# Linear Time Sorting

## Introduction

Sorting is an essential operation in computer science that involves arranging elements into a specific order, such as numerical or alphabetical order. Various sorting algorithms have been developed, each with time and efficiency indicators. Linear time sorting is a subset of sorting algorithms with a significant advantage: they can sort a given set of elements in linear time, the runtime increases linearly with the input size.

The best-known linear time sorting algorithm is descending sort. Computational sorting is particularly efficient when the range of input elements is known and relatively small. This eliminates the need to compare elements, the main time-consuming operation in many other sorting algorithms. Using input-domain knowledge, computational sorting achieves linear time complexity. A numeric sort first scans the input array to determine the count of each element. It then uses these numbers to calculate the correct positions of the elements in the ordered result table. The algorithm consists of the following steps:

1. To determine the range, identify the minimum and maximum values of the input array.
2. Create a worksheet initialized with the range size and zeros.
3. Iterate over the input array and increment each element found.
4. Modify the worksheet by calculating the cumulative total to obtain the correct positions for each element.
5. Create an output array the same size as the input array.
6. Move the input array again, placing each element in the correct position in the output array based on the worksheet.
7. The result table now contains sorted elements.

The main advantage of descending sort is that it achieves a linear time complexity of O(n), which makes it very efficient for large input sizes. However, its applicability is limited to scenarios where the choice of input elements is known in advance and relatively small.

It is important to note that other sorting algorithms, such as quicksort or merge, typically have a time complexity of O(n log n), which is considered efficient for many practical applications. Linear time sorting algorithms, such as numerical sorting, provide an alternative when certain constraints or properties of the input allow linear time complexity to be used.

## History

Linear time sorting algorithms have a rich history in computer science. The development of linear time order can be traced back to the middle of the 20th century, and the contributions of scientists and mathematicians were significant. One of the earliest linear time sorting algorithms is the bucket sort, proposed by Harold H. Seward in 1954. A bucket sort divides the input elements into a finite number of buckets and then sorts each bucket separately. This algorithm has linear time complexity if the distribution of input elements is relatively uniform.

In 1959, Kenneth E. Iverson introduced a radix algorithm that achieves linear time complexity. Radix sorts elements by their numbers or signs from least significant to most significant. It uses robust sorting algorithms, such as numeric or bucket sort, to sort the elements at each digit location. Radix sorting became popular in the era of punch cards and early computer systems. However, the best-known linear time sorting algorithm is an enumeration, introduced by Harold H. Seward and Peter Elias in 1954 and later independently rediscovered by Harold H. "Bobby" Johnson in 1961. Numerical sorting has received considerable attention.

This is particularly effective when the range of input elements is known and relatively small. The history of linear time sorting continued with the development of other specialized algorithms. For example, 1987, Hanan Samet proposed binary distribution tree sorting, a linear time sorting algorithm for multidimensional data. Over the years, researchers have continued to study and improve linear scheduling algorithms, focusing on specific scenarios and constraints. Although algorithms such as quicksort and merge are more widely used for their efficiency in more scenarios, linear-time sorting algorithms provide valuable alternatives when certain circumstances allow the linear-time complexity to be exploited. In general, the history of linear time sorting is characterized by searching for efficient algorithms that can sort large data sets in linear time, overcoming the limitations of comparison-based sorting algorithms. The contributions of various researchers paved the way for developing and understanding these specialized sorting techniques.

## Types of Linear Time Sorting

There are several different linear time sorting algorithms. The two main types are count-based algorithms and radix-based algorithms. Here are the most common linear time sorting algorithms, classified based on the following types:

### Counting-Based Algorithms

1. Counting-Based Sort:Counting-Based is a non-comparative sorting algorithm. It counts the occurrence of each particular element in the input array and uses this information to determine the correct position of each element in the sorted output array. Counting-Based sort assumes that the input elements are integers or can be added to integers.

### Radix-Based Algorithms

1. Radix Sort: Radix Sort is a non-comparison-basedsorting algorithm that sorts elements by their numbers or characters. It counts each number or sign in the elements from the least significant number to the most significant Radical sorting assumes that the input elements are integers or strings.
2. Bucket Sort:Bucket Sort is a variant of Radix Sort that divides elements into fixed groups based on their range or distribution. Each segment is sorted separately using a different sorting algorithm or recursively bin-sort.
3. MSD (Most Significant Digit) Radix Sort: MSD Radix Sort is a variantof radix sort that starts sorting elements based on their most significant It recursively divides the elements into subgroups based on the value of the current number and applies MSD Radix Sort to each subgroup until all the numbers have been counted.
4. LSD (Least Significant Digit) Radix Sort: LSD Radix Sort is another variantthat starts sorting elements based on their least significant It recursively sorts the elements based on each number from rightmost to leftmost, producing a sorted result. Both count-based and root-based sorting algorithms achieve linear time complexity by exploiting specific properties of the input elements, such as their range or representational structure (e.g., numbers or characters). However, their applicability may vary depending on the characteristics of the input data.

## Advantages of linear time sorting

Linear time sorting algorithms, such as numerical sorting, offer several advantages in specific scenarios.

