Implementing a Huffman Tree1 is a fun afternoon project for anyone interested in learning about data compression. A Huffman Tree is a type of binary tree that sees use in the compression of an arbitrary data file. Developing a command line utility to compress/decompress a file using Huffman coding is a good CS101 challenge.

Breaking It Down Into Steps

This project starts where many do: Wikipedia. The Huffman Coding wiki article gives a nice breakdown with examples of the data structure and associated algorithms. In particular, the “Basic Technique”2 section covers the algorithms for compression and decompression. You need three key data structures to implement the big Compress() and Decompress() routines:

  1. A map mapping characters to their frequency in the input.
  2. A Huffman Tree used to generate an encoding map.
  3. An encodings map mapping characters to their binary code.

The next sections build up each structure and then discuss how they come together to implement Compress()/Decompress().

Constructing the Character Frequency Map

A key aspect of Huffman coding is the derivation of binary codes from the frequency of characters in the input file. The frequency of a character in the input drives the length of a character’s binary code. The higher the character frequency the shorter the binary code and vice versa.

So how do you track character frequency? A regular old map does the trick. The keys of the map are the characters in the input and the mapped to values are the character’s frequency. Below is a snippet of code showing how to construct a frequency map:

using CharFreqMap = std::map<char, uint32_t>;
CharFreqMap char_freqs_; /**< Map of character frequencies in the input. */

RetCode HuffmanCoding::CountCharFrequencies(const std::string& infile) {
    /* read the input file in kReadBuffSize sized chunks */
    std::ifstream infile_stream(infile, std::ios::binary);
    while (infile_stream) {
        infile_stream.read(read_buffer_.data(), read_buffer_.size());
        for (std::streamsize i = 0; i < infile_stream.gcount(); ++i) {
            char_freqs_[read_buffer_[i]]++; /* up the char's frequency */
        }
    }
    return (char_freqs_.empty()) ? RetCode::kEmptyFile : RetCode::kSuccess;
}

The code reads character data into the read_buffer_ buffer in 1 kilobyte chunks. The char_freqs_ map tracks the frequency of each character.

Growing a Huffman Tree

You now have a map of character frequencies. How do you use this frequency map to generate binary codes? There’s an intermediate step. You need to represent your character frequencies in a way that you can later use to generate optimal codes. This is where the infamous Huffman Tree comes into play.

A Huffman Tree is a binary tree. The nodes of a Huffman Tree often have a structure like this:

struct HuffmanNode {
    int character;       /**< Character or kInternalNode value. */
    uint32_t count;      /**< Character frequency. */
    HuffmanNodePtr zero; /**< Huffman tree left subtree. */
    HuffmanNodePtr one;  /**< Huffman tree right subtree. */
};

Here’s a picture of a Huffman Tree for the input text aaaaabbc:

                                   ┌───────────────────────────┐                 
                                   │          Node 1           │                 
                                   ├───────────────────────────┤                 
                                   │character = INTERNAL_MARKER│                 
                                   ├───────────────────────────┤                 
                                   │count = 8                  │                 
                                   └─────────────┬─────────────┘                 
                             ┌─────────0─────────┴───────1──────┐                
                             │                                  │                
                ┌────────────▼──────────────┐       ┌───────────▼───────────────┐
                │          Node 2           │       │          Node 3           │
                ├───────────────────────────┤       ├───────────────────────────┤
                │character = INTERNAL_MARKER│       │character = 'a'            │
                ├───────────────────────────┤       ├───────────────────────────┤
                │count = 3                  │       │count = 5                  │
                └─────────────┬─────────────┘       └───────────────────────────┘
             ┌────────0───────┴───────1───────┐                                  
             │                                │                                  
┌────────────▼──────────────┐    ┌────────────▼──────────────┐                   
│          Node 4           │    │          Node 5           │                   
├───────────────────────────┤    ├───────────────────────────┤                   
│character = 'c'            │    │character = 'b'            │                   
├───────────────────────────┤    ├───────────────────────────┤                   
│count = 1                  │    │count = 2                  │                   
└───────────────────────────┘    └───────────────────────────┘                   

There are two types of nodes in the tree: internal nodes and leaf nodes. The leaf nodes of a Huffman Tree contain an input character and its frequency (denoted as count in the image). The internal nodes of a Huffman Tree often replace the character with some special marker value and contain a count value equal to the sum of the count values of its subtrees. By convention, the edge to the left subtree has a label of zero and the edge to the right subtree has a label of one.

