Stories with Tag succinct data structures

• Succinct Data Structures

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  Posted: 2025-03-06 17:48:37

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  Succinct data structures represent data in space close to the information-theoretic lower bound, while still allowing efficient queries. The blog post explores several examples, starting with representing a bit vector using only one extra bit beyond the raw data, while still supporting constant-time rank and select operations. It then extends this to compressed bit vectors using Elias-Fano encoding and explains how to represent arbitrary sets and sparse arrays succinctly. Finally, it touches on representing trees succinctly, demonstrating how to support various navigation operations efficiently despite the compact representation. Overall, the post emphasizes the power of succinct data structures to achieve substantial space savings without significant performance degradation.

  The blog post "Succinct Data Structures" delves into the fascinating realm of representing data structures in a manner that approaches the information-theoretic lower bound of space complexity while still permitting efficient query operations. This means storing data using close to the minimum number of bits theoretically required to represent the information, without sacrificing the speed of accessing and using that data.

  The author begins by establishing the fundamental concept of information-theoretic lower bounds. This refers to the absolute minimum number of bits needed to differentiate between all possible configurations of a data structure. For example, representing a bit vector of length n requires, at minimum, n bits, while a permutation of n elements necessitates approximately n log n bits (using logarithms base 2). These lower bounds provide a benchmark against which the efficiency of succinct data structures can be measured.

  The post then introduces several classic examples of succinct data structures, beginning with Elias-Fano encoding. This technique efficiently represents a monotonically increasing sequence of integers, a common scenario in various applications. The key idea behind Elias-Fano is to separate the binary representation of each integer into high and low bits, storing them in separate structures optimized for their respective characteristics. This allows for efficient rank and select operations, which are fundamental to many algorithms operating on such sequences.

  The discussion continues with the representation of bit vectors. While storing a bit vector trivially uses n bits, succinct representations aim to support operations like rank (counting the number of set bits up to a given position) and select (finding the position of the k-th set bit) efficiently within a space very close to n bits. These representations often employ ingenious techniques like blocking and precomputed tables to achieve constant-time or near constant-time query operations.

  Next, the post touches upon succinct tree representations. Representing a tree efficiently while supporting navigation operations is crucial in many applications. Several succinct tree representations are mentioned, each using different strategies to encode the tree structure and enable operations like finding the parent, children, or subtree size of a node. These techniques often involve clever bit manipulations and carefully designed auxiliary structures.

  The author emphasizes the importance of operations like rank and select in navigating and utilizing these succinct data structures. These functions become the building blocks for higher-level operations, allowing for efficient querying and manipulation of the underlying data despite its compressed representation.

  Finally, the post briefly discusses practical considerations related to succinct data structures. While achieving theoretical optimality in terms of space is a primary goal, the constant factors associated with the complexities of these structures can impact their practical performance. The author concludes by noting the continuing research and development in this area, suggesting the potential for even more efficient and versatile succinct data structures in the future. The post serves as an excellent introduction to the fundamental concepts and techniques of succinct data structures, illustrating their power and utility in representing large datasets efficiently.

  • data structures
  • succinct data structures
  • algorithms
  • data compression
  • bit manipulation
  • space efficiency
  • memory efficiency
  • information theory
  • coding theory
  • Computer Science

  Summary of Comments ( 27 )
  https://news.ycombinator.com/item?id=43282995

  Verbose tl;dr

  Hacker News users discussed the practicality and performance trade-offs of succinct data structures. Some questioned the real-world benefits given the complexity and potential performance hits compared to simpler, less space-efficient solutions, especially with the abundance of cheap memory. Others highlighted the value in specific niches like bioinformatics and embedded systems where memory is constrained. The discussion also touched on the difficulty of implementing and debugging these structures and the lack of mature libraries in common languages. A compelling comment highlighted the use case of storing large language models efficiently, where succinct data structures can significantly reduce storage requirements and memory access times, potentially enabling new applications on resource-constrained devices. Others noted the theoretical elegance of the approach, even if practical applications remain somewhat niche.

  The Hacker News post "Succinct Data Structures" spawned a moderately active discussion with a mix of practical observations, theoretical considerations, and personal anecdotes.

  Several commenters focused on the practical applications, or lack thereof, of succinct data structures. One commenter questioned the real-world utility outside of specialized domains like bioinformatics, expressing skepticism about their general applicability due to the complexity and constant factors involved. Another agreed, pointing out that the performance gains are often marginal and not worth the added code complexity in most cases. A counterpoint was raised by someone who suggested potential benefits for embedded systems or scenarios with extremely tight memory constraints.

  The discussion also delved into the theoretical aspects of succinctness. One commenter highlighted the connection between succinct data structures and information theory, noting how they push the boundaries of representing data with minimal overhead. Another brought up the trade-off between succinctness and query time, emphasizing that achieving extreme compression often comes at the cost of slower access speeds.

  A few commenters shared their personal experiences and preferences. One admitted finding the concepts fascinating but acknowledged the limited practical use in their day-to-day work. Another expressed a preference for simpler data structures that prioritize readability and maintainability over marginal performance gains.

  A couple of comments also touched on specific data structure implementations. One commenter mentioned Elias-Fano coding as a particularly useful technique for representing sorted sets, while another brought up wavelet trees and their applications in compressed string indexing.

  Overall, the comments reflect a nuanced view of succinct data structures. While acknowledging their theoretical elegance and potential benefits in specific niches, many commenters expressed reservations about their widespread adoption due to complexity and limited practical gains in common scenarios. The discussion highlights the importance of carefully considering the trade-offs between space efficiency, performance, and code complexity when choosing data structures.

