Number of Distinct Values in O(n log n) Time Using Python

Given an array of integers, count how many distinct values exist in the array.

Example:

Input: [1, 5, -3, 1, -4, 2, -4, 7, 7]
Output: 6
Explanation: the distinct values in the array are [1, 5, -3, -4, 2, 7]

Note:

For this lesson, your algorithm should run in O(n log n) time and use O(1) extra space.
(There are faster solutions which we will discuss in future lessons)

Hints:

Let's sort the array in non-decreasing order. In this way all the equal elements will come one after another.

Then, we can traverse the sorted array from left to right using an index i. We should increment the solution if value nums[i] has not been seen before. How can we check this?

nums[i] has never been seen before if nums[i] != nums[i - 1].

Problem Definition

The task is to count the number of distinct values in a given array of integers.

Input: An array of integers.

Output: An integer representing the number of distinct values in the array.

Constraints:

The algorithm should run in O(n log n) time.
The algorithm should use O(1) extra space.

Example:

Input: [1, 5, -3, 1, -4, 2, -4, 7, 7]
Output: 6
Explanation: The distinct values in the array are [1, 5, -3, -4, 2, 7]

Understanding the Problem

The core challenge is to identify and count unique elements in the array. This problem is significant in various applications such as data analysis, where understanding the diversity of data points is crucial. A common pitfall is to use additional data structures like sets, which would violate the O(1) extra space constraint.

Approach

To solve this problem, we can use the following approach:

Sort the array in non-decreasing order. This ensures that all equal elements are adjacent.
Traverse the sorted array and count elements that are different from their predecessors.

Let's discuss why this approach works:

Sorting the array takes O(n log n) time.
Traversing the array takes O(n) time.
We use no extra space apart from a few variables, thus adhering to the O(1) space constraint.

Algorithm

Here is a step-by-step breakdown of the algorithm:

Sort the array.
Initialize a counter to 1 (since the first element is always unique).
Iterate through the sorted array starting from the second element.
For each element, check if it is different from the previous element. If it is, increment the counter.

Code Implementation

def count_distinct_values(nums):
    # Step 1: Sort the array
    nums.sort()
    
    # Step 2: Initialize the count of distinct values
    distinct_count = 1  # The first element is always unique
    
    # Step 3: Traverse the sorted array
    for i in range(1, len(nums)):
        # Step 4: Check if the current element is different from the previous one
        if nums[i] != nums[i - 1]:
            distinct_count += 1
    
    return distinct_count

# Example usage
nums = [1, 5, -3, 1, -4, 2, -4, 7, 7]
print(count_distinct_values(nums))  # Output: 6

Complexity Analysis

Time Complexity: The sorting step takes O(n log n) time, and the traversal step takes O(n) time. Therefore, the overall time complexity is O(n log n).

Space Complexity: The algorithm uses O(1) extra space as we are not using any additional data structures.

Edge Cases

Consider the following edge cases:

An empty array: The output should be 0.
An array with all identical elements: The output should be 1.
An array with all distinct elements: The output should be the length of the array.

Example:

Input: []
Output: 0

Input: [2, 2, 2, 2]
Output: 1

Input: [1, 2, 3, 4, 5]
Output: 5

Testing

To test the solution comprehensively, consider the following test cases:

Simple cases with a few elements.
Cases with negative and positive integers.
Edge cases as discussed above.

Example test cases:

assert count_distinct_values([1, 5, -3, 1, -4, 2, -4, 7, 7]) == 6
assert count_distinct_values([]) == 0
assert count_distinct_values([2, 2, 2, 2]) == 1
assert count_distinct_values([1, 2, 3, 4, 5]) == 5

Thinking and Problem-Solving Tips

When approaching such problems, consider the following tips:

Understand the constraints and requirements thoroughly.
Think about different approaches and their trade-offs.
Start with a naive solution and then optimize it.
Practice similar problems to improve problem-solving skills.

Conclusion

In this blog post, we discussed how to count the number of distinct values in an array using an O(n log n) time algorithm with O(1) extra space. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for improving algorithmic thinking and coding skills.

Additional Resources

For further reading and practice, consider the following resources: