Summary of "ДМ 1 курс - 8 лекция - т. Крафта-МакМиллана, арифметическое кодирование, RLE, MTF, BWT, LZ, LZW"
Summary of the Lecture:
“ДМ 1 курс - 8 лекция - т. Крафта-МакМиллана, арифметическое кодирование, RLE, MTF, BWT, LZ, LZW”
Main Topics Covered
- Kraft-McMillan Inequality and Uniquely Decodable Codes
- Prefix Codes and Their Properties
- Proofs Related to Kraft Inequality
- Huffman Coding and Its Optimality
- Arithmetic Coding: Principles and Encoding/Decoding Process
- Run-Length Encoding (RLE)
- Move-to-Front (MTF) Transformation
- Burrows-Wheeler Transform (BWT)
- Lempel-Ziv (LZ) and LZW Compression Algorithms
- General Observations on Compression and Practical Applications
Detailed Outline
1. Kraft-McMillan Inequality and Uniquely Decodable Codes
- The inequality states that a uniquely decodable code with codeword lengths ( n_i ) over an alphabet of size ( m ) exists if and only if: [ \sum_i m^{-n_i} \leq 1 ]
- The lecture revisits the proof for prefix codes and extends it to uniquely decodable codes.
- For binary codes, the sum of ( 2^{-n_i} ) must be ≤ 1 for unique decodability.
- The proof involves assigning ( 2^{-\text{depth}} ) values to leaves of a prefix code tree and summing these to show the inequality holds.
- Induction on tree depth is used to prove the sum at the root is ≤ 1.
2. Prefix Codes and Their Properties
- Prefix codes are a subset of uniquely decodable codes where no codeword is a prefix of another.
- The existence of prefix codes satisfying Kraft’s inequality implies the existence of uniquely decodable codes.
- Huffman codes are optimal prefix codes.
3. Proofs Related to Kraft Inequality
- A complex proof is presented using formal sums and generating functions (a method from a future semester) to prove the inequality for uniquely decodable codes.
- Formal manipulation of sums and powers of codewords shows that if the sum exceeds 1, contradictions arise due to exponential growth.
- The conclusion is that the sum must be ≤ 1.
4. Huffman Coding and Its Optimality
- Huffman coding is an optimal prefix code for separable codes (each symbol encoded independently).
- It achieves minimal average code length given symbol probabilities.
- However, Huffman coding restricts codeword lengths to integers.
5. Arithmetic Coding
- Arithmetic coding encodes entire sequences as intervals on the number line [0,1).
- The interval is successively subdivided according to symbol probabilities.
- Encoding involves narrowing the interval based on each symbol and representing the final interval by a fraction with a denominator that is a power of two.
- Decoding reverses the process by determining which subinterval the encoded number falls into.
- Arithmetic coding can achieve fractional bits per symbol, improving compression beyond Huffman.
- Theoretical optimality is asymptotic and proven in more advanced courses.
- The length of the encoded message relates to the entropy of the source: [ H = -\sum_i p_i \log_2 p_i ]
- The method assumes independence of symbols and does not use symbol ordering.
6. Run-Length Encoding (RLE)
- RLE compresses sequences by encoding runs of identical symbols as a single symbol plus count.
- Effective for data with many repeated symbols, such as low-color images.
- Often combined with other compression methods for better results.
7. Move-to-Front (MTF) Transformation
- MTF encodes symbols by their position in a dynamically updated list.
- When a symbol is encoded, it is moved to the front of the list.
- This tends to produce many zeros for repeated symbols, which compress well with entropy coders like Huffman or arithmetic coding.
- Good for data with local symbol frequency bias or repeated patterns.
8. Burrows-Wheeler Transform (BWT)
- BWT rearranges a string into runs of similar characters by sorting all cyclic rotations lexicographically.
- The last column of this sorted matrix is the transformed string.
- BWT makes data more amenable to compression by clustering similar characters.
- Decoding is possible by reconstructing the original string from the transformed data.