• Iterated Log Coding

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  Posted: 2025-02-26 07:43:21

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  Iterated Log Coding (ILC) offers a novel approach to data compression by representing integers as a series of logarithmic operations. Instead of traditional methods like Huffman coding or arithmetic coding, ILC leverages the repeated application of the logarithm to achieve potentially superior compression for certain data distributions. It encodes an integer by counting how many times the logarithm base b needs to be applied before the result falls below a threshold. This "iteration count" becomes the core of the compressed representation, supplemented by a fractional value representing the remainder after the final logarithm application. Decoding reverses this process, effectively "exponentiating" the iteration count and incorporating the fractional remainder. While the blog post acknowledges that ILC's practical usefulness requires further investigation, it highlights the theoretical potential and presents a basic implementation in Python.

  This blog post introduces a novel compression technique called "Iterated Log Coding," or "iterlog," which cleverly exploits the predictable nature of sorted integer sequences. The core idea revolves around representing each integer in a sorted sequence not by its absolute value, but by the difference between it and the preceding integer, a concept similar to delta encoding. However, iterlog takes this further by recursively applying this differencing process. Instead of just storing these differences directly, it stores the base-2 logarithm of each difference, rounded up to the nearest integer. This process of taking the log of the differences is iterated until all remaining values become zero.

  The author argues that this approach is particularly well-suited for compressing sorted sequences containing clusters of similar values, a common characteristic of many real-world datasets. When integers are close together, their differences, and consequently the logarithms of these differences, will be small. Representing these small values requires fewer bits, leading to significant compression. The iterated nature of the algorithm further amplifies this effect. As the differences shrink through successive iterations, the logarithms also decrease, leading to progressively smaller representations.

  The blog post details the encoding and decoding processes with illustrative examples. Encoding involves repeatedly calculating differences between consecutive numbers and then their logarithms, storing the rounded-up logarithm values at each level. A key point is that the first number in the sequence and the number of elements in the sequence need to be stored separately as they are not part of the differencing process. The number of iterations required is also implicitly stored by the presence of all zeros in the final iteration. Decoding reverses this process by repeatedly exponentiating and summing the logged differences, effectively reconstructing the original sequence.

  The author acknowledges that iterlog coding is not a general-purpose compression algorithm and might not be suitable for all types of data. It's particularly effective for sorted sequences with clustered values, where the differences between successive elements are small. In situations where the differences are large and vary significantly, iterlog may not offer significant compression advantages or might even increase the data size compared to uncompressed representation. The post concludes with a Python implementation of the iterlog encoding and decoding algorithms, allowing readers to experiment with the technique and evaluate its performance on different datasets. The author invites further exploration and optimization of the idea, suggesting that varying the logarithm base or utilizing alternative rounding strategies could potentially improve compression ratios in specific scenarios.

  • data compression
  • iterated logarithm
  • logarithm
  • coding theory
  • compression algorithm
  • binary coding
  • Elias coding
  • variable-length coding
  • integer compression
  • succinct data structures

  Summary of Comments ( 23 )
  https://news.ycombinator.com/item?id=43181610

  Verbose tl;dr

  Hacker News users generally praised the clarity and novelty of the Iterated Log Coding approach. Several commenters appreciated the author's clear explanation of a complex topic and the potential benefits of the technique for compression, especially in specialized domains like bioinformatics. Some discussed its similarities to Huffman coding and Elias gamma coding, suggesting it falls within a family of variable-length codes optimized for certain data distributions. A few pointed out limitations or offered alternative implementations, including using a lookup table for smaller values of 'n' for performance improvements. The practicality of the method for general-purpose compression was questioned, with some suggesting it might be too niche, while others found it theoretically interesting and a valuable addition to existing compression methods.

  The Hacker News post "Iterated Log Coding" discussing the blog post about a new compression algorithm has generated a moderate amount of discussion, with several commenters engaging with the core ideas presented.

  One of the most compelling threads revolves around the practicality and novelty of the "iterated log" approach. A user points out that the method is reminiscent of Elias gamma coding, a well-established variable-length coding scheme. This sparked further discussion comparing the two methods, with some suggesting that iterated log coding might offer advantages in certain scenarios, particularly when dealing with very large numbers, while others remain skeptical, highlighting the efficiency and existing implementations of Elias gamma coding.

  Another commenter questions the choice of using base-10 logarithms in the examples, suggesting that base-2 logarithms would be more natural in a computational context. This comment prompts a brief discussion about the rationale behind the base choice, with the possibility that base-10 was selected for easier human readability in the illustrative examples.

  Several commenters express interest in seeing benchmarks and comparisons against existing compression algorithms. They emphasize the importance of real-world performance data to evaluate the effectiveness of the proposed method. One user specifically asks about the performance on typical integer sequences found in practice.

  There's also a short exchange regarding the potential applications of this compression method. One commenter suggests it could be useful for compressing indexes in databases.

  Finally, a few commenters delve into the theoretical underpinnings of the algorithm, discussing its relationship to other coding schemes and exploring its potential limitations. One user mentions the connection to prefix codes and how the unique decodability property is ensured.

  Overall, the comments section reveals a mixture of curiosity, skepticism, and cautious optimism towards the iterated log coding approach. While some see it as a potentially interesting idea with specific niche applications, others remain unconvinced of its practical value compared to established techniques. The prevailing sentiment appears to be a desire for more empirical evidence and comparisons to solidify the claims made in the original blog post.