Notice how the root to leaf path for the highest frequency character, a, is shorter than the root to leaf paths for the lower frequency chars. This is no coincidence. You traverse a Huffman Tree such that you obtain character to binary string mappings where the most frequent characters have the most compact representation.

So how do you build the tree from the frequency map? The wiki article provides an algorithm for constructing an optimal Huffman Tree:

  1. Start with as many leaves as there are symbols.
  2. Enqueue all leaf nodes into the first queue (by probability in increasing order so that the least likely item is in the head of the queue).
  3. While there is more than one node in the queues:
    • Dequeue the two nodes with the lowest weight by examining the fronts of both queues.
    • Create a new internal node, with the two just-removed nodes as children (either node can be either child) and the sum of their weights as the new weight.
    • Enqueue the new node into the rear of the second queue.
  4. The remaining node is the root node; the tree has now been generated.

Below is a C++ implementation of the algorithm description:

using HuffmanNodePtr = std::shared_ptr<HuffmanNode>;

void HuffmanCoding::BuildEncodingTree() {
    auto HuffmanNodePtrGreater = [](const HuffmanNodePtr a,
                                    const HuffmanNodePtr b) {
        return (a->count > b->count);
    };
    std::priority_queue<HuffmanNodePtr, std::vector<HuffmanNodePtr>,
                        decltype(HuffmanNodePtrGreater)>
        encoding_queue;

    /* load the initial nodes with their chars and freqs */
    for (const auto& [character, frequency] : char_freqs_) {
        encoding_queue.push(
            std::make_shared<HuffmanNode>(character, frequency));
    }

    /* follow the algorithm described in
     * https://en.wikipedia.org/wiki/Huffman_coding under the "Compression"
     * section */
    while (encoding_queue.size() != 1) {
        HuffmanNodePtr first = encoding_queue.top();
        encoding_queue.pop();
        HuffmanNodePtr second = encoding_queue.top();
        encoding_queue.pop();

        HuffmanNodePtr new_node = std::make_shared<HuffmanNode>(
            kInternalNode, first->count + second->count, first, second);

        encoding_queue.push(new_node);
    }
    encoding_root_ = encoding_queue.top(); /* save off the root of the tree */
}

When BuildEncodingTree() terminates, encoding_root_ will point to the root node of the Huffman Tree.

Building a Codebook

It’s the moment you’ve been waiting for: code generation. You probably already guessed how this works. To generate a character’s code, all you need to do is traverse the Huffman Tree. As you walk down from the root to each leaf, you bookkeep the path taken using 0’s to indicate left subtree traversals and 1’s for the right subtree traversals. When you hit a leaf node, you save off the node’s character value and the bit string generated up to that node.

Here’s a snippet showing how to recursively construct character encodings:

using EncodingMap = std::map<char, std::string>;
EncodingMap encodings_; /**< Map of character to binary string encodings. */

void HuffmanCoding::BuildEncodingMap(HuffmanNodePtr root,
                                     std::string encoding) {
    if (root->character != kInternalNode) { /* reached a leaf node */
        encodings_[static_cast<char>(root->character)] = encoding;
        return;
    }
    BuildEncodingMap(root->zero, encoding + "0"); /* recurse into ltree */
    BuildEncodingMap(root->one, encoding + "1");  /* recurse into rtree */
}

At the end of this routine, encodings_ will contain a mapping of each character in the input to a binary string. Using the example text aaaaabbc given in the previous section, the encodings_ map would look like

CharacterEncoding
a1
b01
c10

The original text required one byte per character or 8 bytes of storage. Using this codebook, you could store the text string using the code 11111010110. This would require only two bytes to store the same information. Why two bytes? Because you can only write in units of bytes to an output file meaning you have to pad the bit string with 5 zeroes on the right (for example, 11111010 110 -> 11111010 11000000).

Compression

With codebook in hand, compression boils down to converting an input stream into a coded bit stream. Then, write out the contents of the bit stream byte-by-byte to an output file.

