## The Mastery of Computer Programming: Primary Algorithms - Sykalo Eugene 2023

# Average and worst case analysis

Analysis of Algorithms

## Asymptotic Notation

Asymptotic notation is a way of expressing the limiting behavior of a function as the input size approaches infinity. This is useful for analyzing the time and space complexity of algorithms, as it allows us to compare the efficiency of different algorithms without considering the specific details of their implementation.

### Big O Notation

Big O notation (O) is used to describe the upper bound of the growth rate of a function. In other words, it describes the worst-case scenario for the behavior of the function. For example, if we say that an algorithm has a time complexity of O(n), we mean that the worst-case running time of the algorithm grows no faster than a linear function of the input size n.

### Big Omega Notation

Big Omega notation (Ω) is used to describe the lower bound of the growth rate of a function. In other words, it describes the best-case scenario for the behavior of the function. For example, if we say that an algorithm has a time complexity of Ω(n^2), we mean that the best-case running time of the algorithm grows no slower than a quadratic function of the input size n.

### Big Theta Notation

Big Theta notation (Θ) is used to describe the tight bound of the growth rate of a function. In other words, it describes the behavior of the function within a constant factor. For example, if we say that an algorithm has a time complexity of Θ(n), we mean that the running time of the algorithm grows at the same rate as a linear function of the input size n, within a constant factor.

Asymptotic notation is useful because it allows us to compare the efficiency of different algorithms without considering the specific details of their implementation. By analyzing the asymptotic behavior of an algorithm, we can determine which algorithm is most efficient for a given problem, and we can estimate the time and space complexity of the algorithm for large input sizes.

## Average Case Analysis

Average case analysis involves calculating the expected complexity of an algorithm given a distribution of inputs. This is useful because the worst-case complexity of an algorithm may be much higher than its average-case complexity, and so by analyzing the average-case complexity we can get a better understanding of how the algorithm will perform in practice.

To calculate the average-case complexity of an algorithm, we need to know the probability distribution of inputs and the cost of executing the algorithm for each input. We can then calculate the expected cost of executing the algorithm by taking the weighted average of the costs for each input, using the probabilities as weights.

For example, suppose we have an algorithm that takes an array of n integers as input and returns the sum of the integers. The worst-case complexity of this algorithm is O(n), because we may need to add up all n integers in the array. However, the average-case complexity of the algorithm depends on the distribution of integers in the array. If the integers are uniformly distributed, then the expected value of each integer is (b+a)/2, where a and b are the smallest and largest integers in the array, respectively. Therefore, the expected cost of the algorithm is (n*(b+a)/2), which is still O(n).

It's important to note that calculating the average-case complexity of an algorithm can be difficult, because we may not know the probability distribution of inputs or the cost of executing the algorithm for each input. In practice, we often make simplifying assumptions or use statistical methods to estimate these values. Additionally, the average-case complexity may not always be a good predictor of performance, especially if the distribution of inputs is skewed or if the worst-case inputs are particularly common.

## Worst Case Analysis

Worst case analysis involves calculating the maximum complexity of an algorithm for any input of size n. This is useful because it gives us an upper bound on the running time of the algorithm, which is important for applications where performance is critical.

To calculate the worst-case complexity of an algorithm, we need to identify the input that causes the algorithm to take the longest amount of time to execute. This can be difficult in some cases, especially if the input is complex or the algorithm has many branches or loops. In general, we want to find the input that maximizes the number of operations performed by the algorithm.

For example, suppose we have an algorithm that takes an array of n integers as input and returns the maximum integer in the array. The worst-case complexity of this algorithm is O(n), because we may need to compare each integer in the array to the current maximum integer.

It's important to note that worst-case analysis can be overly pessimistic in some cases, especially if the worst-case input is unlikely to occur in practice. In these cases, it may be more appropriate to analyze the average-case complexity of the algorithm or to use a hybrid approach that takes both worst-case and average-case complexity into account.

## Examples

Average and worst case analysis can be applied to a wide range of problems in computer programming. Here are a few real-life examples:

**Sorting algorithms**: Sorting algorithms are a classic example of the importance of algorithmic efficiency. There are many different sorting algorithms, each with its own time and space complexity characteristics. By using average and worst case analysis, we can determine which sorting algorithm is most efficient for a given problem, and we can estimate the time and space complexity of the algorithm for large input sizes.**Database queries**: Database queries can involve complex operations on large datasets. By analyzing the average and worst case complexity of different query strategies, we can determine which strategy is most efficient for a given dataset and query.**Machine learning algorithms**: Machine learning algorithms often involve complex computations on large datasets. By analyzing the average and worst case complexity of different machine learning algorithms, we can determine which algorithm is most efficient for a given dataset and problem.**Network protocols**: Network protocols involve complex interactions between different devices and network layers. By analyzing the average and worst case complexity of different network protocols, we can determine which protocol is most efficient for a given network topology and traffic pattern.

## Limitations

While average and worst case analysis are useful for understanding the performance characteristics of algorithms, they do have some limitations.

One limitation is that they assume a uniform distribution of inputs, which may not always be the case in practice. In reality, the distribution of inputs may be skewed, with some inputs occurring much more frequently than others. In these cases, the average-case complexity may not be a good predictor of performance.

Another limitation is that they only consider the input size and do not take into account other factors that may affect performance, such as the architecture of the computer or the specifics of the implementation. For example, two algorithms with the same worst-case complexity may perform very differently on different hardware.

Additionally, worst-case analysis can be overly pessimistic in some cases, especially if the worst-case input is unlikely to occur in practice. In these cases, it may be more appropriate to analyze the average-case complexity of the algorithm or to use a hybrid approach that takes both worst-case and average-case complexity into account.

Finally, it's important to remember that algorithmic efficiency is not the only factor to consider when designing and implementing software. Other factors, such as maintainability, readability, and robustness, are also important and may sometimes be at odds with algorithmic efficiency.