and then find the largest number that has a frequency of 1.

Intuition

The intuition behind the solution is to keep track of the frequency of each element in the array and then search for the elements with a frequency of 1, starting from the largest potential number and moving downwards. The search stops when we find the first number that meets this criterion, as that would be the largest unique number.

To apply this solution approach efficiently:

1. We use the Counter class from Python's collections module to quickly count the frequency of each integer in the input array.
2. After we have the frequency of each number, we iterate from the maximum possible integer value down to 0. This ensures that the first integer we find with a frequency of 1 is the largest such integer.
3. We use a generator expression within the next function which goes through the numbers in the decreasing order checking for the condition cnt[x] == 1.
4. If no integer with a frequency of 1 is found, the next function returns -1 as specified by its default parameter.

With this approach, we can efficiently solve the problem in linear time with respect to the number of elements in the array, which is quite optimal for this type of problem.

Learn more about Sorting patterns.

Solution Approach

The implementation consists of a few straightforward steps:

1. Import the Counter from the collections module. The Counter is a subclass of dict specifically designed to count hashable objects. It's an unordered collection where elements are stored as dictionary keys and their counts are stored as dictionary values.

2. The cnt = Counter(nums) creates a Counter object with the frequency of each integer from the nums array. For example, if nums is [1, 2, 2, 3], cnt would be Counter({2: 2, 1: 1, 3: 1}).

3. The core of the implementation is the line return next((x for x in range(1000, -1, -1) if cnt[x] == 1), -1). This line uses a generator expression within the next function.

   • The generator expression gives us a way to iterate through each integer from 1000 down to 0 (range(1000, -1, -1)). We're starting from 1000 because, according to the constraints of the problem, the values in nums will not exceed 1000.

   • For each number x in this range, we check if cnt[x] == 1. This condition is true only for numbers that occur exactly once in nums. If the condition is met, x is yielded by the generator.

   • The next function is used to find the first item in the sequence that satisfies the condition. If such an item is found, it's returned immediately, making the process efficient because we don't need to count or iterate through the whole range if we've already found our largest unique number.

   • The second argument to next is -1, which acts as the default value returned if the generator does not yield any value (which would be the case if there are no unique numbers).

This compact and efficient implementation bypasses the need for sorting or additional loops, as it directly makes use of the Counter to access the frequency and uses the range function in descending order to find the largest unique number.

Example Walkthrough

To illustrate the solution approach, let's take a small example. Suppose our input array is [4, 6, 2, 6, 4].

1. First, we import the Counter class from the collections module.
2. We then create a Counter object to count the frequency of each integer in our array:

   from collections import Counter
   nums = [4, 6, 2, 6, 4]
   cnt = Counter(nums)  # Counter({4: 2, 6: 2, 2: 1})

   After creating the Counter object, it shows that both 4 and 6 occur twice, and 2 occurs once.
3. We then proceed to find the highest unique integer in nums by iterating from the highest possible value (1000) to the lowest in the array, checking if the frequency equals 1. For this example, our array doesn't go up to 1000, so we would just be interested in the range from the highest value in nums which is 6, down to the smallest 2:

   highest_unique = next((x for x in range(6, 1, -1) if cnt[x] == 1), -1)

4. In our range, we start checking from 6 to 2:
   • Check if cnt[6] == 1: This is False as cnt[6] is 2.
   • Move to the next value, 5, but it does not exist in our Counter, so move on.
   • Check if cnt[4] == 1: This is False as cnt[4] is 2.
   • Check if cnt[3] == 1: As 3 is not in our Counter, we move on.
   • Check if cnt[2] == 1: This is True as cnt[2] is 1.
5. Since we have found that 2 is the largest value that occurs exactly once, we don't need to check any further. We can now return 2 as the result.

The next function will yield 2 and since it's the first number that satisfies our condition, this is the value that would be returned from our function call.

If no such unique value is found, the default -1 will be returned, signaling that there are no elements that occur exactly once.

Thus, in our small example, the function would return 2 as the highest value integer that appears exactly once.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code consists of two parts: creating the counter and finding the largest unique number.

1. Creating the Counter: The Counter function from collections module is used to count the frequency of each element in the input list nums. The time complexity of this operation is O(n), where n is the length of the input list nums, as it requires a single pass over all elements to count their frequencies.

2. Finding the Largest Unique Number: The generator expression inside next iterates from 1000 to 0, which is a constant range, and checks if the count of each number is exactly 1. The worst-case time complexity of this operation is O(1) because we're iterating over a fixed range independent of the input size.

Combining both, the overall time complexity is O(n + 1), which simplifies to O(n) because asymptotic analysis drops constant terms.

Space Complexity

The space complexity also consists of two parts: the space used by the Counter and the space for the generator expression.

1. Counter Space: The Counter object will hold at most n unique numbers and their counts, so in the worst case, where all numbers in nums are unique, the space complexity is O(n).

2. Generator Expression Space: The generator expression does not create an additional list; it simply iterates over the range and yields values one by one. Therefore, its space complexity is O(1).

Overall, the space complexity of the algorithm is O(n), dominated by the space required for the Counter.

Learn more about how to find time and space complexity quickly using problem constraints.