Algorithm Efficiency Evaluation
In algorithm design, we pursue the following two objectives:
- Finding a Solution to the Problem: The algorithm must reliably find the correct solution to the problem within the specified input range.
- Seeking the Optimal Solution: There may be multiple solutions to the same problem, and we aim to find the most efficient algorithm.
In other words, once the problem can be solved, the efficiency of the algorithm becomes the main criterion for evaluating its quality. This efficiency is measured in two dimensions:
- Time Efficiency: The amount of time the algorithm takes to run.
- Space Efficiency: The amount of memory space the algorithm occupies.
In summary, our goal is to design data structures and algorithms that are both fast and space-efficient. Effectively evaluating algorithm efficiency is crucial because it allows us to compare different algorithms, guiding the design and optimization process.
There are two main methods for evaluating efficiency: empirical testing and theoretical estimation.
Empirical Testing
Suppose we have algorithms A and B, both of which solve the same problem, and we need to compare the efficiency of these two algorithms. The most direct method is to run these two algorithms on a computer and monitor their execution time and memory usage. This evaluation method reflects the real-world situation but also has significant limitations.
On one hand, it’s difficult to eliminate interference from the testing environment. Hardware configurations can affect an algorithm’s performance. For example, an algorithm with high parallelism is better suited to run on a multi-core CPU, while an algorithm with intensive memory operations will perform better with high-performance memory. In other words, test results may vary across different machines. This means we would need to test on various machines and average the results, which is unrealistic.
On the other hand, conducting comprehensive tests is resource-intensive. As the input data size changes, the efficiency of the algorithm may vary. For instance, algorithm A might have a shorter runtime than algorithm B with small input data; however, with larger input data, the results could be the opposite. Therefore, to draw convincing conclusions, we would need to test various sizes of input data, which requires significant computational resources.
Theoretical Estimation
Due to the limitations of empirical testing, we can instead consider evaluating algorithm efficiency through calculations. This estimation method is known as asymptotic complexity analysis, or simply complexity analysis.
Complexity analysis reflects the relationship between the resources (time and space) required to run the algorithm and the size of the input data. It describes the growth trend of the time and space required for an algorithm to execute as the input data size increases. This definition may seem convoluted, so let’s break it down into three key points:
- “Time and space resources” correspond to time complexity and space complexity.
- “As the input data size increases” means that complexity reflects the relationship between the algorithm’s efficiency and the size of the input data.
- “Growth trend of time and space” indicates that complexity analysis focuses not on the specific values of the runtime or space usage but on the “rate of increase” of time or space.
Complexity analysis overcomes the drawbacks of empirical testing, as seen in the following aspects:
- It does not require actual code execution, making it more energy-efficient.
- It is independent of the testing environment, so the analysis results apply to all platforms.
- It can reflect the algorithm’s efficiency across different data sizes, especially the performance with large data volumes.
Complexity analysis provides us with a “yardstick” for evaluating algorithm efficiency, allowing us to assess the time and space resources required to execute an algorithm and compare the efficiency of different algorithms.
Complexity is a mathematical concept and can be somewhat abstract for beginners, making it relatively challenging to learn. From this perspective, complexity analysis may not be the best topic to introduce first. However, when discussing the characteristics of a data structure or algorithm, it’s almost unavoidable to analyze its speed and space usage.
In summary, it is recommended that before diving into the study of data structures and algorithms, you should first gain a preliminary understanding of complexity analysis to be able to perform basic complexity analysis of simple algorithms.
