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Huffman Coding

geeksforgeeks Huffman Coding | Greedy Algo-3

Prefix Codes, means the codes (bit sequences) are assigned in such a way that the code assigned to one character is not the prefix of code assigned to any other character. This is how Huffman Coding makes sure that there is no ambiguity when decoding the generated bitstream.

Let us understand prefix codes with a counter example. Let there be four characters a, b, c and d, and their corresponding variable length codes be 00, 01, 0 and 1. This coding leads to ambiguity because code assigned to c is the prefix of codes assigned to a and b. If the compressed bit stream is 0001, the de-compressed output may be “cccd” or “ccb” or “acd” or “ab”.See this for applications of Huffman Coding.

There are mainly two major parts in Huffman Coding 1) Build a Huffman Tree from input characters. 2) Traverse the Huffman Tree and assign codes to characters.

Steps to build Huffman Tree

Input is an array of unique characters along with their frequency of occurrences and output is Huffman Tree.

1. Create a leaf node for each unique character and build a min heap of all leaf nodes (Min Heap is used as a priority queue. The value of frequency field is used to compare two nodes in min heap. Initially, the least frequent character is at root)

2. Extract two nodes with the minimum frequency from the min heap.

3. Create a new internal node with a frequency equal to the sum of the two nodes frequencies. Make the first extracted node as its left child and the other extracted node as its right child. Add this node to the min heap.

4. Repeat steps#2 and #3 until the heap contains only one node. The remaining node is the root node and the tree is complete.

Let us understand the algorithm with an example:

// C program for Huffman Coding 
#include <stdio.h> 
#include <stdlib.h> 

// This constant can be avoided by explicitly 
// calculating height of Huffman Tree 
#define MAX_TREE_HT 100 

// A Huffman tree node 
struct MinHeapNode { 

    // One of the input characters 
    char data; 

    // Frequency of the character 
    unsigned freq; 

    // Left and right child of this node 
    struct MinHeapNode *left, *right; 
}; 

// A Min Heap: Collection of 
// min-heap (or Huffman tree) nodes 
struct MinHeap { 

    // Current size of min heap 
    unsigned size; 

    // capacity of min heap 
    unsigned capacity; 

    // Array of minheap node pointers 
    struct MinHeapNode** array; 
}; 

// A utility function allocate a new 
// min heap node with given character 
// and frequency of the character 
struct MinHeapNode* newNode(char data, unsigned freq) 
{ 
    struct MinHeapNode* temp 
        = (struct MinHeapNode*)malloc
(sizeof(struct MinHeapNode)); 

    temp->left = temp->right = NULL; 
    temp->data = data; 
    temp->freq = freq; 

    return temp; 
} 

// A utility function to create 
// a min heap of given capacity 
struct MinHeap* createMinHeap(unsigned capacity) 

{ 

    struct MinHeap* minHeap 
        = (struct MinHeap*)malloc(sizeof(struct MinHeap)); 

    // current size is 0 
    minHeap->size = 0; 

    minHeap->capacity = capacity; 

    minHeap->array 
        = (struct MinHeapNode**)malloc(minHeap-> 
capacity * sizeof(struct MinHeapNode*)); 
    return minHeap; 
} 

// A utility function to 
// swap two min heap nodes 
void swapMinHeapNode(struct MinHeapNode** a, 
                    struct MinHeapNode** b) 

{ 

    struct MinHeapNode* t = *a; 
    *a = *b; 
    *b = t; 
} 

// The standard minHeapify function. 
void minHeapify(struct MinHeap* minHeap, int idx) 

{ 

    int smallest = idx; 
    int left = 2 * idx + 1; 
    int right = 2 * idx + 2; 

    if (left < minHeap->size && minHeap->array[left]-> 
freq < minHeap->array[smallest]->freq) 
        smallest = left; 

    if (right < minHeap->size && minHeap->array[right]-> 
freq < minHeap->array[smallest]->freq) 
        smallest = right; 

    if (smallest != idx) { 
        swapMinHeapNode(&minHeap->array[smallest], 
                        &minHeap->array[idx]); 
        minHeapify(minHeap, smallest); 
    } 
} 

