# Efficient Huffman Coding for Sorted Input | Greedy Algo-4

• Difficulty Level : Medium
• Last Updated : 20 Dec, 2022

We recommend to read following post as a prerequisite for this.
Greedy Algorithms | Set 3 (Huffman Coding)

Time complexity of the algorithm discussed in above post is O(nLogn). If we know that the given array is sorted (by non-decreasing order of frequency), we can generate Huffman codes in O(n) time. Following is a O(n) algorithm for sorted input.
1. Create two empty queues.
2. Create a leaf node for each unique character and Enqueue it to the first queue in non-decreasing order of frequency. Initially second queue is empty.
3. Dequeue two nodes with the minimum frequency by examining the front of both queues. Repeat the following steps two times
1. If second queue is empty, dequeue from first queue.
2. If first queue is empty, dequeue from second queue.
3. Else, compare the front of two queues and dequeue the minimum.
4. Create a new internal node with frequency equal to the sum of the two nodes frequencies. Make the first Dequeued node as its left child and the second Dequeued node as right child. Enqueue this node to second queue.
5. Repeat steps#3 and #4 while there is more than one node in the queues. The remaining node is the root node and the tree is complete.

## C++

 // C++ Program for Efficient Huffman Coding for Sorted input#include using namespace std; // This constant can be avoided by explicitly// calculating height of Huffman Tree#define MAX_TREE_HT 100 // A node of huffman treeclass QueueNode {public:    char data;    unsigned freq;    QueueNode *left, *right;}; // Structure for Queue: collection// of Huffman Tree nodes (or QueueNodes)class Queue {public:    int front, rear;    int capacity;    QueueNode** array;}; // A utility function to create a new QueuenodeQueueNode* newNode(char data, unsigned freq){    QueueNode* temp = new QueueNode[(sizeof(QueueNode))];    temp->left = temp->right = NULL;    temp->data = data;    temp->freq = freq;    return temp;} // A utility function to create a Queue of given capacityQueue* createQueue(int capacity){    Queue* queue = new Queue[(sizeof(Queue))];    queue->front = queue->rear = -1;    queue->capacity = capacity;    queue->array = new QueueNode*[(queue->capacity                                   * sizeof(QueueNode*))];    return queue;} // A utility function to check if size of given queue is 1int isSizeOne(Queue* queue){    return queue->front == queue->rear           && queue->front != -1;} // A utility function to check if given queue is emptyint isEmpty(Queue* queue) { return queue->front == -1; } // A utility function to check if given queue is fullint isFull(Queue* queue){    return queue->rear == queue->capacity - 1;} // A utility function to add an item to queuevoid enQueue(Queue* queue, QueueNode* item){    if (isFull(queue))        return;    queue->array[++queue->rear] = item;    if (queue->front == -1)        ++queue->front;} // A utility function to remove an item from queueQueueNode* deQueue(Queue* queue){    if (isEmpty(queue))        return NULL;    QueueNode* temp = queue->array[queue->front];    if (queue->front        == queue               ->rear) // If there is only one item in queue        queue->front = queue->rear = -1;    else        ++queue->front;    return temp;} // A utility function to get from of queueQueueNode* getFront(Queue* queue){    if (isEmpty(queue))        return NULL;    return queue->array[queue->front];} /* A function to get minimum item from two queues */QueueNode* findMin(Queue* firstQueue, Queue* secondQueue){    // Step 3.a: If first queue is empty, dequeue from    // second queue    if (isEmpty(firstQueue))        return deQueue(secondQueue);     // Step 3.b: If second queue is empty, dequeue from    // first queue    if (isEmpty(secondQueue))        return deQueue(firstQueue);     // Step 3.c: Else, compare the front of two queues and    // dequeue minimum    if (getFront(firstQueue)->freq        < getFront(secondQueue)->freq)        return deQueue(firstQueue);     return deQueue(secondQueue);} // Utility function to check if this node is leafint isLeaf(QueueNode* root){    return !(root->left) && !(root->right);} // A utility function to print an array of size nvoid printArr(int arr[], int n){    int i;    for (i = 0; i < n; ++i)        cout << arr[i];    cout << endl;} // The main function that builds Huffman treeQueueNode* buildHuffmanTree(char data[], int freq[],                            int size){    QueueNode *left, *right, *top;     // Step 1: Create two empty queues    Queue* firstQueue = createQueue(size);    Queue* secondQueue = createQueue(size);     // Step 2:Create a leaf node for each unique character    // and Enqueue it to the first queue in non-decreasing    // order of frequency. Initially second queue is empty    for (int i = 0; i < size; ++i)        enQueue(firstQueue, newNode(data[i], freq[i]));     // Run while Queues contain more than one node. Finally,    // first queue will be empty and second queue will    // contain only one node    while (        !(isEmpty(firstQueue) && isSizeOne(secondQueue))) {        // Step 3: Dequeue two nodes with the minimum        // frequency by examining the front of both queues        left = findMin(firstQueue, secondQueue);        right = findMin(firstQueue, secondQueue);         // Step 4: Create a new internal node with frequency        // equal to the sum of the two nodes frequencies.        // Enqueue this node to second queue.        top = newNode('\$', left->freq + right->freq);        top->left = left;        top->right = right;        enQueue(secondQueue, top);    }     return deQueue(secondQueue);} // Prints huffman codes from the root of Huffman Tree. It// uses arr[] to store codesvoid printCodes(QueueNode* 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 Treevoid HuffmanCodes(char data[], int freq[], int size){    // Construct Huffman Tree    QueueNode* 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 codeint 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;} // This code is contributed by rathbhupendra

