# What is Block Sort

**Block sort**is a family of sorting algorithms that operate on fixed-size blocks of elements rather than individual elements. In other words, instead of comparing and sorting individual elements, block sort algorithms compare and sort entire blocks of elements.

One popular block sort algorithm is called "merge sort". In merge sort, the list of elements to be sorted is recursively divided into smaller sub-lists until each sub-list consists of a single element. Then, these sub-lists are merged together in pairs, with each pair being sorted in the process. This process is repeated until the entire list is sorted.

Another example of a block sort algorithm is "quick sort with median-of-three partitioning", which works by selecting a "pivot" element from the block of elements and partitioning the block into two sub-blocks - one containing all elements less than the pivot, and the other containing all elements greater than or equal to the pivot. This process is then recursively applied to the two sub-blocks until the entire block is sorted.

Block sort algorithms are often used in situations where sorting large amounts of data is required, such as in database systems or data analysis applications.

## Who invented it?

There isn't a single person credited with inventing the block sort algorithm, as it is a general approach to sorting that has been developed and refined by many researchers and computer scientists over the years.

One early example of a block sort algorithm is the "tape sort" algorithm, which was developed in the 1950s for sorting data stored on magnetic tape. Tape sort operated on fixed-size blocks of data, rather than individual records, and was an early example of the block sort approach.

Other important contributions to the development of block sort algorithms include the work of Donald Knuth, who introduced the "Algorithm M" block sort algorithm in his book "The Art of Computer Programming", and the work of Robert Sedgewick, who developed several block sort algorithms for use in his book "Algorithms in C".

Today, block sort algorithms continue to be an important tool in computer science and are used in a wide range of applications, from sorting large data sets to implementing compression algorithms and other data processing tasks.

One early example of a block sort algorithm is the "tape sort" algorithm, which was developed in the 1950s for sorting data stored on magnetic tape. Tape sort operated on fixed-size blocks of data, rather than individual records, and was an early example of the block sort approach.

Other important contributions to the development of block sort algorithms include the work of Donald Knuth, who introduced the "Algorithm M" block sort algorithm in his book "The Art of Computer Programming", and the work of Robert Sedgewick, who developed several block sort algorithms for use in his book "Algorithms in C".

Today, block sort algorithms continue to be an important tool in computer science and are used in a wide range of applications, from sorting large data sets to implementing compression algorithms and other data processing tasks.

## Pseudocode

```
function block_sort(list, block_size):
// Divide the list into blocks of size `block_size`
blocks = []
for i = 0 to length of list, increment by `block_size`:
block = list[i:i+block_size]
blocks.append(block)
// Sort each block using a comparison-based sorting algorithm (e.g. quicksort)
for i = 0 to length of blocks:
sort(blocks[i])
// Merge the sorted blocks back together into a single sorted list
result = []
while length of blocks > 0:
min_block = find_minimum_block(blocks)
result.append(min_block.pop(0))
if length of min_block > 0:
blocks.add(min_block)
else:
remove min_block from blocks
return result
function find_minimum_block(blocks):
min_block = blocks[0]
for block in blocks:
if block[0] < min_block[0]:
min_block = block
return min_block
```

This implementation divides the input list into blocks of size block_size, sorts each block using a comparison-based sorting algorithm (e.g. quicksort), and then merges the sorted blocks back together into a single sorted list. The find_minimum_block() function is used to identify the next smallest element to add to the sorted list, which is determined by comparing the first element of each block.

