Welcome back! As we progress through our course on Advanced Data Structures - Stacks and Queues in C++, we focus on leveraging queues to tackle algorithmic challenges often encountered in technical interviews. With their orderly structure, queues are excellent for representing sequential processes and managing streaming data. In this lesson, we'll explore two problems that highlight complex queue manipulations. Let's get started and decode these intriguing interview problems, ensuring that the concepts are thoroughly understood with additional examples and detailed explanations.

Let's begin with the concept of queue interleaving. Imagine you're orchestrating a dance sequence where dancers from two groups must perform in an alternating pattern. Our first computational task involves rearranging a list of elements, ensuring that if we start with an order like $a_1$, $a_2$, ..., $a_{n/2}$, $b_1$, $b_2$, ..., $b_{n/2}$, we end up with a sequence $a_1$, $b_1$, $a_2$, $b_2$, ..., $a_{n/2}$, $b_{n/2}$. This organization method mirrors real-life situations, such as merging traffic from two lanes onto a single-lane road, ensuring each car takes its turn from each lane.

We will utilize two auxiliary queues, similar to having two sub-lines in the dance sequence or two lanes on the road, to hold the divided sections of the original queue. We maintain a clean and memory-efficient interleaving without needing extra arrays by systematically dequeuing elements from these and enqueuing them back into the original queue.

First, consider a queue composed of dancers (or elements). We want to divide this queue into two groups, with the first half entering the firstHalf queue and the second half into the secondHalf queue. This way, we can alternately choose a dancer from each group and form a new, interleaved queue.

Let's construct our division:

C++
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1#include <queue>
2#include <iostream>
3
4std::queue<int> firstHalf;
5std::queue<int> secondHalf;
6
7// Assume 'queue' is the original queue with 'n' elements
8int n = queue.size();
9
10for (int i = 0; i < n / 2; i++)
11{
12    firstHalf.push(queue.front());
13    queue.pop();
14}
15
16while (!queue.empty())
17{
18    secondHalf.push(queue.front());
19    queue.pop();
20}

By iterating over the original queue, we distribute the elements into two separate queues, simulating the splitting of dancers into two groups. With the first group ready, we proceed to the second, ensuring a balanced division.

With both groups lined up, we alternately take a member from each group, thus combining them into the interwoven order:

C++
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1while (!firstHalf.empty() || !secondHalf.empty())
2{
3    if (!firstHalf.empty())
4    {
5        queue.push(firstHalf.front());
6        firstHalf.pop();
7    }
8    if (!secondHalf.empty())
9    {
10        queue.push(secondHalf.front());
11        secondHalf.pop();
12    }
13}

Imagine this as a dance coordinator calling out to each group in turn, forming a new sequence. This approach ensures no auxiliary arrays are needed, thus elegantly solving the problem using only the queues.

C++
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1#include <iostream>
2#include <queue>
3
4void interleaveQueue(std::queue<int>& queue)
5{
6    if (queue.size() % 2 != 0)
7    {
8        throw std::invalid_argument("The queue must contain an even number of elements.");
9    }
10
11    std::queue<int> firstHalf;
12    std::queue<int> secondHalf;
13
14    int n = queue.size();
15
16    for (int i = 0; i < n / 2; i++)
17    {
18        firstHalf.push(queue.front());
19        queue.pop();
20    }
21
22    while (!queue.empty())
23    {
24        secondHalf.push(queue.front());
25        queue.pop();
26    }
27
28    while (!firstHalf.empty() || !secondHalf.empty())
29    {
30        if (!firstHalf.empty())
31        {
32            queue.push(firstHalf.front());
33            firstHalf.pop();
34        }
35        if (!secondHalf.empty())
36        {
37            queue.push(secondHalf.front());
38            secondHalf.pop();
39        }
40    }
41}
42
43int main()
44{
45    std::queue<int> queue;
46    queue.push(1);
47    queue.push(2);
48    queue.push(3);
49    queue.push(4);
50    queue.push(5);
51    queue.push(6);
52
53    interleaveQueue(queue);
54
55    std::cout << "Interleaved queue:";
56    while (!queue.empty())
57    {
58        std::cout << " " << queue.front();
59        queue.pop();
60    }
61    return 0;
62}

This code includes the interleaveQueue function, which interleaves a given queue and checks that the number of elements is even before processing. The main function demonstrates its usage, including an example where a queue of integers is interleaved and printed.

Now, let's shift our attention to the second problem: computing a moving average from a data stream. This problem is common in technical interviews and requires real-time decision-making, akin to a trader monitoring livestock prices for quick buying or selling decisions. Our task is to calculate the average of the last k items in a stream of data, a critical operation for trend analysis in data analytics.

Alternatively, consider a fitness tracking app that updates a user's average heart rate over the last 10 minutes. The app computes the average heart rate readings to showcase the most recent state of health, updating this information with each new reading received.

A queue presents an efficient solution. It maintains a sliding window of the most recent k elements, mirroring our fitness app's ongoing cycle of heart rate readings, where fresh readings replace old data.

Imagine creating a class MovingAverage that emulates our fitness tracking app's backend, tasked with dynamically providing the average of the last k heart rate readings:

C++
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1#include <queue>
2
3class MovingAverage
4{
5    int size;
6    std::queue<int> window;
7    double sum;
8
9public:
10    MovingAverage(int size) : size(size), sum(0.0) {}
11
12    double next(int val)
13    {
14        if (window.size() == size)
15        {
16            sum -= window.front();
17            window.pop();
18        }
19
20        window.push(val);
21        sum += val;
22        return sum / window.size();
23    }
24};

In the next method, once the window reaches its maximum capacity (comparable to reaching the 10-minute mark in our app), we discard the oldest reading before adding the new one. We add the new value to our sum and then calculate the average by dividing the sum by the current window's size — much like updating the app display with the latest heart rate average.

Here is the complete code for you to test, imagining live data streaming in and out, with the heartbeat average updated after each new entry:

C++
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1#include <iostream>
2#include <queue>
3
4class MovingAverage
5{
6    int size;
7    std::queue<int> window;
8    double sum;
9
10public:
11    MovingAverage(int size) : size(size), sum(0.0) {}
12
13    double next(int val)
14    {
15        if (window.size() == size)
16        {
17            sum -= window.front();
18            window.pop();
19        }
20
21        window.push(val);
22        sum += val;
23        return sum / window.size();
24    }
25};
26
27int main()
28{
29    MovingAverage m(3);
30    std::cout << m.next(1) << std::endl;  // returns 1.0
31    std::cout << m.next(10) << std::endl; // returns 5.5
32    std::cout << m.next(3) << std::endl;  // returns 4.66667
33    std::cout << m.next(5) << std::endl;  // returns 6.0
34    return 0;
35}

Try integrating this code into a C++ environment, running it, and visualizing how the moving average changes dynamically, akin to our app's heart rate tracker.

Throughout this lesson, we've unlocked the potential of queues to streamline complex data manipulations, whether it’s interleaving queues or maintaining a sliding window for moving averages from streaming data. Both problems demonstrated how queues minimize redundancy and maximize efficiency — two highly prized qualities in technical interviews.

After mastering these techniques and understanding the rationale for using queues, you've added robust tactics to your coding repertoire, setting you up for success in practical scenarios. Using real-life analogies, such as dance sequences and fitness apps, helps make these abstract concepts more tangible.

It's time to put these theories into practice. Your next challenge awaits with hands-on exercises where you'll apply our freshly-acquired queue knowledge. These exercises will be both a test and an opportunity to perfect the art of manipulating queues to solve algorithmic puzzles. Are you ready? Let the coding begin!