Introduction to Block Sort

Block sort is a sorting algorithm that sorts an array by dividing it into blocks of fixed size, sorting each block individually, and then merging the sorted blocks back into a single sorted array. Block sort is a good choice for sorting large datasets that do not fit in memory. It can efficiently sort data in blocks that fit in memory, and then merge the sorted blocks together to obtain the final sorted array.

Approach: The block sort approach is as follows:

• Divide the input array into blocks of fixed size. 
• Sort each block using a comparison-based sorting algorithm (e.x., quicksort or mergesort).
• Merge the sorted blocks back into a single sorted array using a priority queue or min-heap.

Illustration:

Let the given array be: [1, 7, 8, 2, 3, 5, 4, 6]

• Let's choose a block size of 3. So we will divide the input into blocks of 3 integers each:
  • Block 1: [1, 7, 8]
  • Block 2: [2, 3, 5]
  • Block 3: [4, 6]
     
• Now we sort each block individually using a comparison-based sorting algorithm. Let's use quicksort for this example. Here are the sorted blocks:
  • Block 1: [1, 7, 8]
  • Block 2: [2, 3, 5]
  • Block 3: [4, 6]
     
• Now we merge the sorted blocks together. We create a min-heap that contains the first element from each block:
  min-heap: [1, 2, 4]
   
• We extract the minimum element from the heap, add it to the output array, and replace it with the next element from the block it came from. Continue until the heap is empty.

Iteration Details

1. Extract 1 (minimum) from the heap, add it to the output:
   • Output: [1]
   • Replace 1 with the next element 7 from Block 1.
   • Min-Heap: [2,7,4]
2. Extract 2 (minimum) from the heap, add it to the output:
   • Output: [1,2]
   • Replace 2 with the next element 3 from Block 2.
   • Min-Heap: [3,7,4]
3. Extract 3 (minimum) from the heap, add it to the output:
   • Output: [1,2,3]
   • Replace 3 with the next element 5 from Block 2.
   • Min-Heap: [4,7,5]
4. Extract 4 (minimum) from the heap, add it to the output:
   • Output: [1,2,3,4]
   • Replace 4 with the next element 6 from Block 3.
   • Min-Heap: [5,7,6]
5. Extract 5 (minimum) from the heap, add it to the output:
   • Output: [1,2,3,4,5]
   • Replace 5 with the next element none (Block 2 is exhausted).
   • Min-Heap: [6,7]
6. Extract 6 (minimum) from the heap, add it to the output:
   • Output: [1,2,3,4,5,6]
   • Replace 6 with the next element none (Block 3 is exhausted).
   • Min-Heap: [7]
7. Extract 7 (minimum) from the heap, add it to the output:
   • Output: [1,2,3,4,5,6,7]
   • Replace 7 with the next element 8 from Block 1.
   • Min-Heap: [8]
8. Extract 8 (minimum) from the heap, add it to the output:
   • Output: [1,2,3,4,5,6,7,8]
   • Replace 8 with the next element none (Block 1 is exhausted).
   • Min-Heap: []
     
     
   • Final Output

   The sorted array is: [1,2,3,4,5,6,7,8]

#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;

vector<int> blockSort(vector<int> arr, int blockSize)
{
    vector<vector<int> > blocks;

    // Divide the input array into blocks of size blockSize
    for (int i = 0; i < arr.size(); i += blockSize) {
        vector<int> block;

        for (int j = i; j < i + blockSize && j < arr.size();
             j++) {
            block.push_back(arr[j]);
        }

        // Sort each block and append it to the list of
        // sorted blocks
        sort(block.begin(), block.end());
        blocks.push_back(block);
    }

    // Merge the sorted blocks into a single sorted list
    vector<int> result;
    while (!blocks.empty()) {
        // Find the smallest element in the first block of
        // each sorted block
        int minIdx = 0;
        for (int i = 1; i < blocks.size(); i++) {
            if (blocks[i][0] < blocks[minIdx][0]) {
                minIdx = i;
            }
        }

        // Remove the smallest element and append it to the
        // result list
        result.push_back(blocks[minIdx][0]);
        blocks[minIdx].erase(blocks[minIdx].begin());