Here is an implementation of the Compress() routine:

RetCode HuffmanCoding::Compress(const std::string& uncompressed_filepath,
                                const std::string& compressed_filepath) {
    /* verify uncompressed_filepath points to an existing file */
    std::filesystem::path uncompressed_path(uncompressed_filepath);
    if (!std::filesystem::exists(uncompressed_filepath)) {
        return RetCode::kFileDoesNotExist;
    }

    /* scan the uncompressed file once to compute char frequencies */
    RetCode retcode = CountCharFrequencies(uncompressed_filepath);
    if (RetCode::kSuccess != retcode) {
        return retcode;
    }

    BuildEncodingTree();                  /* construct the huffman code tree */
    BuildEncodingMap(encoding_root_, ""); /* construct char to bit string map */
    Encode(uncompressed_filepath, compressed_filepath); /* compress the data */

    return retcode;
}

You can see that Compress() just does some file checks and then builds up the data structures previously discussed. Encode() is where the actual translation happens. The code for Encode() is a bit ugly:

void HuffmanCoding::Encode(const std::string& infile,
                           const std::string& outfile) {
    /* controls for writing compressed data byte by byte */
    const int kBitsPerByte = 8;
    uint8_t currbyte = 0;
    int bitcount = 0;

    std::ofstream outfile_stream(outfile, std::ios::out | std::ios::binary);
    WriteHeader(outfile_stream); /* write the compressed files' header first */

    std::ifstream infile_stream(infile, std::ios::in);
    while (infile_stream) {
        /* read uncompressed data */
        infile_stream.read(read_buffer_.data(), read_buffer_.size());

        /* encode the chars in the buffer */
        for (std::streamsize i = 0; i < infile_stream.gcount(); ++i) {
            /* since the smallest unit we can write to a file is a byte not a
             * bit, the code below constructs a byte from the bits in an
             * encoding and then writes the byte to the output file */
            for (const char& bit : encodings_.at(read_buffer_[i])) {
                uint8_t ibit = (bit == '1') ? 1 : 0;
                currbyte = (currbyte << 1) | ibit;
                bitcount++;
                if (bitcount == kBitsPerByte) {
                    outfile_stream.write(reinterpret_cast<char*>(&currbyte),
                                         sizeof(currbyte));
                    currbyte = 0;
                    bitcount = 0;
                }
            }
        }
    }

    if (bitcount) { /* the very last character didn't land on the byte boundary
                       so we need to pad it with zeroes before writing it out to
                       file */
        while (bitcount != kBitsPerByte) {
            currbyte <<= 1;
            bitcount++;
        }
        outfile_stream.write(reinterpret_cast<char*>(&currbyte),
                             sizeof(currbyte));
    }
}

Encode() reads the input file in 1kb chunks. The encodings_ codebook makes it possible to find each character’s binary code. You iterate each binary code bit-by-bit appending each bit to the variable currbyte. When currbyte’s bitcount hits 8, you write currbyte out to file. This process repeats until you have processed all characters in the input. The if (bitcount) clause at the end handles the edge case previously discussed where you need to append a couple of zeroes to a binary code to make it a complete byte before writing to the file.

You might have noticed a call to WriteHeader(). You’ll see the purpose of WriteHeader() in the next section.

Decompression

Assuming you have the Huffman Tree used to compress a file available, decompressing the contents of the file requires only a tree traversal. Imagine the compressed file is a bit stream. You can navigate the tree from the root using the current bit in the stream to guide whether you step into the left subtree or right subtree. When you encounter a leaf node, write the character of that node to an output file and then reset to the root of the tree.

As always, the devils in the details. For this tree traversal to work, you need to know the following bits of information:

  • How to reconstruct the Huffman Tree.
  • How many characters were in original input file.

Two Birds With One Stone

An easy way of reconstructing the tree is to write out the character frequency table to the beginning of the file in a header section. Writing the whole table isn’t particularly efficient for small input files given that the header will be significantly larger than the compressed data. However, as the input grows, the overhead of the header becomes negligible.

Included in the header is a magic number3. That magic number forms the first few bytes of the compressed file and helps identify a file as a Huffman coded file.

Here’s how that header might look like in memory:

Huffman Header

Below is the header generation code in all its glory:

void HuffmanCoding::WriteHeader(std::ofstream& os) const {
    os.write(reinterpret_cast<const char*>(&kHuffmanFmtIdentifier),
             sizeof(kHuffmanFmtIdentifier));

    std::size_t num_chars = char_freqs_.size();
    os.write(reinterpret_cast<char*>(&num_chars), sizeof(num_chars));

    for (const auto& [character, frequency] : char_freqs_) {
        os.write(&character, sizeof(character));
        os.write(reinterpret_cast<const char*>(&frequency), sizeof(frequency));
    }
}

WriteHeader() solves the problems encountered earlier: how to reconstruct the tree and how many characters were in the uncompressed file. Using the frequency table parsed from a compressed file’s header, you can run the BuildEncodingTree() routine just as before. A quick sum of the frequencies in the char_freqs_ maps reveals how many characters were in the original input.

