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

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 array can contain both positive and negative integers.
The algorithm should run in O(n log n) time complexity.
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 efficiently. 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 can increase space complexity.

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 a naive solution and its limitations:

Naive Solution: Use a set to store unique elements and return its size. This approach has O(n) time complexity but uses O(n) space, which violates the space constraint.

Optimized Solution: By sorting the array first, we can achieve O(n log n) time complexity and use O(1) extra space by counting distinct elements during a single traversal of the sorted array.

Algorithm

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

Sort the array in non-decreasing order.
Initialize a counter for distinct elements.
Traverse the sorted array and compare each element with its predecessor. Increment the counter if the current element is different from the previous one.

Code Implementation

#include <iostream>
#include <vector>
#include <algorithm>

int countDistinctValues(std::vector<int>& nums) {
    // Step 1: Sort the array
    std::sort(nums.begin(), nums.end());
    
    // Step 2: Initialize the counter for distinct values
    int distinctCount = 0;
    
    // Step 3: Traverse the sorted array
    for (size_t i = 0; i < nums.size(); ++i) {
        // If it's the first element or different from the previous one, it's distinct
        if (i == 0 || nums[i] != nums[i - 1]) {
            ++distinctCount;
        }
    }
    
    return distinctCount;
}

int main() {
    std::vector<int> nums = {1, 5, -3, 1, -4, 2, -4, 7, 7};
    std::cout << "Number of distinct values: " << countDistinctValues(nums) << std::endl;
    return 0;
}

Complexity Analysis

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

Space Complexity: The algorithm uses O(1) extra space as it sorts the array in place and uses a constant amount of additional memory for the counter.

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.

Examples:

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 a variety of test cases:

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

Use a testing framework like Google Test for C++ to automate and manage test cases efficiently.

Thinking and Problem-Solving Tips

When approaching such problems:

Understand the problem constraints and requirements thoroughly.
Consider both time and space complexity when designing your solution.
Break down the problem into smaller, manageable steps.
Practice solving similar problems to improve your problem-solving skills.

Conclusion

In this blog post, we discussed how to count the number of distinct values in an array using an efficient algorithm with O(n log n) time complexity and O(1) extra space. Understanding and solving such problems is crucial for developing strong problem-solving skills in programming.

Keep practicing and exploring further to enhance your understanding and proficiency in algorithms and data structures.

Additional Resources

For further reading and practice, consider the following resources: