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---
# Chapter 2.   Complexity analysis
# Chapter 2.   Complexity Analysis
![Complexity analysis](../assets/covers/chapter_complexity_analysis.jpg){ class="cover-image" }
!!! abstract
Complexity analysis is like a space-time navigator in the vast universe of algorithms.
Complexity analysis is like a space-time guide in the vast universe of algorithms.
It guides us in exploring deeper within the dimensions of time and space, seeking more elegant solutions.
It leads us to explore deeply within the two dimensions of time and space, seeking more elegant solutions.
## Chapter contents
- [2.1   Algorithm efficiency assessment](performance_evaluation.md)
- [2.2   Iteration and recursion](iteration_and_recursion.md)
- [2.3   Time complexity](time_complexity.md)
- [2.4   Space complexity](space_complexity.md)
- [2.1   Algorithm Efficiency Evaluation](performance_evaluation.md)
- [2.2   Iteration and Recursion](iteration_and_recursion.md)
- [2.3   Time Complexity](time_complexity.md)
- [2.4   Space Complexity](space_complexity.md)
- [2.5   Summary](summary.md)
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# 2.1   Algorithm efficiency assessment
# 2.1   Algorithm Efficiency Evaluation
In algorithm design, we pursue the following two objectives in sequence.
In algorithm design, we pursue the following two levels of objectives sequentially.
1. **Finding a Solution to the Problem**: The algorithm should reliably find the correct solution within the specified range of inputs.
2. **Seeking the Optimal Solution**: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.
1. **Finding a solution to the problem**: The algorithm must reliably obtain the correct solution within the specified input range.
2. **Seeking the optimal solution**: Multiple solutions may exist for the same problem, and we hope to find an algorithm that is as efficient as possible.
In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating an algorithm, which includes the following two dimensions.
In other words, under the premise of being able to solve the problem, algorithm efficiency has become the primary evaluation criterion for measuring the quality of algorithms. It includes the following two dimensions.
- **Time efficiency**: The speed at which an algorithm runs.
- **Space efficiency**: The size of the memory space occupied by an algorithm.
- **Time efficiency**: The length of time the algorithm runs.
- **Space efficiency**: The size of memory space the algorithm occupies.
In short, **our goal is to design data structures and algorithms that are both fast and memory-efficient**. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.
In short, **our goal is to design data structures and algorithms that are "both fast and memory-efficient"**. Effectively evaluating algorithm efficiency is crucial, because only in this way can we compare various algorithms and guide the algorithm design and optimization process.
There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.
Efficiency evaluation methods are mainly divided into two types: actual testing and theoretical estimation.
## 2.1.1   Actual testing
## 2.1.1   Actual Testing
Suppose we have algorithms `A` and `B`, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms, monitor and record their runtime and memory usage. This assessment method reflects the actual situation, but it has significant limitations.
Suppose we now have algorithm `A` and algorithm `B`, both of which can solve the same problem, and we need to compare the efficiency of these two algorithms. The most direct method is to find a computer, run these two algorithms, and monitor and record their running time and memory usage. This evaluation approach can reflect the real situation, but it also has considerable limitations.
On one hand, **it's difficult to eliminate interference from the testing environment**. Hardware configurations can affect algorithm performance. For example, an algorithm with a high degree of parallelism is better suited for running on multi-core CPUs, while an algorithm that involves intensive memory operations performs better with high-performance memory. The test results of an algorithm may vary across different machines. This means testing across multiple machines to calculate average efficiency becomes impractical.
On one hand, **it is difficult to eliminate interference factors from the testing environment**. Hardware configuration affects the performance of algorithms. For example, if an algorithm has a high degree of parallelism, it is more suitable for running on multi-core CPUs; if an algorithm has intensive memory operations, it will perform better on high-performance memory. In other words, the test results of an algorithm on different machines may be inconsistent. This means we need to test on various machines and calculate average efficiency, which is impractical.
On the other hand, **conducting a full test is very resource-intensive**. Algorithm efficiency varies with input data size. For example, with smaller data volumes, algorithm `A` might run faster than `B`, but with larger data volumes, the test results may be the opposite. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires excessive computational resources.
On the other hand, **conducting complete testing is very resource-intensive**. As the input data volume changes, the algorithm will exhibit different efficiencies. For example, when the input data volume is small, the running time of algorithm `A` is shorter than algorithm `B`; but when the input data volume is large, the test results may be exactly the opposite. Therefore, to obtain convincing conclusions, we need to test input data of various scales, which requires a large amount of computational resources.
## 2.1.2   Theoretical estimation
## 2.1.2   Theoretical Estimation
Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as <u>asymptotic complexity analysis</u>, or simply <u>complexity analysis</u>.
Since actual testing has considerable limitations, we can consider evaluating algorithm efficiency through calculations alone. This estimation method is called <u>asymptotic complexity analysis</u>, or <u>complexity analysis</u> for short.
Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. **It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases**. This definition might sound complex, but we can break it down into three key points to understand it better.
Complexity analysis can reflect the relationship between the time and space resources required for algorithm execution and the input data scale. **It describes the growth trend of the time and space required for algorithm execution as the input data scale increases**. This definition is somewhat convoluted, so we can break it down into three key points to understand.
- "Time and space resources" correspond to <u>time complexity</u> and <u>space complexity</u>, respectively.
- "As the size of input data increases" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.
- "The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied, but on the "rate" at which time or space increases.
- "As the input data scale increases" means that complexity reflects the relationship between algorithm running efficiency and input data scale.
- "Growth trend of time and space" indicates that complexity analysis focuses not on the specific values of running time or occupied space, but on how "fast" time or space grows.
**Complexity analysis overcomes the disadvantages of actual testing methods**, reflected in the following aspects:
**Complexity analysis overcomes the drawbacks of the actual testing method**, reflected in the following aspects.
- It does not require actually running the code, making it more environmentally friendly and energy efficient.
- It is independent of the testing environment and applicable to all operating platforms.
- It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.
- It does not need to actually run the code, making it more environmentally friendly and energy-efficient.
- It is independent of the testing environment, and the analysis results are applicable to all running platforms.
- It can reflect algorithm efficiency at different data volumes, especially algorithm performance at large data volumes.
!!! tip
If you're still confused about the concept of complexity, don't worry. We will cover it in detail in subsequent chapters.
If you are still confused about the concept of complexity, don't worry—we will introduce it in detail in subsequent chapters.
Complexity analysis provides us with a "ruler" to evaluate the efficiency of an algorithm, enabling us to measure the time and space resources required to execute it and compare the efficiency of different algorithms.
Complexity analysis provides us with a "ruler" for evaluating algorithm efficiency, allowing us to measure the time and space resources required to execute a certain algorithm and compare the efficiency between different algorithms.
Complexity is a mathematical concept that might be abstract and challenging for beginners. From this perspective, complexity analysis might not be the most suitable topic to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.
Complexity is a mathematical concept that may be relatively abstract for beginners, with a relatively high learning difficulty. From this perspective, complexity analysis may not be very suitable as the first content to be introduced. However, when we discuss the characteristics of a certain data structure or algorithm, it is difficult to avoid analyzing its running speed and space usage.
In summary, it is recommended to develop a basic understanding of complexity analysis before diving deep into data structures and algorithms, **so that you can perform complexity analysis on simple algorithms**.
In summary, it is recommended that before diving deep into data structures and algorithms, **you first establish a preliminary understanding of complexity analysis so that you can complete complexity analysis of simple algorithms**.
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# 2.5 &nbsp; Summary
### 1. &nbsp; Key review
### 1. &nbsp; Key Review
**Algorithm Efficiency Assessment**
- Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.
- We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources.
- Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.
- Time efficiency and space efficiency are the two primary evaluation metrics for measuring algorithm performance.
- We can evaluate algorithm efficiency through actual testing, but it is difficult to eliminate the influence of the testing environment, and it consumes substantial computational resources.
- Complexity analysis can eliminate the drawbacks of actual testing, with results applicable to all running platforms, and it can reveal algorithm efficiency under different data scales.
**Time Complexity**
- Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.
- Worst-case time complexity is denoted using big-$O$ notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations $T(n)$ as $n$ approaches infinity.
- Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.
- Common time complexities, arranged from low to high, include $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$, among others.
- The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.
- Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.
- Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but may fail in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.
- Worst-case time complexity is represented using Big $O$ notation, corresponding to the asymptotic upper bound of a function, reflecting the growth level of the number of operations $T(n)$ as $n$ approaches positive infinity.
- Deriving time complexity involves two steps: first, counting the number of operations, then determining the asymptotic upper bound.
- Common time complexities arranged from low to high include $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$.
- The time complexity of some algorithms is not fixed, but rather depends on the distribution of input data. Time complexity is divided into worst-case, best-case, and average-case time complexity. Best-case time complexity is rarely used because input data generally needs to satisfy strict conditions to achieve the best case.
- Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires statistical analysis of input data distribution and the combined mathematical expectation.
**Space Complexity**
- Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume.
- The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.
- We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation.
- Common space complexities, arranged from low to high, include $O(1)$, $O(\log n)$, $O(n)$, $O(n^2)$, and $O(2^n)$, among others.
- Space complexity serves a similar purpose to time complexity, used to measure the trend of algorithm memory usage as data volume increases.
- The memory space related to algorithm execution can be divided into input space, temporary space, and output space. Typically, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.
- We typically only focus on worst-case space complexity, which is the space complexity of an algorithm under worst-case input data and worst-case runtime.
- Common space complexities arranged from low to high include $O(1)$, $O(\log n)$, $O(n)$, $O(n^2)$, and $O(2^n)$.
### 2. &nbsp; Q & A
**Q**: Is the space complexity of tail recursion $O(1)$?
Theoretically, the space complexity of a tail-recursive function can be optimized to $O(1)$. However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of $O(n)$.
Theoretically, the space complexity of tail recursive functions can be optimized to $O(1)$. However, most programming languages (such as Java, Python, C++, Go, C#, etc.) do not support automatic tail recursion optimization, so the space complexity is generally considered to be $O(n)$.
**Q**: What is the difference between the terms "function" and "method"?
**Q**: What is the difference between the terms function and method?
A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.
A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object, is implicitly passed to the object that invokes it, and can operate on data contained in class instances.
Here are some examples from common programming languages:
The following examples use several common programming languages for illustration.
- C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.
- C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with structures are equivalent to methods in other programming languages.
- Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.
- C++ and Python support both procedural programming (functions) and object-oriented programming (methods).
**Q**: Does the "Common Types of Space Complexity" figure reflect the absolute size of occupied space?
**Q**: Does the diagram for "common space complexity types" reflect the absolute size of occupied space?
No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.
No, the diagram shows space complexity, which reflects growth trends rather than the absolute size of occupied space.
If you take $n = 8$, you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.
Assuming $n = 8$, you might find that the values of each curve do not correspond to the functions. This is because each curve contains a constant term used to compress the value range into a visually comfortable range.
In practice, since we usually don't know the "constant term" complexity of each method, it's generally not possible to choose the best solution for $n = 8$ based solely on complexity. However, for $n = 8^5$, it's much easier to choose, as the growth trend becomes dominant.
In practice, because we generally do not know what the "constant term" complexity of each method is, we usually cannot select the optimal solution for $n = 8$ based on complexity alone. But for $n = 8^5$, the choice is straightforward, as the growth trend already dominates.
**Q**: Are there situations where algorithms are designed to sacrifice time (or space) based on actual use cases?
In practical applications, most situations choose to sacrifice space for time. For example, with database indexes, we typically choose to build B+ trees or hash indexes, occupying substantial memory space in exchange for efficient queries of $O(\log n)$ or even $O(1)$.
In scenarios where space resources are precious, time may be sacrificed for space. For example, in embedded development, device memory is precious, and engineers may forgo using hash tables and choose to use array sequential search to save memory usage, at the cost of slower searches.
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