Decoding Data

You can decompose the Decompression() routine into three separate parts:

  1. Reading the header.
  2. Building the encoding tree.
  3. Decoding the input bit stream.

You already have a routine to write the header. The function that reads the header in is nearly identical. Just replace stream writes with reads. Once you read the header, building the encoding tree requires calling BuildEncodingTree(). Decoding the bit stream is the only new thing here.

Here’s the code that implements the concept:

void HuffmanCoding::DecodeStream(const std::vector<bool>& bitstream,
                                 std::ofstream& os) {
    /* take a tally of how many chars we need to decode */
    uint32_t num_chars = 0;
    for (const auto& kv : char_freqs_) {
        num_chars += kv.second;
    }

    /* repeatedly traverse the huffman tree decoding characters along the way */
    uint32_t num_chars_decoded = 0;
    HuffmanNodePtr node = encoding_root_;
    std::size_t i = 0;
    while ((i < bitstream.size()) && (num_chars_decoded != num_chars)) {
        node = (bitstream[i]) ? node->one : node->zero;

        if (!node->zero && !node->one) { /* reached a leaf node */
            os << static_cast<char>(node->character);
            node = encoding_root_;
            num_chars_decoded++;
        }
        i++;
    }
}

One weak aspect of this code is that DecodeStream() expects the entire input bit stream at once. That is, all bits (represented as bool types) are in memory and buffered. If the file is large enough, the bitstream vector may well not fit in memory. For this project, it’s reasonable to keep it simple and not worry about multi gigabyte files. A better approach would be to read the data, perhaps in page sized chunks, and create a parser object that tracks where in the decoding process it is.

Similar to Compress(), the Decompress() routine is a wrapper around the Decode() routine:

RetCode HuffmanCoding::Decode(const std::string& infile,
                              const std::string& outfile) {
    std::ifstream infile_stream(infile, std::ios::in | std::ios::binary);
    RetCode retcode = ReadHeader(infile_stream); /* read in char frequencies */
    if (RetCode::kSuccess != retcode) {          /* invalid header */
        return retcode;
    }

    BuildEncodingTree(); /* construct the encoding tree */

    /* build up a bit vector from the compressed file's binary content */
    const int kNumBitsInByte = 8;
    std::vector<bool> bitstream;
    while (infile_stream) {
        infile_stream.read(read_buffer_.data(), read_buffer_.size());
        for (std::streamsize i = 0; i < infile_stream.gcount(); ++i) {
            for (int j = 0; j < kNumBitsInByte; ++j) {
                uint8_t mask = 1 << (kNumBitsInByte - j - 1);
                bitstream.push_back(read_buffer_[i] & mask);
            }
        }
    }

    /* reconstruct the message by traversing the huffman tree */
    std::ofstream outfile_stream(outfile);
    DecodeStream(bitstream, outfile_stream);

    return retcode;
}

RetCode HuffmanCoding::Decompress(const std::string& compressed_filepath,
                                  const std::string& uncompressed_filepath) {
    /* verify compressed_filepath points to an existing file */
    std::filesystem::path compressed_path(compressed_filepath);
    if (!std::filesystem::exists(compressed_filepath)) {
        return RetCode::kFileDoesNotExist;
    }

    return Decode(compressed_filepath, uncompressed_filepath);
}

Conclusion

Putting it all together, you have a utility capable of compressing and decompressing any image, text, executable, etc. using Huffman coding. The implementation isn’t the most robust or efficient with regards to space/time. The header could be significantly smaller4. You probably shouldn’t pass in any files that don’t fit in memory. That said, the core concepts are there. Playing around with the tool, you’ll find some files compress down to 50% of their original size!

The complete project source with build instructions, usage, etc. is available on GitHub under huffman.


  1. This posts title image shows a Huffman Tree generated from the text “this is an example of a Huffman tree”. The source of the image is of course Wikipedia↩︎

  2. “Huffman Coding: Basic Technique” ↩︎

  3. Like with most things in computing, magic number has multiple meanings. The second definition in the “Magic number (programming)” wiki is the one referenced here: “A constant numerical or text value used to identify a file format or protocol”. ↩︎

  4. Checkout “Canonical Huffman Codes” to learn about a coding strategy that could lead to significantly smaller headers. Up to 1 bit per Huffman node, 8-bits per character. That’s a big savings over writing out the whole frequency table. ↩︎