## Sample Code

```
// C++ code snippet
vector
``` block_sort(vector list, int block_size) {
// Divide the list into blocks of size `block_size`
vector> blocks;
for (int i = 0; i < list.size(); i += block_size) {
vector block(list.begin() + i, list.begin() + i + block_size);
blocks.push_back(block);
}
// Sort each block using a comparison-based sorting algorithm (e.g. quicksort)
for (int i = 0; i < blocks.size(); i++) {
sort(blocks[i].begin(), blocks[i].end());
}
// Merge the sorted blocks back together into a single sorted list
vector result;
while (blocks.size() > 0) {
vector min_block = blocks[0];
int min_index = 0;
for (int i = 1; i < blocks.size(); i++) {
if (blocks[i][0] < min_block[0]) {
min_block = blocks[i];
min_index = i;
}
}
result.push_back(min_block[0]);
min_block.erase(min_block.begin());
if (min_block.size() > 0) {
blocks[min_index] = min_block;
} else {
blocks.erase(blocks.begin() + min_index);
}
}
return result;
}
int main() {
vector list = { 5, 3, 8, 4, 2, 7, 1, 6 };
int block_size = 3;
vector sorted_list = block_sort(list, block_size);
// Print the sorted list
for (int i = 0; i < sorted_list.size(); i++) {
cout << sorted_list[i] << " ";
}
cout << endl;
return 0;
}

```
# Python code snippet
def block_sort(lst, block_size):
# Divide the list into blocks of size `block_size`
blocks = [lst[i:i+block_size] for i in range(0, len(lst), block_size)]
# Sort each block using a comparison-based sorting algorithm (e.g. quicksort)
for i in range(len(blocks)):
blocks[i].sort()
# Merge the sorted blocks back together into a single sorted list
result = []
while len(blocks) > 0:
min_block = blocks[0]
min_index = 0
for i in range(1, len(blocks)):
if blocks[i][0] < min_block[0]:
min_block = blocks[i]
min_index = i
result.append(min_block.pop(0))
if len(min_block) > 0:
blocks[min_index] = min_block
else:
blocks.pop(min_index)
return result
# Example usage
lst = [5, 3, 8, 4, 2, 7, 1, 6]
block_size = 3
sorted_lst = block_sort(lst, block_size)
print(sorted_lst)
```

```
import java.util.*;
public class BlockSort {
public static List
``` blockSort(List list, int blockSize) {
// Divide the list into blocks of size `blockSize`
List> blocks = new ArrayList<>();
for (int i = 0; i < list.size(); i += blockSize) {
List block = list.subList(i, Math.min(i + blockSize, list.size()));
blocks.add(block);
}
// Sort each block using a comparison-based sorting algorithm (e.g. quicksort)
for (int i = 0; i < blocks.size(); i++) {
Collections.sort(blocks.get(i));
}
// Merge the sorted blocks back together into a single sorted list
List result = new ArrayList<>();
while (blocks.size() > 0) {
List minBlock = blocks.get(0);
int minIndex = 0;
for (int i = 1; i < blocks.size(); i++) {
if (blocks.get(i).get(0) < minBlock.get(0)) {
minBlock = blocks.get(i);
minIndex = i;
}
}
result.add(minBlock.get(0));
minBlock.remove(0);
if (minBlock.size() > 0) {
blocks.set(minIndex, minBlock);
} else {
blocks.remove(minIndex);
}
}
return result;
}
public static void main(String[] args) {
List list = Arrays.asList(5, 3, 8, 4, 2, 7, 1, 6);
int blockSize = 3;
List sortedList = blockSort(list, blockSize);
// Print the sorted list
System.out.println(sortedList);
}
}

## Time and Space Complexity

- The time complexity of block sort algorithm is O(n log n), where n is the size of the input list. This is because the algorithm first divides the input list into blocks of size B, where B is a constant factor, which takes O(n/B) time. Then each block is sorted using a comparison-based sorting algorithm, such as quicksort, which has an average time complexity of O(B log B). Since there are n/B blocks, the total time complexity of sorting all the blocks is O((n/B) * B log B) = O(n log B). Finally, the sorted blocks are merged using a simple linear scan of the blocks, which takes O(n) time. Therefore, the overall time complexity of block sort is O(n log n).
- The space complexity of block sort is O(n + B), where n is the size of the input list and B is the block size. The space complexity is dominated by the space required to store the sorted blocks in memory, which is O(n + B) in the worst case. The worst case occurs when all the blocks have exactly B elements, which requires additional space to store the partially-filled block at the end of the list. The space complexity of the algorithm is generally considered to be good, especially when the block size is small compared to the size of the input list.

## Advantages

- Memory efficiency: Block sort algorithm uses a fixed-size block to sort the input list, which reduces the amount of memory required to perform the sort compared to some other algorithms like merge sort and quicksort that require extra memory for their operations.
- Good performance on large datasets: Block sort can be faster than some other sorting algorithms, such as quicksort or heapsort, on large datasets with a large number of elements.

## Disadvantages

- Sensitive to block size: The performance of block sort is highly dependent on the choice of block size. If the block size is too small, the algorithm may become inefficient because of the overhead of sorting small blocks. If the block size is too large, the algorithm may also become inefficient because the blocks may be too large to fit in memory.
- Not a stable sort: Block sort is not a stable sort, which means that the order of equal elements in the input list may not be preserved in the sorted output list.
- Worst-case time complexity: The worst-case time complexity of block sort is O(n^2), which occurs when the input list is already sorted or nearly sorted. This is because the blocks may contain many elements that are already in sorted order, which leads to inefficient comparisons and merging.