## C

 // C Program for Efficient Huffman Coding for Sorted input#include #include  // This constant can be avoided by explicitly calculating// height of Huffman Tree#define MAX_TREE_HT 100 // A node of huffman treestruct QueueNode {    char data;    unsigned freq;    struct QueueNode *left, *right;}; // Structure for Queue: collection of Huffman Tree nodes (or// QueueNodes)struct Queue {    int front, rear;    int capacity;    struct QueueNode** array;}; // A utility function to create a new Queuenodestruct QueueNode* newNode(char data, unsigned freq){    struct QueueNode* temp = (struct QueueNode*)malloc(        sizeof(struct QueueNode));    temp->left = temp->right = NULL;    temp->data = data;    temp->freq = freq;    return temp;} // A utility function to create a Queue of given capacitystruct Queue* createQueue(int capacity){    struct Queue* queue        = (struct Queue*)malloc(sizeof(struct Queue));    queue->front = queue->rear = -1;    queue->capacity = capacity;    queue->array = (struct QueueNode**)malloc(        queue->capacity * sizeof(struct QueueNode*));    return queue;} // A utility function to check if size of given queue is 1int isSizeOne(struct Queue* queue){    return queue->front == queue->rear           && queue->front != -1;} // A utility function to check if given queue is emptyint isEmpty(struct Queue* queue){    return queue->front == -1;} // A utility function to check if given queue is fullint isFull(struct Queue* queue){    return queue->rear == queue->capacity - 1;} // A utility function to add an item to queuevoid enQueue(struct Queue* queue, struct QueueNode* item){    if (isFull(queue))        return;    queue->array[++queue->rear] = item;    if (queue->front == -1)        ++queue->front;} // A utility function to remove an item from queuestruct QueueNode* deQueue(struct Queue* queue){    if (isEmpty(queue))        return NULL;    struct QueueNode* temp = queue->array[queue->front];    if (queue->front        == queue               ->rear) // If there is only one item in queue        queue->front = queue->rear = -1;    else        ++queue->front;    return temp;} // A utility function to get from of queuestruct QueueNode* getFront(struct Queue* queue){    if (isEmpty(queue))        return NULL;    return queue->array[queue->front];} /* A function to get minimum item from two queues */struct QueueNode* findMin(struct Queue* firstQueue,                          struct Queue* secondQueue){    // Step 3.a: If first queue is empty, dequeue from    // second queue    if (isEmpty(firstQueue))        return deQueue(secondQueue);     // Step 3.b: If second queue is empty, dequeue from    // first queue    if (isEmpty(secondQueue))        return deQueue(firstQueue);     // Step 3.c:  Else, compare the front of two queues and    // dequeue minimum    if (getFront(firstQueue)->freq        < getFront(secondQueue)->freq)        return deQueue(firstQueue);     return deQueue(secondQueue);} // Utility function to check if this node is leafint isLeaf(struct QueueNode* root){    return !(root->left) && !(root->right);} // A utility function to print an array of size nvoid printArr(int arr[], int n){    int i;    for (i = 0; i < n; ++i)        printf("%d", arr[i]);    printf("\n");} // The main function that builds Huffman treestruct QueueNode* buildHuffmanTree(char data[], int freq[],                                   int size){    struct QueueNode *left, *right, *top;     // Step 1: Create two empty queues    struct Queue* firstQueue = createQueue(size);    struct Queue* secondQueue = createQueue(size);     // Step 2:Create a leaf node for each unique character    // and Enqueue it to the first queue in non-decreasing    // order of frequency. Initially second queue is empty    for (int i = 0; i < size; ++i)        enQueue(firstQueue, newNode(data[i], freq[i]));     // Run while Queues contain more than one node. Finally,    // first queue will be empty and second queue will    // contain only one node    while (        !(isEmpty(firstQueue) && isSizeOne(secondQueue))) {        // Step 3: Dequeue two nodes with the minimum        // frequency by examining the front of both queues        left = findMin(firstQueue, secondQueue);        right = findMin(firstQueue, secondQueue);         // Step 4: Create a new internal node with frequency        // equal to the sum of the two nodes frequencies.        // Enqueue this node to second queue.        top = newNode('\$', left->freq + right->freq);        top->left = left;        top->right = right;        enQueue(secondQueue, top);    }     return deQueue(secondQueue);} // Prints huffman codes from the root of Huffman Tree.  It// uses arr[] to store codesvoid printCodes(struct QueueNode* 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 Treevoid HuffmanCodes(char data[], int freq[], int size){    //  Construct Huffman Tree    struct QueueNode* 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 functionsint 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;}

