External Sorting

External sorting is a term for a class of sorting algorithms that can handle massive amounts of data. External sorting is required when the data being sorted does not fit into the main memory of a computing device (usually RAM) and instead, must reside in the slower external memory (usually a hard drive). 

External sorting typically uses a hybrid sort-merge strategy. In the sorting phase, chunks of data small enough to fit in the main memory are read, sorted, and written out to a temporary file. In the merge phase, the sorted sub-files are combined into a single larger file.

Example: 

The external merge sort algorithm, which sorts chunks that each fit in RAM, then merges the sorted chunks together. We first divide the file into runs such that the size of a run is small enough to fit into the main memory. Then sort each run in the main memory using the merge sort sorting algorithm. Finally merge the resulting runs together into successively bigger runs, until the file is sorted.

When We do External Sorting?

• When the unsorted data is too large to perform sorting in computer internal memory then we use external sorting.
• In external sorting we use the secondary device. in a secondary storage device, we use the tape disk array. 
• when data is large like in merge sort and quick sort.
• Quick Sort: best average runtime.
• Merge Sort: Best Worse case time.
• To perform sort-merge, join operation on data.
• To perform order by the query.
• To select duplicate element.
• Where we need to take large input from the user.
  

Examples:

1. Merge sort
2. Tape sort
3. Polyphase sort
4. External radix
5. External merge

Prerequisites: MergeSort, Merge K Sorted Arrays:

Input:

input_file: Name of input file. input.txt
output_file: Name of output file, output.txt
run_size: Size of a run (can fit in RAM)
num_ways: Number of runs to be merged

To solve the problem follow the below idea:

The idea is straightforward, All the elements cannot be sorted at once as the size is very large. So the data is divided into chunks and then sorted using merge sort. The sorted data is then dumped into files. As such a huge amount of data cannot be handled altogether. Now After sorting the individual chunks. Sort the whole array by using the idea of merging k sorted arrays.

Follow the below steps to solve the problem:

• Read input_file such that at most ‘run_size’ elements are read at a time. Do following for the every run read in an array.
• Sort the run using MergeSort.
• Store the sorted array in a file. Let’s say ‘i’ for i^th file.
• Merge the sorted files using the approach discussed merge k sorted arrays

Below is the implementation of the above approach.

CPP

// C++ program to implement
// external sorting using
// merge sort
#include <bits/stdc++.h>
using namespace std;

struct MinHeapNode {

    // The element to be stored
    int element;

    // index of the array from which
    // the element is taken
    int i;
};

// Prototype of a utility function
// to swap two min heap nodes
void swap(MinHeapNode* x, MinHeapNode* y);

// A class for Min Heap
class MinHeap {

    // pointer to array of elements in heap
    MinHeapNode* harr;

    // size of min heap
    int heap_size;

public:

    // Constructor: creates a min
    // heap of given size
    MinHeap(MinHeapNode a[], int size);

    // to heapify a subtree with
    // root at given index
    void MinHeapify(int);

    // to get index of left child
    // of node at index i
    int left(int i) { return (2 * i + 1); }

    // to get index of right child
    // of node at index i
    int right(int i) { return (2 * i + 2); }

    // to get the root
    MinHeapNode getMin() { return harr[0]; }

    // to replace root with new node
    // x and heapify() new root
    void replaceMin(MinHeapNode x)
    {
        harr[0] = x;
        MinHeapify(0);
    }
};

// Constructor: Builds a heap from
// a given array a[] of given size
MinHeap::MinHeap(MinHeapNode a[], int size)
{
    heap_size = size;
    harr = a; // store address of array
    int i = (heap_size - 1) / 2;
    while (i >= 0) {
        MinHeapify(i);
        i--;
    }
}

// A recursive method to heapify
// a subtree with root
// at given index. This method
// assumes that the
// subtrees are already heapified
void MinHeap::MinHeapify(int i)
{
    int l = left(i);
    int r = right(i);
    int smallest = i;

    if (l < heap_size && harr[l].element < harr[i].element)
        smallest = l;

    if (r < heap_size
        && harr[r].element < harr[smallest].element)
        smallest = r;

    if (smallest != i) {
        swap(&harr[i], &harr[smallest]);
        MinHeapify(smallest);
    }
}

// A utility function to swap two elements
void swap(MinHeapNode* x, MinHeapNode* y)
{
    MinHeapNode temp = *x;
    *x = *y;
    *y = temp;
}

// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
    int i, j, k;
    int n1 = m - l + 1;
    int n2 = r - m;

    /* create temp arrays */
    int L[n1], R[n2];

    /* Copy data to temp arrays L[] and R[] */
    for (i = 0; i < n1; i++)
        L[i] = arr[l + i];
    for (j = 0; j < n2; j++)
        R[j] = arr[m + 1 + j];

    /* Merge the temp arrays back into arr[l..r]*/
    // Initial index of first subarray
    i = 0;

    // Initial index of second subarray
    j = 0;

    // Initial index of merged subarray
    k = l;
    while (i < n1 && j < n2) {
        if (L[i] <= R[j])
            arr[k++] = L[i++];
        else
            arr[k++] = R[j++];
    }

    /* Copy the remaining elements of L[],
        if there are any */
    while (i < n1)
        arr[k++] = L[i++];

    /* Copy the remaining elements of R[],
        if there are any */
    while (j < n2)
        arr[k++] = R[j++];
}