// A utility function to check 
// if size of heap is 1 or not 
int isSizeOne(struct MinHeap* minHeap) 
{ 

    return (minHeap->size == 1); 
} 

// A standard function to extract 
// minimum value node from heap 
struct MinHeapNode* extractMin(struct MinHeap* minHeap) 

{ 

    struct MinHeapNode* temp = minHeap->array[0]; 
    minHeap->array[0] 
        = minHeap->array[minHeap->size - 1]; 

    --minHeap->size; 
    minHeapify(minHeap, 0); 

    return temp; 
} 

// A utility function to insert 
// a new node to Min Heap 
void insertMinHeap(struct MinHeap* minHeap, 
                struct MinHeapNode* minHeapNode) 

{ 

    ++minHeap->size; 
    int i = minHeap->size - 1; 

    while (i && minHeapNode->freq < minHeap->array[(i - 1) / 2]->freq) { 

        minHeap->array[i] = minHeap->array[(i - 1) / 2]; 
        i = (i - 1) / 2; 
    } 

    minHeap->array[i] = minHeapNode; 
} 

// A standard function to build min heap 
void buildMinHeap(struct MinHeap* minHeap) 

{ 

    int n = minHeap->size - 1; 
    int i; 

    for (i = (n - 1) / 2; i >= 0; --i) 
        minHeapify(minHeap, i); 
} 

// A utility function to print an array of size n 
void printArr(int arr[], int n) 
{ 
    int i; 
    for (i = 0; i < n; ++i) 
        printf("%d", arr[i]); 

    printf("\n"); 
} 

// Utility function to check if this node is leaf 
int isLeaf(struct MinHeapNode* root) 

{ 

    return !(root->left) && !(root->right); 
} 

// Creates a min heap of capacity 
// equal to size and inserts all character of 
// data[] in min heap. Initially size of 
// min heap is equal to capacity 
struct MinHeap* createAndBuildMinHeap(char data[], int freq[], int size) 

{ 

    struct MinHeap* minHeap = createMinHeap(size); 

    for (int i = 0; i < size; ++i) 
        minHeap->array[i] = newNode(data[i], freq[i]); 

    minHeap->size = size; 
    buildMinHeap(minHeap); 

    return minHeap; 
} 

// The main function that builds Huffman tree 
struct MinHeapNode* buildHuffmanTree(char data[], int freq[], int size) 

{ 
    struct MinHeapNode *left, *right, *top; 

    // Step 1: Create a min heap of capacity 
    // equal to size. Initially, there are 
    // modes equal to size. 
    struct MinHeap* minHeap = createAndBuildMinHeap(data, freq, size); 

    // Iterate while size of heap doesn't become 1 
    while (!isSizeOne(minHeap)) { 

        // Step 2: Extract the two minimum 
        // freq items from min heap 
        left = extractMin(minHeap); 
        right = extractMin(minHeap); 

        // Step 3: Create a new internal 
        // node with frequency equal to the 
        // sum of the two nodes frequencies. 
        // Make the two extracted node as 
        // left and right children of this new node. 
        // Add this node to the min heap 
        // '$' is a special value for internal nodes, not used 
        top = newNode('$', left->freq + right->freq); 

        top->left = left; 
        top->right = right; 

        insertMinHeap(minHeap, top); 
    } 

    // Step 4: The remaining node is the 
    // root node and the tree is complete. 
    return extractMin(minHeap); 
} 

// Prints huffman codes from the root of Huffman Tree. 
// It uses arr[] to store codes 
void printCodes(struct MinHeapNode* root, int arr[], int top) 

{ 

    // Assign 0 to left edge and recur 
    if (root->left) { 

        arr[top] = 0; 
        printCodes(root->left, arr, top + 1); 
    } 

    // Assign 1 to right edge and recur 
    if (root->right) { 

        arr[top] = 1; 
        printCodes(root->right, arr, top + 1); 
    } 

    // If this is a leaf node, then 
    // it contains one of the input 
    // characters, print the character 
    // and its code from arr[] 
    if (isLeaf(root)) { 

        printf("%c: ", root->data); 
        printArr(arr, top); 
    } 
} 

// The main function that builds a 
// Huffman Tree and print codes by traversing 
// the built Huffman Tree 
void HuffmanCodes(char data[], int freq[], int size) 

