Let's consider three files with sizes of 10, 20, and 30 units each. Let's assume we merge the files in a suboptimal way, starting by combining the 20 and 30-sized files incurring a cost of 50, then merging this outcome with the 10-sized file. In this scenario, the total cost would be 50 + 60 = 110. Conversely, if we first merge the 10 and 20-sized files at a cost of 30 and then merge this result with the 30-sized file, the total cost would be 30 + 60 = 90. This basic example highlights the cost variation resulting from different merge sequences, emphasizing the importance of an optimal merging strategy.
The total cost is determined by the cumulative merge operations. As previously explained, this issue becomes relevant whenever there is a need for extensive merging, like in external sorting algorithms and operating systems tasks that entail merging files. By employing the most efficient merge strategy, we can ensure that these operations are executed with minimal computational burden.
Program:
Let's consider an example to demonstrate the best merge strategy in the C programming language.
#include <stdio.h>
#include <stdlib.h>
// Define a structure to represent a file
typedef struct File {
int size; // Size of the file
} File;
// A function to swap two elements in an array
void swap(File* a, File* b) {
File temp = *a;
*a = *b;
*b = temp;
}
// A utility function to heapify a subtree rooted at index i in a min-heap
// n is the size of the heap
void heapify(File files[], int n, int i) {
int smallest = i; // Initialize smallest as root
int left = 2 * i + 1; // Left child index
int right = 2 * i + 2; // Right child index
// If the left child is smaller than the root
if (left < n && files[left].size < files[smallest].size) {
smallest = left;
}
// If the right child is smaller than the smallest so far
if (right < n && files[right].size < files[smallest].size) {
smallest = right;
}
// If the smallest is not the root
if (smallest != i) {
swap(&files[i], &files[smallest]); // Swap the root with the smallest
heapify(files, n, smallest); // Recursively heapify the affected subtree
}
}
//Function to build a min-heap from the Array of files
void buildMinHeap(File files[], int n) {
for (int i = n / 2 - 1; i >= 0; i--) {
heapify(files, n, i);
}
}
// A utility function to extract the smallest element from the heap
File extractMin(File files[], int* n) {
File minFile = files[0]; // The root is the minimum element
files[0] = files[*n - 1]; // Move the last element to root
(*n)--; // Reduce heap size
heapify(files, *n, 0); // Heapify the root
return minFile;
}
// A utility function to insert a new file into the heap
void insertHeap(File files[], int* n, File newFile) {
(*n)++; // Increase the size of the heap
int i = *n - 1;
files[i] = newFile; // Insert the new file at the end of the heap
// Fix the min-heap property if violated
while (i != 0 && files[(i - 1) / 2].size > files[i].size) {
swap(&files[i], &files[(i - 1) / 2]);
i = (i - 1) / 2;
}
}
//Function to calculate the optimal merge pattern cost
int calculateOptimalMergeCost(File files[], int n) {
int totalCost = 0; // Initialize total cost
// Build a min-heap from the Array of files
buildMinHeap(files, n);
// Continue merging files until only one file remains
while (n > 1) {
// Extract the two smallest files
File file1 = extractMin(files, &n);
File file2 = extractMin(files, &n);
// Merge them and calculate the cost
int mergeCost = file1.size + file2.size;
totalCost += mergeCost;
// Create a new merged file and insert it back into the heap
File mergedFile;
mergedFile.size = mergeCost;
insertHeap(files, &n, mergedFile);
}
return totalCost; // Return the total merge cost
}
// A function to print the Array of file sizes
void printFileSizes(File files[], int n) {
printf("File sizes: ");
for (int i = 0; i < n; i++) {
printf("%d ", files[i].size);
}
printf("\n");
}
// A function to initialize an array of files with given sizes
void initializeFiles(File files[], int sizes[], int n) {
for (int i = 0; i < n; i++) {
files[i].size = sizes[i];
}
}
// Main Function
int main() {
int sizes[] = {5, 10, 15, 20, 25, 30, 35}; //Array of file sizes
int n = sizeof(sizes) / sizeof(sizes[0]); // Number of files
// Allocate memory for an array of File structures
File* files = (File*)malloc(n * sizeof(File));
// Initialize the files with their sizes
initializeFiles(files, sizes, n);
// Print the file sizes before merging
printf("Before merging:\n");
printFileSizes(files, n);
// Calculate the optimal merge pattern cost
int totalCost = calculateOptimalMergeCost(files, n);
// Print the total cost
printf("Optimal merge cost: %d\n", totalCost);
// Free the allocated memory
free(files);
return 0;
}
Output:
Before merging:
File sizes: 5 10 15 20 25 30 35
Optimal merge cost: 370
Explanation:
In this instance, the following code addresses the Optimal Merge Pattern dilemma by employing a min-heap. This method significantly lowers the total merge expense by combining files with varying sizes. The objective is to merge these files in a manner that minimizes the computational cost, which is directly related to the total sum of file sizes.
This issue can be resolved with optimal efficiency through the application of a greedy algorithm. This method operates by consistently merging the two smallest files initially, thereby minimizing cost increment at each stage. This approach mirrors the construction of Huffman Trees, a technique commonly employed in data compression tasks. Typically, the implementation involves leveraging a min-heap or priority queue for effective execution. The algorithm entails inserting all file sizes into the heap, iteratively extracting the two smallest values, merging them, and reinserting the resulting size back into the heap. This process continues until only a single file remains within the heap.
A File Structure is established to save the dimensions of each file, providing adaptability for potential expansion to include additional attributes for a file.
Heap Functions:
- heapify: The array of files maintains the heap property, the smallest element of which is at the root. The algorithm rearranges the heap in case of disturbance.
- extractMin: It removes and returns the smallest file from the heap, essentially the file that has the least size.
- insertHeap: It puts a newly merged file back into the heap, whih maintain the min-heap property.
- buildMinHeap: It converts an unsorted array to a min-heap so that we can efficiently extract the files' smallest sizes.
- Optimal Merge Cost Calculation: The central algorithm is the function calculateOptimalMergeCost. It removes from the heap the two smallest files, merges them, and re-inserts the result back into the heap. This process is repeated until only one file remains, amassing the merge cost each time a pair of files is merged.
- The initializeFiles sets up the file sizes.
- The printFileSizes helps visualize the process.
- Heap Operations: Creating the initial heap takes O(n) time, where n = number of files.
- Merge Procedure: Because the extraction and insertion into the min heap each takes O(log n) due to the maintenance of its heap structure, and there are n − 1 merges, the total time complexity for the code is O(n log n). This efficiency can be attributed to the fact that the min heap always guarantees that the smallest two files will be merged at each step.
Helper Functions:
Complexity Analysis:
Time Complexity:
Space Complexity:
The space complexity of this code is O(n) due to the Array of size n storing file sizes. The heap's structure also necessitates an equivalent amount of space. Additional auxiliary variables for file sizes and merge costs occupy some extra constant space, but the total space utilization increases to O(n).