1. Efficient for large input sizes:The time complexity of linear time sorting algorithms is O(n), which means that the running time increases linearly with the input size. This makes them very efficient for large data sets compared to comparison-based sorting algorithms such as quicksort or merge algorithms, which typically have a time complexity of O(n log n).
2. No comparison operations:Linear-time sorting algorithms, such as enumeration sort, do not rely on elementary comparison Instead, they use specific attributes or information about the input elements, such as their extent or distribution. This feature makes them advantageous when the cost of comparison is high, such as for complex objects or expensive comparison operations.
3. Suitability to specific input properties: Linear-time sorting algorithms often have specific requirements or assumptions about the input elements. For example, to calculate a sort order, you need to know the range of input elements in advance. When these conditions are met, linear time sorting algorithms can offer significant performance advantages over general sorting algorithms.
4. Stable sort:Many linear-time sorting algorithms, including numerical and radix sort, are inherently stable. Consistency means elements with duplicate keys or values maintain relative order in the sorted output. This can be critical when sorting objects or records with multiple attributes or when preserving the original order of elements of equal value is essential.
5. Ease of use: Linear-timesorting algorithms such as enumeration sorting are often relatively easy to implement compared to more complex comparison-based sorting algorithms. They can be easier to understand and debug, making them suitable for situations where simplicity and clarity are desired.

## Disadvantages of linear time sorting

Although linear scheduling algorithms have their advantages, they also have certain limitations and disadvantages:

1. Constraining input requirements: Linear time sorting algorithms often have specific requirements or assumptions about the input elements. For example, to calculate a sort order, you need to know the range of input elements in advance. This restriction limits their applicability to situations where these conditions are met. Memory requirements may become impractical or exceed available resources if the range is extensive or unknown.
2. Additional space requirements: Some linear time sorting algorithms, such as numerical sort, require additional space to store other arrays or data structures. The space required is often proportional to the number of input elements. This can be a disadvantage when memory usage is an issue, especially when dealing with large data sets or limited memory resources.
3. Lack of Versatility:Linear time sorting algorithms are specialized algorithms designed for specific scenarios or constraints. They may need to be more suitable and efficient for general sorting tasks or different input distributions. Comparison-based sorting algorithms such as quicksort or merge are more versatile and can handle a broader range of input range.
4. Inefficient for small ranges or sparse data:Linear-time sorting algorithms such as enumeration are most efficient when the range of input elements is small and densely distributed. If the range is extensive or the data is sparse (i.e., only a few distinct values), the algorithm may save time and effort processing empty or sparsely populated portions of the input range.
5. Limited to specificdata types: Linear-time sorting algorithms, such as enumeration sort, are primarily designed to sort non-negative integers or key-value objects. They may not be suitable for sorting other data types, such as floating-point numbers, strings, or complex data structures. Adapting linear time sorting algorithms to handle different data types or custom comparison functions may require additional preprocessing or modifications.

When choosing a sorting algorithm, it is essential to carefully consider the input data's specifics and the sorting problem's requirements. While linear scheduling algorithms offer advantages in specific scenarios, they are only sometimes the most appropriate or efficient choice.

## Applications of Linear time sorting algorithms

Linear time sorting algorithms are efficient and have many applications in various fields. Here are some typical applications of linear time order:

1. Sorting Small Range Integers:Linear time sorting algorithms such as count sort and radix sort is ideal for sorting arrays of integers when the range of values is These algorithms achieve linear time complexity by making assumptions about the input data, allowing them to bypass comparison-based sorting.
2. String sorting:Linear time sorting algorithms can also be applied to sort strings efficiently. By taking unique properties of strings, such as their length or characters, algorithms like Radix Sort can achieve linear time complexity when sorting strings.
3. Database Functions:Sorting is an essential function of Linear time sorting algorithms can efficiently sort large data sets based on specific columns or fields. This enables faster query processing and better performance in database operations.
4. Creating Histograms:Histograms are essential for various statistical and data analysis tasks. Linear time sorting algorithms, such as numerical sorting, can generate histograms by efficiently counting the occurrences of elements in a dataset.
5. External sorting: The external sorting technique is used in scenarios where the data cannot fit entirelyin memory. Linear time sorting algorithms such as External Radix Sort or External Counting Sort can efficiently sort large data sets stored on disk or other external storage devices.
6. Event Scheduling:Linear time sorting algorithms can schedule events based on their start or end times. Sorting events in ascending order makes identifying conflicts, overlapping periods, or finding the next available period easy.
7. Analyzing log files:Analyzing log files is a common task in system administration and debugging. Linear time sorting algorithms can be used to sort logs based on timestamps, making it easier to identify patterns, anomalies or search for specific events.
8. Data Compression:Sorting plays an essential role in various data compression techniques. Algorithms such as Burrows-Wheeler Transform (BWT) or Move-To-Front Transform (MTF) rely on linear time ordering to rearrange data to improve compression efficiency. These are just a few examples of applications of linear time sorting algorithms.

## Implementation of Linear Time Sorting in C++

Here's an example of a program implementing Counting Sort, which is a linear time sorting algorithm:

Sample Output

```Sorted array: 1 2 2 3 3 4 8
```

This indicates that the input array has been sorted in ascending order using the Counting Sort algorithm, resulting in the sorted array [1, 2, 2, 3, 3, 4, 8].

In this C++ program, the counting sort function takes a reference to the vector arr and runs the counting sort routine. It finds the table's maximum value to determine the worksheet's size. It then counts each element's occurrence and calculates the worksheet's prefix sum. Then, it creates a result vector and puts the elements in order according to the worksheet. Finally, it copies the sorted elements back into the original array. In the primary function, the example array {4, 2, 2, 8, 3, 3, 1} is sorted by the enumeration sort algorithm and printed as a sorted matrix. Note that the program uses libraries to work with vectors and find the maximum element of an array using the max_element function.

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