/* l is for left index and r is right index of the
   sub-array of arr to be sorted */
void mergeSort(int arr[], int l, int r)
{
    if (l < r) {
        // Same as (l+r)/2, but avoids overflow for
        // large l and h
        int m = l + (r - l) / 2;

        // Sort first and second halves
        mergeSort(arr, l, m);
        mergeSort(arr, m + 1, r);

        merge(arr, l, m, r);
    }
}

FILE* openFile(char* fileName, char* mode)
{
    FILE* fp = fopen(fileName, mode);
    if (fp == NULL) {
        perror("Error while opening the file.\n");
        exit(EXIT_FAILURE);
    }
    return fp;
}

// Merges k sorted files. Names of files are assumed
// to be 1, 2, 3, ... k
void mergeFiles(char* output_file, int n, int k)
{
    FILE* in[k];
    for (int i = 0; i < k; i++) {
        char fileName[2];

        // convert i to string
        snprintf(fileName, sizeof(fileName), "%d", i);

        // Open output files in read mode.
        in[i] = openFile(fileName, "r");
    }

    // FINAL OUTPUT FILE
    FILE* out = openFile(output_file, "w");

    // Create a min heap with k heap
    // nodes. Every heap node
    // has first element of scratch
    // output file
    MinHeapNode* harr = new MinHeapNode[k];
    int i;
    for (i = 0; i < k; i++) {
        // break if no output file is empty and
        // index i will be no. of input files
        if (fscanf(in[i], "%d ", &harr[i].element) != 1)
            break;

        // Index of scratch output file
        harr[i].i = i;
    }
    // Create the heap
    MinHeap hp(harr, i);

    int count = 0;

    // Now one by one get the
    // minimum element from min
    // heap and replace it with
    // next element.
    // run till all filled input
    // files reach EOF
    while (count != i) {
        // Get the minimum element
        // and store it in output file
        MinHeapNode root = hp.getMin();
        fprintf(out, "%d ", root.element);

        // Find the next element that
        // will replace current
        // root of heap. The next element
        // belongs to same
        // input file as the current min element.
        if (fscanf(in[root.i], "%d ", &root.element) != 1) {
            root.element = INT_MAX;
            count++;
        }

        // Replace root with next
        // element of input file
        hp.replaceMin(root);
    }

    // close input and output files
    for (int i = 0; i < k; i++)
        fclose(in[i]);

    fclose(out);
}

// Using a merge-sort algorithm,
// create the initial runs
// and divide them evenly among
// the output files
void createInitialRuns(char* input_file, int run_size,
                       int num_ways)
{
    // For big input file
    FILE* in = openFile(input_file, "r");

    // output scratch files
    FILE* out[num_ways];
    char fileName[2];
    for (int i = 0; i < num_ways; i++) {
        // convert i to string
        snprintf(fileName, sizeof(fileName), "%d", i);

        // Open output files in write mode.
        out[i] = openFile(fileName, "w");
    }

    // allocate a dynamic array large enough
    // to accommodate runs of size run_size
    int* arr = (int*)malloc(run_size * sizeof(int));

    bool more_input = true;
    int next_output_file = 0;

    int i;
    while (more_input) {
        // write run_size elements
        // into arr from input file
        for (i = 0; i < run_size; i++) {
            if (fscanf(in, "%d ", &arr[i]) != 1) {
                more_input = false;
                break;
            }
        }

        // sort array using merge sort
        mergeSort(arr, 0, i - 1);

        // write the records to the
        // appropriate scratch output file
        // can't assume that the loop
        // runs to run_size
        // since the last run's length
        // may be less than run_size
        for (int j = 0; j < i; j++)
            fprintf(out[next_output_file], "%d ", arr[j]);

        next_output_file++;
    }

    // close input and output files
    for (int i = 0; i < num_ways; i++)
        fclose(out[i]);

    fclose(in);
}

// For sorting data stored on disk
void externalSort(char* input_file, char* output_file,
                  int num_ways, int run_size)
{
    // read the input file,
    // create the initial runs,
    // and assign the runs to
    // the scratch output files
    createInitialRuns(input_file, run_size, num_ways);

    // Merge the runs using
    // the K-way merging
    mergeFiles(output_file, run_size, num_ways);
}

// Driver code
int main()
{
    // No. of Partitions of input file.
    int num_ways = 10;

    // The size of each partition
    int run_size = 1000;

    char input_file[] = "input.txt";
    char output_file[] = "output.txt";

    FILE* in = openFile(input_file, "w");

    srand(time(NULL));

    // generate input
    for (int i = 0; i < num_ways * run_size; i++)
        fprintf(in, "%d ", rand());

    fclose(in);

    externalSort(input_file, output_file, num_ways,
                 run_size);

    return 0;
}

Time Complexity: O(N * log N). 

• Time taken for merge sort is O(runs * run_size * log run_size), which is equal to O(N log run_size). 
• To merge the sorted arrays the time complexity is O(N * log runs). 
• Therefore, the overall time complexity is O(N * log run_size + N * log runs). 
• Since log run_size + log runs = log run_size*runs = log N, the result time complexity will be O(N * log N).

Auxiliary space: O(run_size). run_size is the space needed to store the array.

Note: This code won’t work on an online compiler as it requires file creation permissions. When running in a local machine, it produces a sample input file “input.txt” with 10000 random numbers. It sorts the numbers and puts the sorted numbers in a file “output.txt”. It also generates files with names 1, 2, .. to store sorted runs.