- Used in modern compressors like bzip2.
- Efficient implementations use suffix arrays or suffix trees (beyond the course scope).
9. Lempel-Ziv (LZ) and LZW Compression Algorithms
- LZ77 and LZ78 are dictionary-based compression algorithms.
- They encode repeated substrings by referencing previous occurrences (back-references).
- LZW builds a dictionary of substrings dynamically during encoding.
- Encoding involves outputting codes for dictionary entries and adding new entries as the input is processed.
- Decoding reconstructs the dictionary in parallel.
- Dictionary size grows during encoding; practical implementations limit dictionary size (e.g., ( 2^{16} ) entries).
- Widely used in practical compression tools (e.g., GIF, early ZIP).
10. General Observations on Compression and Practical Applications
- The lecture emphasizes the difference between mathematically provable optimal codes and heuristic methods used in practice.
- Real-world compressors combine these techniques with domain-specific transformations (e.g., psychoacoustic models for audio).
- Compression is often a trade-off between theoretical optimality and practical efficiency.
- The lecture ends with an overview of how these methods compose into modern compression pipelines.
Methodologies and Instructions Highlighted
Proving Kraft Inequality for Prefix Codes
- Assign ( 2^{-\text{depth}} ) to each leaf.
- Sum values at child nodes and assign to parent nodes.
- Use induction to show sum at root ≤ 1.
Arithmetic Coding Encoding Steps
- Start with interval [0,1).
- For each symbol, subdivide current interval according to symbol probabilities.
- Narrow interval to subinterval corresponding to the symbol.
- Final code is a fraction inside the last interval with denominator ( 2^k ).
- Output binary representation of this fraction.
Arithmetic Coding Decoding Steps
- Start with code fraction and interval [0,1).
- Determine which subinterval the code falls into.
- Output corresponding symbol.
- Narrow interval to that subinterval.
- Repeat until all symbols decoded.
Run-Length Encoding
- Detect consecutive runs of identical symbols.
- Encode run as symbol + count.
Move-to-Front Encoding
- Maintain ordered list of symbols.
- For each symbol, output its position in the list.
- Move that symbol to the front.
Burrows-Wheeler Transform
- Form all cyclic rotations of input string.
- Sort rotations lexicographically.
- Output last column of sorted rotations.
- Decode using inverse BWT algorithm.
LZW Encoding
- Initialize dictionary with all single symbols.
- Find longest prefix in dictionary matching current input.
- Output dictionary code for prefix.
- Add prefix + next symbol to dictionary.
- Repeat until input exhausted.
LZW Decoding
- Initialize dictionary same as encoder.
- Read code, output corresponding string.
- Add new entries as encoder did.
- Handle special cases where code refers to not-yet-added entry.
Speakers / Sources
- The lecture is delivered by a university professor (name not explicitly stated).
- References to “Sasha” (likely a teaching assistant or co-lecturer).
- Mention of historical figures and algorithms: Kraft, McMillan, Huffman, Burrows, Wheeler, Lempel, Ziv, Welch.
- No other distinct speakers identified.
Summary
This lecture provides an in-depth exploration of foundational concepts in coding theory and data compression. It rigorously proves the Kraft-McMillan inequality and its implications for prefix and uniquely decodable codes. It introduces Huffman coding as an optimal prefix code and then transitions to arithmetic coding, which allows fractional bits per symbol and approaches theoretical entropy limits.
The lecture then moves to practical compression techniques: Run-Length Encoding for repeated symbols, Move-to-Front to exploit locality, and the Burrows-Wheeler Transform for rearranging data to improve compressibility. Finally, it covers dictionary-based algorithms LZ and LZW, explaining their adaptive nature and widespread use.
Throughout, the lecture balances formal mathematical proofs with heuristic and practical considerations, illustrating how theoretical limits inform real-world compression tools.
If you need explanations or summaries of any specific part or algorithm in more detail, feel free to ask!
Category
Educational
