External Sorting - GeeksforGeeks

17 Sep, 2021
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 do not fit into the main memory of a computing device (usually RAM) and instead, they 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 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.One example of external sorting is 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 main memory. Then sort each run in main memory using merge sort sorting algorithm. Finally merge the resulting runs together into successively bigger runs, until the file is sorted. 
Prerequisite for the algorithm/code: MergeSort: Used for sort individual runs (a run is part of file that is small enough to fit in main memory) Merge K Sorted Arrays: Used to merge sorted runs.Below are the steps used in C++ implementation. Inputs:  
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
Solution: The idea is very simple, 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 huge amount of data cannot be handled altogether. Now After sorting the individual chunks. Sort the whole array by using the idea of merge k sorted arrays.Algorithm:  
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. Lets say ‘i’ for ith file.Merge the sorted files using the approach discussed merge k sorted arrays
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. Lets say ‘i’ for ith file.
Merge the sorted files using the approach discussed merge k sorted arrays
Following is C++ implementation of the above steps.  
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 nodesvoid swap(MinHeapNode* x, MinHeapNode* y); // A class for Min Heapclass 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 sizeMinHeap::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 heapifiedvoid 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 elementsvoid 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, ... kvoid 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 filesvoid 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 diskvoid 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 program to test aboveint 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;}
Complexity Analysis:  
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 online compiler as it requires file creation permissions. When run local machine, it produces 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.References:  
https://en.wikipedia.org/wiki/External_sorting
http://web.eecs.utk.edu/~leparker/Courses/CS302-Fall06/Notes/external-sorting2.html
This article is contributed by Aditya Goel. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above 
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