        // If the block is now empty, remove it from the
        // list of sorted blocks
        if (blocks[minIdx].empty()) {
            blocks.erase(blocks.begin() + minIdx);
        }
    }

    return result;
}

int main()
{
    // Original arr
    vector<int> arr = { 1, 7, 8, 2, 3, 5, 4, 6 };
    cout << "Input: ";
    for (int i = 0; i < arr.size(); i++) {
        cout << arr[i] << " ";
    }
    cout << endl;

    // Select box size
    int blockSize = 3;

    // Function call
    vector<int> sortedArr = blockSort(arr, blockSize);

    // Output the sorted array
    cout << "Output: ";
    for (int i = 0; i < sortedArr.size(); i++) {
        cout << sortedArr[i] << " ";
    }
    cout << endl;

    return 0;
}

// Java Implementation

import java.util.ArrayList;
import java.util.Collections;
import java.util.List;

public class GFG {
    public static List<Integer> blockSort(List<Integer> arr,
                                          int blockSize)
    {

        List<List<Integer> > blocks = new ArrayList<>();

        // Divide the input array into blocks of size
        // blockSize
        for (int i = 0; i < arr.size(); i += blockSize) {

            List<Integer> block = new ArrayList<>();

            for (int j = i;
                 j < i + blockSize && j < arr.size(); j++) {
                block.add(arr.get(j));
            }

            // Sort each block and append it to the list of
            // sorted blocks
            Collections.sort(block);
            blocks.add(block);
        }

        // Merge the sorted blocks into a single sorted list
        List<Integer> result = new ArrayList<>();
        while (!blocks.isEmpty()) {

            // Find the smallest element in the first block
            // of each sorted block
            int minIdx = 0;

            for (int i = 1; i < blocks.size(); i++) {
                if (blocks.get(i).get(0)
                    < blocks.get(minIdx).get(0)) {
                    minIdx = i;
                }
            }

            // Remove the smallest element and append it to
            // the result list
            result.add(blocks.get(minIdx).remove(0));

            // If the block is now empty, remove it from the
            // list of sorted blocks
            if (blocks.get(minIdx).isEmpty()) {
                blocks.remove(minIdx);
            }
        }

        return result;
    }

    // Driver code
    public static void main(String[] args)
    {

        // Original arr
        List<Integer> arr = new ArrayList<>();
        arr.add(1);
        arr.add(7);
        arr.add(8);
        arr.add(2);
        arr.add(3);
        arr.add(5);
        arr.add(4);
        arr.add(6);
        System.out.println("Input: " + arr);

        // Select box size
        int blockSize = 3;

        // Function call
        System.out.println("Output: "
                           + blockSort(arr, blockSize));
    }
}

# Python Implementation


def block_sort(arr, block_size):

    # Create an empty list to
    # hold the sorted blocks
    blocks = []

    # Divide the input array into
# blocks of size block_size

    for i in range(0, len(arr), block_size):

        block = arr[i:i + block_size]

        # Sort each block and append
        # it to the list of sorted blocks
        blocks.append(sorted(block))

    # Merge the sorted blocks into
    # a single sorted list
    result = []
    while blocks:

        # Find the smallest element in
        # the first block of
        # each sorted block
        min_idx = 0
        for i in range(1, len(blocks)):
            if blocks[i][0] < blocks[min_idx][0]:
                min_idx = i

        # Remove the smallest element and
        # append it to the result list
        result.append(blocks[min_idx].pop(0))

        # If the block is now empty, remove
        # it from the list of sorted blocks
        if len(blocks[min_idx]) == 0:
            blocks.pop(min_idx)
    return result


# Original arr
arr = [1, 7, 8, 2, 3, 5, 4, 6]
print('Input: ', arr)

# Select box size
block_size = 3

# Function call
print('Output:', block_sort(arr, block_size))

using System;
using System.Collections.Generic;
using System.Linq;

public class GFG
{
    static List<int> BlockSort(List<int> arr, int blockSize)
    {
        List<List<int>> blocks = new List<List<int>>();
              // Divide the input array into blocks of size blockSize
    for (int i = 0; i < arr.Count; i += blockSize)
    {
        List<int> block = new List<int>();

        for (int j = i; j < i + blockSize && j < arr.Count; j++)
        {
            block.Add(arr[j]);
        }

        // Sort each block and append it to the list of sorted blocks
        block.Sort();
        blocks.Add(block);
    }

