Algorithms are the backbone of efficient software, yet they often seem like an abstract concept reserved for Academia. But why are we discussing them? Because, in reality, every line of code we write has an impact on performance, and understanding algorithm complexity is key to writing efficient and scalable software.

In this article, we’ll break down the calculation of algorithm complexity, explore what affects performance in code, and provide practical examples to help you grasp these concepts. We’ll also discuss when complexity matters most and when it’s less of a concern in everyday programming.

This article is not intended to be exhaustive but aims to offer a few key insights that can assist programmers in their daily work. The goal is to provide an overview of where to focus and how to approach these topics effectively. A bibliography is included at the end for further exploration.

The Basics of Algorithm Complexity: What Is It and How Do We Measure It?

Algorithm complexity is a way to describe how the runtime or space requirements of an algorithm grow as the size of the input increases. This is often expressed using Big O notation, which provides a high-level understanding of the algorithm’s efficiency. 

For example:

• O(1): Constant time. The algorithm’s runtime doesn’t change with the size of the input.
• O(n): Linear time. The runtime grows linearly with the size of the input.
• O(log n): Logarithmic time. The runtime grows logarithmically as the input size increases.

So to put more clearly, the idea is what sort of behavior would a mathematical function that has the input size as a parameter and the number of microprocessor clock cycles as an output. Will that function be constant, taking the same time no matter the size of the input? Will it be linear, where a bigger input results in a proportionally higher number of cycles, hence consuming even more time? Or will it be of polynomial growth like O(n²) or O(n³), in which 100 input items would result in 1,000 or 10,000 operations accordingly?

Understanding these distinctions is crucial because they help you anticipate how your code will perform as data scales. But remember, complexity calculations often assume ideal conditions, which don’t always reflect real-world scenarios.

Factors Affecting Code Performance: Beyond Big O

While Big O notation gives us a theoretical framework, real-world performance depends on several other factors:

• Control Structures: Loops, conditionals, and branching (e.g., `for`, `while`) can significantly affect runtime. Nested loops can quickly increase complexity from linear to quadratic or worse.

• Hardware and System Architecture: The speed of your CPU, memory hierarchy, and even network latency can influence how your algorithm performs in practice.

• Programming Language and Compiler Optimizations: Different languages and compilers optimize code in various ways, impacting the actual performance of your algorithm.

For instance, a simple nested loop can drastically slow down your code if not carefully considered.

Practical Examples of Algorithm Complexity

Let’s look at a few examples to understand how different algorithms compare in terms of complexity.

This algorithm runs in linear time because it may need to check each element in the array before finding the target.

O(n log n): QuickSort

QuickSort is one of the fastest sorting algorithms, operating in O(n log n) time on average. It’s commonly used in practice because of its efficiency with large datasets.

O(n²): Dijkstra’s Algorithm

Dijkstra’s algorithm for finding the shortest path in a graph has a time complexity of O(n²) in its simplest form, though optimizations can bring it down to O(V log V + E), where V is the number of vertices and E is the number of edges.

O(n!): The Traveling Salesman Problem

The brute-force solution to the Traveling Salesman Problem, which calculates every possible route, runs in O(n!) time—an impractical approach for large inputs.

Case Study: Optimizing an Algorithm for Finding Triplets

Let’s look at an example of optimizing an algorithm from O(n³) to O(n²). The objective here is to write an algorithm that, given an array, returns the indexes of three numbers in that array that sum up to a specific target. For instance:

Given the following Array: 

Index	0	1	2	3	4	5	6
Value	1	6	8	5	4	7	3

We want to know which 3 elements sum: 13

So in this case the elements in index 1 (that contains a 6), the element on index 4 (that contains a 4) and element on index 6 make the expected sum 13. So the expected answer should be: 1st and 5th.

Inefficient Implementation of a solution O(n³)

This version contains three nested loops, resulting in cubic time complexity. It evaluates every possible combination of three elements, which significantly slows down as the array size increases. For instance, with an array of 1,000,000 elements, the algorithm would need to perform 1,000,000³ operations, equating to 1 quadrillion (1 x 10¹⁸) operations. Clearly, this approach doesn’t scale well.

A slightly more optimized version O(n²)

By sorting the array and then scanning it from both ends, we reduce the complexity to O(n²), which makes it much more efficient for larger datasets. For example, with an input array of 1,000,000 elements, we would need 1,000,000² (or 1 trillion) operations to get the result. While this is a significant improvement over the previous version, we are still not fully optimizing computing power.

This can still be optimized!

So we arrive to the final and most efficient implementation. This version is much more efficient, with a time complexity of O(n). This means that if the input consists of 1,000,000 elements, the computer would need roughly the same number of operations—1,000,000—to reach a solution. In terms of complexity, we’ve improved performance by several orders of magnitude, going from 1 quadrillion (1 x 10¹⁸) operations to just 1 million (1 x 10⁶). This in a real world scenario would mean the difference between something that can be resolved and something that cannot be resolved, running in seconds, minutes or even hours, vs something that might take years to compute.

Quiz: Which Snippet Is More Efficient?

Here’s a quick quiz to test your understanding. Which of these code snippets is more efficient? Take your time to choose the right solution.

Example A

Example B

Example C

The correct answer is Example A, which runs in O(n) time. Example B, which is an implementation of Bubble Sort, runs in O(n²), and Example C runs in O(n³), making them less efficient.

Efficiency Matters, But Context Is Key

While it’s important to understand and consider algorithm complexity, it’s equally important to know when to apply this knowledge. In day-to-day development, many algorithms you write will be trivial, and the focus should be on clarity and maintainability.

However, for critical paths in your code—especially in performance-sensitive applications like gaming or real-time processing—algorithmic efficiency can make a significant difference. At Zarego, we prioritize writing efficient, clean code in everything we do. If you’re a developer who cares about code efficiency and would like to be part of a team that values these practices, consider joining our team at Zarego. Explore career opportunities with us.

The key takeaway? Be mindful of complexity, optimize where it counts, and always strive for efficient, maintainable code.

Bibliography

• The Art of Computer Programming (Donald Knuth, ISBN 0137935102)
• Algorithms in C, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (Robert Sedgewick, ISBN 0201756080)
• C Programming Language (Kernighan and Ritchie, ISBN 0131103628)