{ 
    // Construct Huffman Tree 
    struct MinHeapNode* root 
        = buildHuffmanTree(data, freq, size); 

    // Print Huffman codes using 
    // the Huffman tree built above 
    int arr[MAX_TREE_HT], top = 0; 

    printCodes(root, arr, top); 
} 

// Driver program to test above functions 
int main() 
{ 

    char arr[] = { 'a', 'b', 'c', 'd', 'e', 'f' }; 
    int freq[] = { 5, 9, 12, 13, 16, 45 }; 

    int size = sizeof(arr) / sizeof(arr[0]); 

    HuffmanCodes(arr, freq, size); 

    return 0; 
} 
// C++ program for Huffman Coding 
#include <iostream> 
#include <cstdlib> 
using namespace std; 

// This constant can be avoided by explicitly 
// calculating height of Huffman Tree 
#define MAX_TREE_HT 100 

// A Huffman tree node 
struct MinHeapNode { 

    // One of the input characters 
    char data; 

    // Frequency of the character 
    unsigned freq; 

    // Left and right child of this node 
    struct MinHeapNode *left, *right; 
}; 

// A Min Heap: Collection of 
// min-heap (or Huffman tree) nodes 
struct MinHeap { 

    // Current size of min heap 
    unsigned size; 

    // capacity of min heap 
    unsigned capacity; 

    // Attay of minheap node pointers 
    struct MinHeapNode** array; 
}; 

// A utility function allocate a new 
// min heap node with given character 
// and frequency of the character 
struct MinHeapNode* newNode(char data, unsigned freq) 
{ 
    struct MinHeapNode* temp 
        = (struct MinHeapNode*)malloc
(sizeof(struct MinHeapNode)); 

    temp->left = temp->right = NULL; 
    temp->data = data; 
    temp->freq = freq; 

    return temp; 
} 

// A utility function to create 
// a min heap of given capacity 
struct MinHeap* createMinHeap(unsigned capacity) 

{ 

    struct MinHeap* minHeap 
        = (struct MinHeap*)malloc(sizeof(struct MinHeap)); 

    // current size is 0 
    minHeap->size = 0; 

    minHeap->capacity = capacity; 

    minHeap->array 
        = (struct MinHeapNode**)malloc(minHeap-> 
capacity * sizeof(struct MinHeapNode*)); 
    return minHeap; 
} 

// A utility function to 
// swap two min heap nodes 
void swapMinHeapNode(struct MinHeapNode** a, 
                    struct MinHeapNode** b) 

{ 

    struct MinHeapNode* t = *a; 
    *a = *b; 
    *b = t; 
} 

// The standard minHeapify function. 
void minHeapify(struct MinHeap* minHeap, int idx) 

{ 

    int smallest = idx; 
    int left = 2 * idx + 1; 
    int right = 2 * idx + 2; 

    if (left < minHeap->size && minHeap->array[left]-> 
freq < minHeap->array[smallest]->freq) 
        smallest = left; 

    if (right < minHeap->size && minHeap->array[right]-> 
freq < minHeap->array[smallest]->freq) 
        smallest = right; 

    if (smallest != idx) { 
        swapMinHeapNode(&minHeap->array[smallest], 
                        &minHeap->array[idx]); 
        minHeapify(minHeap, smallest); 
    } 
} 

// A utility function to check 
// if size of heap is 1 or not 
int isSizeOne(struct MinHeap* minHeap) 
{ 

    return (minHeap->size == 1); 
} 

// A standard function to extract 
// minimum value node from heap 
struct MinHeapNode* extractMin(struct MinHeap* minHeap) 

{ 

    struct MinHeapNode* temp = minHeap->array[0]; 
    minHeap->array[0] 
        = minHeap->array[minHeap->size - 1]; 

    --minHeap->size; 
    minHeapify(minHeap, 0); 

    return temp; 
} 

// A utility function to insert 
// a new node to Min Heap 
void insertMinHeap(struct MinHeap* minHeap, 
                struct MinHeapNode* minHeapNode) 

