Complexity Analysis and Optimization in C++

Welcome to the lesson! Today, our journey will ride through the captivating world of Complexity Analysis and techniques for optimization. These fundamental concepts are crucial for every programmer, especially those seeking to build efficient and scalable programs. Having a firm understanding of how code impacts system resources enables us to optimize it for better performance. Isn't it fascinating how we can tailor our code to be more efficient? So, buckle up and let's get started!

First things first, let's remind ourselves of what Complexity Analysis is. Simply put, Complexity Analysis is a way of determining how our data input size affects the performance of our program, most commonly in terms of time and space. In more technical terms, it’s a theoretical measure of the execution of an algorithm, particularly the time or memory needed, given the problem size n, which is usually the number of items. Interested in how it works?

Let's take, for example, a linear search function that looks for a value x in an array of size n. In the worst-case scenario, the function has to traverse the entire array, thus taking time proportional to n. We would say that this function has a time complexity of O(n).

#include <iostream>
#include <vector>

int linear_search(int x, const std::vector<int>& lst) {
    for (size_t i = 0; i < lst.size(); ++i) {
        if (lst[i] == x) {
            return i;
        }
    }
    return -1;
}

int main() {
    std::vector<int> lst = {1, 2, 3, 4, 5, 6, 7, 8, 9};
    int index = linear_search(5, lst);
    // index becomes 4
    std::cout << "Index: " << index << std::endl;
    return 0;
}

Now that we have refreshed our understanding of complexity analysis, let's delve into some basic examples of optimization. Optimization involves tweaking your code to make it more efficient by improving its runtime or reducing the space it uses.

An easy example of optimization could be replacing iterative statements (like a for loop) with built-in functions or using simple mathematical formulas whenever possible. Consider two functions, each returning the sum of numbers from 1 to an input number n.

The first one uses a for loop:

#include <iostream>

long long sum_numbers(int n) {
    long long total = 0;
    for (int i = 1; i <= n; ++i) {
        total += i;
    }
    return total;
}

int main() {
    long long result = sum_numbers(1000000);
    std::cout << "Sum: " << result << std::endl;
    return 0;
}

The second one uses a simple mathematical formula:

#include <iostream>

long long sum_numbers(int n) {
    return static_cast<long long>(n) * (n + 1) / 2;
}

int main() {
    long long result = sum_numbers(1000000);
    std::cout << "Sum: " << result << std::endl;
    return 0;
}

While both functions yield the same result, the second one is much more efficient. It doesn't need to iterate through all the numbers between 1 and n. This is a classic example of optimization, where we've reimagined our approach to solving a problem in a way that uses fewer resources.

Note that although all the values between 1 and n are of type int, we must consider that the result of the operations can exceed the limits of the int type. Therefore, we perform conversions to long long.