## Python3

 # Python3 program for Efficient Huffman Coding# for Sorted input # Class for the nodes of the Huffman treeclass QueueNode:         def __init__(self, data = None, freq = None,                 left = None, right = None):        self.data = data        self.freq = freq        self.left = left        self.right = right     # Function to check if the following    # node is a leaf node    def isLeaf(self):        return (self.left == None and                self.right == None) # Class for the two Queuesclass Queue:         def __init__(self):        self.queue = []     # Function for checking if the    # queue has only 1 node    def isSizeOne(self):        return len(self.queue) == 1     # Function for checking if    # the queue is empty    def isEmpty(self):        return self.queue == []     # Function to add item to the queue    def enqueue(self, x):        self.queue.append(x)     # Function to remove item from the queue    def dequeue(self):        return self.queue.pop(0) # Function to get minimum item from two queuesdef findMin(firstQueue, secondQueue):         # Step 3.1: If second queue is empty,    # dequeue from first queue    if secondQueue.isEmpty():        return firstQueue.dequeue()     # Step 3.2: If first queue is empty,    # dequeue from second queue    if firstQueue.isEmpty():        return secondQueue.dequeue()     # Step 3.3:  Else, compare the front of    # two queues and dequeue minimum    if (firstQueue.queue[0].freq <        secondQueue.queue[0].freq):        return firstQueue.dequeue()     return secondQueue.dequeue() # The main function that builds Huffman treedef buildHuffmanTree(data, freq, size):         # Step 1: Create two empty queues    firstQueue = Queue()    secondQueue = Queue()     # Step 2: Create a leaf node for each unique    # character and Enqueue it to the first queue    # in non-decreasing order of frequency.    # Initially second queue is empty.    for i in range(size):        firstQueue.enqueue(QueueNode(data[i], freq[i]))     # Run while Queues contain more than one node.    # Finally, first queue will be empty and    # second queue will contain only one node    while not (firstQueue.isEmpty() and               secondQueue.isSizeOne()):                            # Step 3: Dequeue two nodes with the minimum        # frequency by examining the front of both queues        left = findMin(firstQueue, secondQueue)        right = findMin(firstQueue, secondQueue)         # Step 4: Create a new internal node with        # frequency equal to the sum of the two        # nodes frequencies. Enqueue this node        # to second queue.        top = QueueNode("\$", left.freq + right.freq,                        left, right)        secondQueue.enqueue(top)     return secondQueue.dequeue() # Prints huffman codes from the root of# Huffman tree. It uses arr[] to store codesdef printCodes(root, arr):         # Assign 0 to left edge and recur    if root.left:        arr.append(0)        printCodes(root.left, arr)        arr.pop(-1)     # Assign 1 to right edge and recur    if root.right:        arr.append(1)        printCodes(root.right, arr)        arr.pop(-1)     # If this is a leaf node, then it contains    # one of the input characters, print the    # character and its code from arr[]    if root.isLeaf():        print(f"{root.data}: ", end = "")        for i in arr:            print(i, end = "")                     print() # The main function that builds a Huffman# tree and print codes by traversing the# built Huffman treedef HuffmanCodes(data, freq, size):         # Construct Huffman Tree    root = buildHuffmanTree(data, freq, size)     # Print Huffman codes using the Huffman    # tree built above    arr = []    printCodes(root, arr) # Driver codearr = ["a", "b", "c", "d", "e", "f"]freq = [5, 9, 12, 13, 16, 45]size = len(arr) HuffmanCodes(arr, freq, size) # This code is contributed by Kevin Joshi