{ 

    ++minHeap->size; 
    int i = minHeap->size - 1; 

    while (i && minHeapNode->freq < minHeap->array[(i - 1) / 2]->freq) { 

        minHeap->array[i] = minHeap->array[(i - 1) / 2]; 
        i = (i - 1) / 2; 
    } 

    minHeap->array[i] = minHeapNode; 
} 

// A standard function to build min heap 
void buildMinHeap(struct MinHeap* minHeap) 

{ 

    int n = minHeap->size - 1; 
    int i; 

    for (i = (n - 1) / 2; i >= 0; --i) 
        minHeapify(minHeap, i); 
} 

// A utility function to print an array of size n 
void printArr(int arr[], int n) 
{ 
    int i; 
    for (i = 0; i < n; ++i) 
        cout<< arr[i]; 

    cout<<"\n"; 
} 

// Utility function to check if this node is leaf 
int isLeaf(struct MinHeapNode* root) 

{ 

    return !(root->left) && !(root->right); 
} 

// Creates a min heap of capacity 
// equal to size and inserts all character of 
// data[] in min heap. Initially size of 
// min heap is equal to capacity 
struct MinHeap* createAndBuildMinHeap(char data[], int freq[], int size) 

{ 

    struct MinHeap* minHeap = createMinHeap(size); 

    for (int i = 0; i < size; ++i) 
        minHeap->array[i] = newNode(data[i], freq[i]); 

    minHeap->size = size; 
    buildMinHeap(minHeap); 

    return minHeap; 
} 

// The main function that builds Huffman tree 
struct MinHeapNode* buildHuffmanTree(char data[], int freq[], int size) 

{ 
    struct MinHeapNode *left, *right, *top; 

    // Step 1: Create a min heap of capacity 
    // equal to size. Initially, there are 
    // modes equal to size. 
    struct MinHeap* minHeap = createAndBuildMinHeap(data, freq, size); 

    // Iterate while size of heap doesn't become 1 
    while (!isSizeOne(minHeap)) { 

        // Step 2: Extract the two minimum 
        // freq items from min heap 
        left = extractMin(minHeap); 
        right = extractMin(minHeap); 

        // Step 3: Create a new internal 
        // node with frequency equal to the 
        // sum of the two nodes frequencies. 
        // Make the two extracted node as 
        // left and right children of this new node. 
        // Add this node to the min heap 
        // '$' is a special value for internal nodes, not used 
        top = newNode('$', left->freq + right->freq); 

        top->left = left; 
        top->right = right; 

        insertMinHeap(minHeap, top); 
    } 

    // Step 4: The remaining node is the 
    // root node and the tree is complete. 
    return extractMin(minHeap); 
} 

// Prints huffman codes from the root of Huffman Tree. 
// It uses arr[] to store codes 
void printCodes(struct MinHeapNode* root, int arr[], int top) 

{ 

    // Assign 0 to left edge and recur 
    if (root->left) { 

        arr[top] = 0; 
        printCodes(root->left, arr, top + 1); 
    } 

    // Assign 1 to right edge and recur 
    if (root->right) { 

        arr[top] = 1; 
        printCodes(root->right, arr, top + 1); 
    } 

    // If this is a leaf node, then 
    // it contains one of the input 
    // characters, print the character 
    // and its code from arr[] 
    if (isLeaf(root)) { 

        cout<< root->data <<": "; 
        printArr(arr, top); 
    } 
} 

// The main function that builds a 
// Huffman Tree and print codes by traversing 
// the built Huffman Tree 
void HuffmanCodes(char data[], int freq[], int size) 

{ 
    // Construct Huffman Tree 
    struct MinHeapNode* root 
        = buildHuffmanTree(data, freq, size); 

    // Print Huffman codes using 
    // the Huffman tree built above 
    int arr[MAX_TREE_HT], top = 0; 

    printCodes(root, arr, top); 
} 

// Driver program to test above functions 
int main() 
{ 

    char arr[] = { 'a', 'b', 'c', 'd', 'e', 'f' }; 
    int freq[] = { 5, 9, 12, 13, 16, 45 }; 

    int size = sizeof(arr) / sizeof(arr[0]); 

    HuffmanCodes(arr, freq, size); 

    return 0; 
} 

Huffman-tree-vs-trie