    // Merge the sorted blocks into a single sorted list
    List<int> result = new List<int>();
    while (blocks.Any())
    {
        // Find the smallest element in the first block of each sorted block
        int minIdx = 0;
        for (int i = 1; i < blocks.Count; i++)
        {
            if (blocks[i][0] < blocks[minIdx][0])
            {
                minIdx = i;
            }
        }

        // Remove the smallest element and append it to the result list
        result.Add(blocks[minIdx][0]);
        blocks[minIdx].RemoveAt(0);

        // If the block is now empty, remove it from the list of sorted blocks
        if (!blocks[minIdx].Any())
        {
            blocks.RemoveAt(minIdx);
        }
    }

    return result;
    }

    static public void Main()
    {
        // Original arr
        List<int> arr = new List<int> { 1, 7, 8, 2, 3, 5, 4, 6 };
        Console.Write("Input: ");
        for (int i = 0; i < arr.Count; i++)
        {
            Console.Write(arr[i] + " ");
        }
        Console.WriteLine();

        // Select box size
        int blockSize = 3;

        // Function call
        List<int> sortedArr = BlockSort(arr, blockSize);

        // Output the sorted array
        Console.Write("Output: ");
        for (int i = 0; i < sortedArr.Count; i++)
        {
            Console.Write(sortedArr[i] + " ");
        }
        Console.WriteLine();
    }
}
// This code is contributed by Siddharth Aher

function blockSort(arr, blockSize) {
  let blocks = [];

  // Divide the input array into blocks of size blockSize
  for (let i = 0; i < arr.length; i += blockSize) {
    let block = [];

    for (let j = i; j < i + blockSize && j < arr.length; j++) {
      block.push(arr[j]);
    }

    // Sort each block and append it to the list of sorted blocks
    block.sort((a, b) => a - b);
    blocks.push(block);
  }

  // Merge the sorted blocks into a single sorted list
  let result = [];
  while (blocks.length > 0) {
    // Find the smallest element in the first block of each sorted block
    let minIdx = 0;
    for (let i = 1; i < blocks.length; i++) {
      if (blocks[i][0] < blocks[minIdx][0]) {
        minIdx = i;
      }
    }

    // Remove the smallest element and append it to the result list
    result.push(blocks[minIdx][0]);
    blocks[minIdx].shift();

    // If the block is now empty, remove it from the list of sorted blocks
    if (blocks[minIdx].length === 0) {
      blocks.splice(minIdx, 1);
    }
  }

  return result;
}

// Original arr
let arr = [1, 7, 8, 2, 3, 5, 4, 6];
console.log("Input: " + arr.join(" "));

// Select box size
let blockSize = 3;

// Function call
let sortedArr = blockSort(arr, blockSize);

// Output the sorted array
console.log("Output: " + sortedArr.join(" "));

Input:  [1, 7, 8, 2, 3, 5, 4, 6]
Output: [1, 2, 3, 4, 5, 6, 7, 8]

Time Complexity: O(n*logn)
Auxiliary Space: O(1).

Advantages of Block Sort:

• Block sort has a relatively good worst-case time complexity of O(n*logn).
• Block sort can be efficiently parallelized by sorting each block separately in parallel.
• Block sort can handle large datasets that do not fit in memory by sorting the data in blocks that fit in memory.

Disadvantages of Block Sort:

• Block sort has a relatively high overhead due to the need to divide the input into blocks and then merge the sorted blocks together.
• The choice of block size can affect the performance of block sort. If the block size is too small, there will be more blocks to sort, which increases the overhead. If the block size is too large, the individual blocks may not fit in memory, reducing the efficiency of block sort.

Why Block Sort is better than Other Sorting Algorithms?

• Block sort is often used when sorting very large arrays that cannot fit into memory at once, as it allows the array to be sorted into smaller, more manageable pieces. In contrast, many other sorting algorithms require that the entire array be loaded into memory before sorting can begin.
• The performance characteristics of block sort can vary depending on the specific algorithm used and the size of the blocks. In general, block sorting can be very efficient for sorting large data sets. Other sorting algorithms may have different performance characteristics depending on factors such as the data distribution
• Block sort algorithms can be efficient for sorting large data sets, but they can also be complex to implement and may require specialized hardware or software