## C++14

 // Clean c++ stl code to generate huffman codes if the array// is sorted in non-decreasing order#include using namespace std; // Node structure for creating a binary treestruct Node {    char ch;    int freq;    Node* left;    Node* right;    Node(char c, int f, Node* l = nullptr,         Node* r = nullptr)        : ch(c)        , freq(f)        , left(l)        , right(r){};}; // Find the min freq node between q1 and q2Node* minNode(queue& q1, queue& q2){    Node* temp;     if (q1.empty()) {        temp = q2.front();        q2.pop();        return temp;    }     if (q2.empty()) {        temp = q1.front();        q1.pop();        return temp;    }     if (q1.front()->freq < q2.front()->freq) {        temp = q1.front();        q1.pop();        return temp;    }    else {        temp = q2.front();        q2.pop();        return temp;    }} // Function to print the generated huffman codesvoid printHuffmanCodes(Node* root, string str = ""){    if (!root)        return;    if (root->ch != '\$') {        cout << root->ch << ": " << str << '\n';        return;    }     printHuffmanCodes(root->left, str + "0");    printHuffmanCodes(root->right, str + "1");     return;} // Function to generate huffman codesvoid generateHuffmanCode(vector > v){    if (!v.size())        return;     queue q1;    queue q2;     for (auto it = v.begin(); it != v.end(); ++it)        q1.push(new Node(it->first, it->second));     while (!q1.empty() or q2.size() > 1) {        Node* l = minNode(q1, q2);        Node* r = minNode(q1, q2);        Node* node = new Node('\$', l->freq + r->freq, l, r);        q2.push(node);    }     printHuffmanCodes(q2.front());    return;} int main(){    vector > v        = { { 'a' , 5 },  { 'b' , 9 },  { 'c' , 12 },            { 'd' , 13 }, { 'e' , 16 }, { 'f' , 45 } };    generateHuffmanCode(v);    return 0;}

Output:

f: 0
c: 100
d: 101
a: 1100
b: 1101
e: 111

Time complexity: O(n)
If the input is not sorted, it needs to be sorted first before it can be processed by the above algorithm. Sorting can be done using heap sort or merge-sort both of which run in Theta(nlogn). So, the overall time complexity becomes O(nlogn) for unsorted input.
Auxiliary Space: O(n)

Reference:
http://en.wikipedia.org/wiki/Huffman_coding